Stochastic Processes and Financial Mathematics 9783662647103, 9783662647110

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Mathematics Study Resources 1

Ludger Rüschendorf

Stochastic Processes and Financial Mathematics

Mathematics Study Resources Volume 1 Series Editors Kolja Knauer, Departament de Matemàtiques Informàtic, Universitat de Barcelona, Barcelona, Barcelona, Spain Elijah Liflyand, Department of Mathematics, Bar-Ilan University, Ramat-Gan, Israel

This series comprises direct translations of successful foreign language titles, especially from the German language. Powered by advances in automated translation, these books draw on global teaching excellence to provide students and lecturers with diverse materials for teaching and study.

Ludger Rüschendorf

Stochastic Processes and Financial Mathematics

Ludger Rüschendorf Institute of Mathematical Stochastics University of Freiburg Freiburg, Germany

ISSN 2731-3824 ISSN 2731-3832 (electronic) Mathematics Study Resources ISBN 978-3-662-64710-3 ISBN 978-3-662-64711-0 (eBook) https://doi.org/10.1007/978-3-662-64711-0 This book is a translation of the original German edition ,Stochastische Prozesse und Finanzmathematik“ by Ludger Rüschendorf, published by Springer-Verlag GmbH Germany in 2020. The translation was done with the help of artificial intelligence (machine translation by the service DeepL.com) and in a subsequent editing, improved by the author. Springer Nature works continuously to further the development of tools for the production of books and on the related technologies to support the authors. © Springer-Verlag GmbH Germany, part of Springer Nature 2023 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer-Verlag GmbH, DE part of Springer Nature. The registered company address is: Heidelberger Platz 3, 14197 Berlin, Germany

Contents

1

Option Pricing in Discrete Time Models . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1

2 Skorohod’s Embedding Theorem and Donsker’s Theorem . . . . . . . . . . . . . 2.1 Skorohod’s Embedding Theorem . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 Functional Limit Theorem .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

11 11 19

3 Stochastic Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 Martingales and Predictable Processes. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 Itô Integral for the Brownian Motion . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.1 Extension of the Integral to L2 -Integrands . . . . . . . . . . . . . . . . . . . . 3.2.2 Construction of the Integral for L0 (B) . . . .. . . . . . . . . . . . . . . . . . . . 3.3 Quadratic Variation of Continuous Local Martingales . . . . . . . . . . . . . . . . 3.4 Stochastic Integral of Continuous Local Martingales . . . . . . . . . . . . . . . . . 3.4.1 Stochastic Integral for Continuous L2 -Martingales .. . . . . . . . . . 3.4.2 Extension to the Set of Continuous Local Martingales . . . . . . . 3.4.3 Extension to the Case of Continuous Semimartingales . . . . . . . 3.5 Integration of Semimartingales .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5.1 Decomposition Theorems . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5.2 Stochastic Integral for M2loc . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5.3 Stochastic Integral for Semimartingales .. .. . . . . . . . . . . . . . . . . . . .

27 27 40 43 48 55 71 71 76 79 82 84 86 87

4 Elements of Stochastic Analysis . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 Itô Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2 Martingale Representation .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 Measure Change, Theorem of Girsanov . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.1 Applications of the Girsanov Theorem . . . .. . . . . . . . . . . . . . . . . . . . 4.3.2 The Clark Formula .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4 Stochastic Differential Equations . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4.1 Strong Solution—Weak Solution of Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5 Semigroup, PDE, and SDE Approach to Diffusion Processes . . . . . . . .

95 95 117 132 145 149 152 165 173

5 Option Prices in Complete and Incomplete Markets . . . . . . . . . . . . . . . . . . . . 5.1 The Black–Scholes Model and Risk-Neutral Valuation .. . . . . . . . . . . . . . 5.1.1 Risk-Neutral Valuation of Options .. . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.1.2 Discussion of the Black–Scholes Formula . . . . . . . . . . . . . . . . . . . .

191 192 199 206 v

vi

Contents

5.1.3 Hedging Strategies and Partial Differential Equations .. . . . . . . 216 5.2 Complete and Incomplete Markets .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 219 6 Utility Optimization, Minimum Distance Martingale Measures, and Utility Indifference Pricing . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.1 Utility Optimization and Utility Indifference Pricing . . . . . . . . . . . . . . . . . 6.2 Minimum Distance Martingale Measures.. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3 Duality Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3.1 Minimum Distance Martingale Measures and Minimax Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.3.2 Relationship to Portfolio Optimization . . . .. . . . . . . . . . . . . . . . . . . . 6.4 Utility-Based Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.5 Examples in Exponential Lévy Models .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 6.6 Properties of the Utility Indifference Price . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7 Variance-Minimal Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.1 Hedging in the Martingale Case . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2 Hedging in the Semimartingale Case . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2.1 Föllmer–Schweizer Decomposition and Optimality Equation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2.2 Minimal Martingale Measures and Optimal Strategies . . . . . . .

233 233 235 242 242 250 255 263 269 279 280 284 285 288

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 295 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 299

Introduction

This book provides an introduction to recent topics in stochastic processes and stochastic analysis and combines them with an introduction to the basics of financial mathematics. Through a selective choice of topics and a motivating (not technically elaborate) presentation, it allows one to gain an insight into the fundamental developments, ideas and models of stochastic processes and stochastic analysis and their importance for an understanding of the issues and concepts of financial mathematics. Unlike other textbooks, it does not resort to a highly reductive presentation of the subject (e.g., in the context of discrete time models). The book is particularly suitable as a foundation or companion text to a more advanced 4-hour lecture on these topics in graduate mathematics. The aim is to present the particularly interesting developments and ideas in an extent suitable for a lecture or a seminar. With its motivating form of presentation and linguistic design, it is aimed in particular at students and offers a wealth of illustrative examples and applications. It is therefore not only particularly well suited for self-study but also provides many interesting additions for interested researchers and also lecturers. A prerequisite of the book is a good knowledge of a more advanced probability theory course, in particular of discrete-time processes (martingales, Markov chains) as well as introductory continuous-time processes (Brownian motion, Poisson process, Lévy processes, processes with independent increments, and Markov processes) as taught in many existing good representations of probability theory; for their scope, see Rüschendorf (2016). In Chap. 1, an introduction is given in a non-technical way to the basic principles of the theory of arbitrage-free prices, the hedging principle and the risk-neutral price measure, leading from the binomial price formula by approximation to the Black–Scholes formula. The necessary link from processes in discrete time (binomial model) to those in continuous time (Black–Scholes model) is given by approximation theorems for stochastic processes such as the fundamental Donsker theorem and the associated Skorohod embedding theorem in Chap. 2. Theorems of this type allow interpretation of continuous financial market models using simple discrete models such as the Cox–Ross–Rubinstein model. The first main part of the book in Chap. 3 is devoted to the introduction of the stochastic integral (Itô integral) and the related theory of martingales and semimartingales. K. Itô had introduced this integral for Brownian motion in several papers in 1944–1951. It is thus possible—despite the infinite variation of the vii

viii

paths of Brownian motion—to give integral expressions of the form

Introduction

t

ϕs dBs for

0

a stochastic process (ϕs ) a significant meaning. Also, Itô had recognized that, through the corresponding stochastic differential equations of the form dXt = a(t, Xt )dt + b (t, Xt ) dBt by analogy with the ordinary differential equations, a fundamental tool is given to construct, by local properties, in the above example the drift a and the volatility b, an associated stochastic model; in the above case a diffusion process; compare the works of Itô and McKean (1974) and Itô (1984). Semimartingales are a generalization of the class of diffusion models described by a (local) drift, a continuous (local) diffusion part (martingale part) and a jump part. They form a fundamental class of models in particular of importance also for the relevant models of financial mathematics. The second main part of the textbook in Chap. 4 is devoted to stochastic analysis, its importance for model building and its analysis. Essential building blocks are the partial integration formula and the Itô formula with numerous applications, e.g., to Lévy’s characterization of Brownian motion. The stochastic exponential is a solution of a fundamental stochastic differential equation and shows its importance in the characterization of equivalent measure changes (Girsanov’s theorem). Together with the martingale representation theorem (completeness theorem) and the Kunita– Watanabe decomposition theorem, these form the foundation for arbitrage-free pricing theory in financial mathematics in the third part of the textbook. A detailed section in this part is devoted to stochastic differential equations. In particular, a comparison of the different approaches to diffusion processes is presented, namely 1. the Markov process (semigroup) approach, 2. the PDE approach (Kolmogorov equations) and 3. the approach by stochastic differential equations (Itô theory). Moreover, the particularly fruitful connection between stochastic and partial differential equations (Dirichlet problem, Cauchy problem, Feynman–Kac representation) is also elaborated. Chapter 5 introduces the foundations of general arbitrage-free pricing theory. The foundations are the first and second fundamental theorems of asset pricing and the associated risk-neutral valuation formula, which is based on valuation by means of equivalent martingale measures. For their construction, Girsanov’s theorem proves to be very useful. For the standard options, it can be used to determine the corresponding pricing formulas (Black–Scholes formulas) in a simple way. The determination of the corresponding hedging strategies leads in the Black–Scholes model (geometric Brownian motion) to a class of partial differential equations, the Black–Scholes differential equations, going back to the fundamental contributions of Black and Scholes. The connection with stochastic calculus goes back to a derivation by Merton in 1969, which led to the development of a general theory of arbitrage-free pricing for continuous-time price processes and the important role of equivalent martingale measures for this purpose by Harrison, Kreps and Pliska in the period 1979–1983. Central topics of this general theory are the completeness and incompleteness of market models, the determination of associated arbitrage-free price intervals via

Introduction

ix

the equivalent martingale measures and the corresponding sub- or super-hedging strategies (optional decomposition theorem). In incomplete models, the no-arbitrage principle does not uniquely identify an arbitrage-free price. Therefore, additional criteria must be applied to select an arbitrage-free price. Chapter 6 gives an introduction to the determination (resp. selection) of option prices via minimum distance martingales as well as to pricing and hedging via utility functions. In addition, the problem of portfolio optimization is treated. For exponential Lévy models, these methods are characterized in detail for a number of standard utility functions. Chapter 7 is devoted to the determination of optimal hedging strategies by the variance-minimal strategy criterion. In incomplete market models, not every claim H is hedgeable. Therefore, a natural question is: How good is H hedgeable? The answer to this question is based on the Föllmer–Schweizer decomposition, a generalization of the Kunita–Watanabe decomposition, and the associated minimal martingale measure. This textbook is based on lectures by the author on stochastic processes and financial mathematics, given repeatedly over many years since about the mid1990s, and on related transcripts and notes by Sascha Frank and Georg Hoerder (2007, unpublished), Anna Barz (2007, unpublished) and Janine Kühn (2013, unpublished). Chapter 6 is based in large part on the elaboration of Sandrine Gümbel (2015). We would like to express our sincere thanks to all of them. Special thanks also go to Monika Hattenbach for preparing some parts of the text as well as for the habitual exquisite final text corrections and text layout.

1

Option Pricing in Discrete Time Models

This chapter gives an introduction to basic concepts and ideas of option pricing in the context of models in discrete time. In this frame the basic no-arbitrage principle and the determination of the fair (arbitrage-free) price resulting from it by means of a suitable hedging argument can be explained in a simple way. This approach makes it possible to introduce essential concepts and methods with little technical effort and,allows for example, to derive the Black–Scholes formula by means of an approximation argument. Let (St )t ≥0 be the price process of a security (stock) that is traded in a market. Further, let there be in the market a risk-free, fixed-interest investment (Bt )t ≥0 (bond), with interest factor r ≥ 0. The value of the bond at time t = 0 is therefore 1 (1+r)t of the value at time t: B0 =

1 · Bt . (1 + r)t

Securities traded in the market are those on underlying commodities (e.g. wheat, oil, gold) as well as derivatives, i.e. contracts (functionals) on underlying commodities. Derivatives include, for example, forwards, futures and options such as puts and calls. Forwards and Futures Forwards are contracts that give the market participant the right to buy or sell an underlying or financial asset at a time T in the future or in a future time period [T , T  ] at a fixed price K. A distinction is made between • long position: Entering into a contract to buy • short position: Entering into a sales contract

© Springer-Verlag GmbH Germany, part of Springer Nature 2023 L. Rüschendorf, Stochastic Processes and Financial Mathematics, Mathematics Study Resources 1, https://doi.org/10.1007/978-3-662-64711-0_1

1

2

1 Option Pricing in Discrete Time Models

Forwards and futures involve hedging for their settlement, i.e. the market participant has both the right and the obligation to buy or sell. While futures are traded on financial markets, forwards are based on an individual agreement between the participants without market intervention. Options, Call, Put A call option (call for short) gives the buyer of the option the right to buy a financial asset at a future date T at an agreed price. However, the buyer is not obliged to exercise the contract. In contrast, a put option (put for short) gives the buyer the right to sell a financial asset at a fixed price at a time point T (Fig. 1.1). A European Call is the right (but not the obligation) to buy a security at time T at price K. The value of a European Call at time T is   Y = ST − K + . As an alternative there is the American Call. This American Call gives the buyer the right to buy a security at any time in the time interval τ in the time interval [0, T ] at the price K. The time τ is a stopping time from a mathematical point of view. An American Call therefore has the value at the time of execution   Y = Sτ − K + . A put denotes the right to sell a security at time T at a specified price K. The value of the put at time T is therefore   Y = K − ST + .

Fig. 1.1 European Call, Y = (ST − K)+

1 Option Pricing in Discrete Time Models

3

The question arises: What is a correct premium for a call Y ? The classic answer to this is the expected value   EY = E ST − K + Black and Scholes, in 1973, introduced a compelling new method to calculate the premium. In the following this premium is called the fair price or Black–Scholes price. The basic idea of the Black–Scholes pricing principle can be explained by means of a simple pricing model. Simple Pricing Model Be S0 = 100 the price of the security at the time t = 0. We consider a one-period discrete model. Up to time T = 1 the price of the security can with probability p = 0.4 rise to S1a = 130 and with probability p = 0.6 fall to S1b = 80.

A call secures the buyer the right to purchase the security at time T = 1 at the price (strike) K = 110. Accordingly, the call has at T = 1 the value Y = (S1 − K)+ . As an alternative, there is a risk-free investment at the interest rate r = 0.05. With the classic valuation, the price of the call t = 1 would be determined as follows: Ep Y = 20 · 0.4 = 8 According to this, the value of the call at t = 0 is C0 =

1 8 · Ep Y = = 7.62. 1+r 1.05

The Black–Scholes formula gives a different valuation: C0BS = 9.52. This valuation method is based on the following consideration: Determine a portfolio of shares of the stock and of the bond, which generates at T = 1 the same payoff as the call. Such a portfolio is called a duplicating portfolio. Once one has determined such a duplicating portfolio, one has already found the fair price for the option. Both instruments, the call and the duplicating portfolio, have at T = 1 the same payoff. Therefore, they should also have the same price. !

price of the call Y = price of the duplicating portfolio.

4

1 Option Pricing in Discrete Time Models

Calculating a duplicating portfolio in the above example: Let π be a portfolio π = (, B), where  denote the stock shares and B the bond shares. Choose specifically  = 0.4 and B = −30.48 Here, the quantity B is negative, denoting a loan. The price of the portfolio at time t = 0 is easy to calculate in this example: V0 (π) =  · 100 + B = 40 − 30.48 = 9.52. The value of the portfolio at the time T = 1 is: V1 (π) =  · S1 + B · (1 + r)  =  =

0.4 · 130 − 30.48 · 1.05,

S1 = 130

0.4 · 80 − 30.48 · 1.05,

S1 = 80

20,

S1 = 130

0,

S1 = 80

=Y The portfolio generates the same payoff as the call Y at time T = 1, i.e., the portfolio duplicates the call Y . Thus, the fair price of the call at time is t = 0 is equal to the price of the portfolio: C0BS = 9.52. If one were to price the call differently, then a risk-free profit (arbitrage) would be possible. This fair pricing is therefore based on the no-arbitrage principle: In a real market model there is no risk free gain possible; NFLVR = No free lunch with vanishing risk This NFLVR principle implies that two market instruments with the same payoff also have the same price. General Discrete Models for Securities Let Sn = (Sn0 , Sn1 , . . . , Snd ) describe the d development of d equity securities; let Sni denote the price of the ith stock at time n, Sn0 = (1 + r)n . The dimension d of a large bank’s securities holdings can be quite large, e.g., in the case of Deutsche Bank d ≈ 500,000.

1 Option Pricing in Discrete Time Models

5

Some basic terms used to describe securities trading are summarized below. A portfolio describes the proportionsof different securities in the portfolio.  A trading strategy  = (n ) = (0n , . . . , dn ) describes the development of the portfolio: in ∼  shares of security no. i at time n. The value of the portfolio at time n is Vn () = n · Sn =

d 

in Sni .

(1.1)

i=0

 is called self-financing, if n · Sn = n+1 · Sn .

(1.2)

Changes to the portfolio result only from regrouping. No additional capital is required for the change. No withdrawal of value takes place. The increase in value is Vn () = Vn+1 () − Vn () = n+1 · Sn . The no-arbitrage principle implies: There is no-arbitrage strategy , i.e. there is no strategy  such that   V0 () = 0, Vn () ≥ 0 and P Vn () > 0 > 0,

(1.3)

or equivalent to it:   V0 () ≤ 0, Vn () ≥ 0 and P Vn () > 0 > 0. Cox–Ross–Rubinstein Model A simple and basic model for price evolution in discrete time is the Cox–Ross– Rubinstein model. This describes the time evolution of the price of a security by subdividing the time interval [0, T ] into n sub-intervals of length h, T = nh. The CRR model is based on the following assumption for price movements in the subintervals (Fig. 1.2).

Fig. 1.2 Cox–Ross–Rubinstein model for d = 1

6

1 Option Pricing in Discrete Time Models

Assumption A The price trend in a step is constant.

The price Sk after k substeps is therefore of the form u d k− S0 . The parameters of this model are u, d, q, n. The basic question is: “What is the value of a call Y := (Sn − K)+ ?” To answer this question, we first consider the development of the value C in one period. Step 1: Development of value in one period

To determine C, the hedging portfolio π = (, B) (with : stocks, B: bonds) is chosen so that the payoff at the end of the period is identical to that of the call. The development of the portfolio π = (, B) in one period is as follows:

From the postulated equality of the payoff, the hedging portfolio has a unique solution, the hedging portfolio (, B) with =

Cu − Cd , (u − d)S

B=

uCd − dCu (u − d)(1 + r)

(1.4)

1 Option Pricing in Discrete Time Models

7

From the no-arbitrage principle, the fair price C of the call thus follows: !

C = S + B uCd − dCu Cu − Cd + (u − d) (u − d)(1 + r)      1+r −d u − (1 + r) = Cu + Cd (1 + r) u−d u−d



 =

=:p ∗





(1.5)

=:1−p ∗

= p∗ Cu + (1 − p∗ )Cd /(1 + r) = Ep ∗

1 Y, 1+r

Y = Y1 , d = 1.

The value development of the price process in a period with respect to the riskneutral measure P ∗ is, with S0 = S Ep∗ S1 = p∗ uS + (1 − p∗ ) dS =

u − (1 + r) 1+r −d uS + dS u−d u−d

= (1 + r)S. Equivalent to this is: Ep ∗

1 S1 = S0 = S. 1+r

(1.6)

With respect to the risk neutral measure P ∗ the expected discounted value of S1 is equal tothat of S0 (equilibrium). Equation (1.6) states that the discounted price S1 is a martingale with respect to P ∗ e. process is S0 , 1+r We note two features of this process: (a) Evolution of the pricing process: With p∗ =

(1+r)−d u−d

it holds: p∗ uS + (1 − p∗ )dS = (1 + r)S

8

1 Option Pricing in Discrete Time Models

1 (b) The fair price C = Ep∗ 1+r S1 is independent of the underlying statistical price model! The value of the call at time t = 0 is equal to the expected value of the discounted call with respect to the risk neutral measure.

Step 2: Two-period model In the two-period-model the evolution of the price process S0 → S1 → S2 and the corresponding evolution of the value process C = C0 → C1 → C2 are as follows:

The value C is obtained by backward induction through iterated application of the first substep. Cu results from Cuu , Cud Cu =

1 (p∗ Cuu + (1 − p∗ )Cud ). 1+r

Cd is obtained similarly from Cud , Cdd , Then the value of C at time t = 0 results from Cu , Cd as follows: C=

1 (p∗ Cu + (1 − p∗ )Cd ). 1+r

1 Option Pricing in Discrete Time Models

9

More generally, by backward induction,one obtains the value C = Cn of the call for n periods. The result is the binomial pricing formula n     j n−j  1 n Cn = d  S −K + (p∗ )j (1 − p∗ )n−j u n j (1 + r) j =0

=Sn

(1.7)

1 = Ep ∗ (Sn − K)+ (1 + r)n Here P ∗ = (p∗ , 1 − p∗ ) is the risk-neutral measure: p∗ u + (1 − p∗ )d = 1 + r. The value of the call Y = (Sn − K)+ is given as the expected discounted value of the call with respect to the risk neutral measure P ∗ . The probabilities for P (Sn = uj d n−j S) are just the binomial probabilities. The development of the price is multiplicative with the factors u, d. The binomial pricing formula can also be rewritten by logarithmic transformation into the form Cn = S n (a, p ) − K(1 + r)−n n (a, p∗ ), where p∗ =

1+r−d u−d ,

p =

u ∗ 1+r p ,

a =



   log SKd n / log du , and +

n (a, p) := P (Xn, p ≥ a), Xn, p ∼ B(n, p). The transition to the additive binomial model can be described as follows. It holds that  Sn Xn,k = S n

log

k=1

u d with Xn,k = ξn,k U + (1 − ξn,k )D, where U = log 1+r > 0 > D = log 1+r , (ξn,k )k ∗ is an i.i.d. B(1, p ) distributed sequence. (Sn ) is therefore an exponential pricing model.

Sn = Se

n k=1

Xn,k

(1.8)

with a sum of i.i.d. terms as exponents. As a result, we have an explicit value formula using the binomial distribution. By choosing the parameters u, d appropriately, we thus obtain an approximation of the value formula by the normal distribution from the central limit theorem.

10

1 2

1 Option Pricing in Discrete Time Models

√t √t Let u = un = eσ n , d = dn = e−σ n , h =  t + 12 μ σ n and it holds:

t n.

Then it follows: p∗ ∼

Cn → C

(1.9)

with  √  C = S (x) − K(1 + r)−t  x − σ t x=

log(S/K(1 + r)t ) 1 √ + σ t. √ 2 σ t

(1.10)

This is the famous Black–Scholes formula. It thus describes approximately the arbitrage-free valuation of a call in a binomial model (Cox–Ross–Rubinstein model). The additive binomial model (as a stochastic process) can be approximated by a Brownian motion (see the Donsker theorem in Chap. 3). Therefore, it is natural to assume that the above pricing formula (Black–Scholes formula) will also result from a suitable derivation of the price of a call in the analogous model in continuous time (Black–Scholes model). This model in continuous time is an exponential Brownian motion (geometric Brownian motion). Analogous pricing formulas in general models in continuous time (such as exponential stable distributions, Weibull distributions, or hyperbolic distributions) by approximation with Cox–Ross–Rubinstein models are given in Rachev and Rüschendorf (1994).

2

Skorohod’s Embedding Theorem and Donsker’s Theorem

Subject of this chapter is a description of the interaction of discrete-time and continuous-time models. Functionals of discrete-time sum processes can be approximatively described by functionals of a continuous-time limit process. Conversely, functionals of the limit process can be simulated by those of discrete-time approximating processes. By this connection, by means of simple laws in the discrete time model, suitable continuous-time models and their regularities can be justified by means of approximation. A particularly important example of this connection is the Donsker theorem and the closely related Skorohod embedding theorem. Hereby the simple binomial model (or more generally the Cox–Ross–Rubinstein model) can be used to approximate a geometric Brownian motion (or more generally an exponential Lévy process) as a suitable continuous-time limit model.

2.1

Skorohod’s Embedding Theorem

We consider an i.i.d. sequence of real random variables (Xi ) with EXi = 0 and EXi2 = 1 and with partial sums Sn := ni=1 Xi . Linear interpolation produces a continuous function from the partial sum sequence (see Fig. 2.1). One can also set the function constant from a fixed time n on constant. Then the curve is rescaled by linear interpolation in the time dimension and by the factor n1 to the interval [0, 1] and rescaled in the spatial dimension with  (n)  the factor √1n . We obtain the random function St , with t ∈ [0, 1]. The statement  (n)  converges on the unit interval to the of Donsker’s theorem is that the process St Brownian motion: 

(n)  0≤t ≤1

St

D   −→ Bt 0≤t ≤1.

© Springer-Verlag GmbH Germany, part of Springer Nature 2023 L. Rüschendorf, Stochastic Processes and Financial Mathematics, Mathematics Study Resources 1, https://doi.org/10.1007/978-3-662-64711-0_2

11

12

2 Skorohod’s Embedding Theorem and Donsker’s Theorem

Sn

√1 n

n 1

Fig. 2.1 Simulation of the Brownian motion

Convergence in distribution of a stochastic process is explained in more detail (n) below. The process (St ) is introduced in the following two steps: 1. continuous interpolation 2. scaling in the temporal and spatial dimension   (St(n) ) reflects the whole partial sum sequence √Skn . Using Donsker’s theorem, one can approximate the behavior of functionals of partial sums by the behavior of functionals of the Brownian motion. The Brownian motion model has the advantage of being analytically tractable. Because of the above connection, the Brownian motion is a universal model that can be used to approximate the behavior of functionals of the partial sum process. In particular, as a consequence of this approximation one obtains the strong law of large numbers the central limit theorem and the law of the iterated logarithm. In the opposite direction, Donsker’s theorem provides a standard method for simulating a Brownian motion through the scaled partial sum sequences. This method is used in many simulation programs. Continuous processes induce measures on the space of continuous functions. One can treat these based on a general theory of convergence in distribution in function spaces. However, there is another way to obtain such approximation theorems in a direct way. This is based on the Skorohod representation (or embedding). Theorem 2.1 (Skorohod Representation) Let X be a real variable with   random EX = 0 and finite second moment EX2 < ∞ and be B = Bt a Brownian motion starting in zero, P = P0 . Then there exists a stopping time τ such that d

Bτ = X

and

Eτ = EX2 .

(2.1)

2.1 Skorohod’s Embedding Theorem

13

The first point is trivial to achieve because lim sup Bt = ∞, resp. lim inf Bt = −∞. Therefore every value is reached by the stopping time τ : stop Bt when X(ω) is reached. Then Bτ = X. Thus the random variable X is reproduced. However, in general Eτ for the above stopping time is not finite. We construct in the following proof an extended stopping time with property (2.1). Proof (1) first case For two numbers a and b with a < 0 < b we define the two-point measure   μa,b {a} :=

  b −a and μa,b {b} := . b−a b−a

  Further let μ0,0 {0} = 1. We assume, in the first case, that the distribution of X is the two-point measure above: Q := P X = μa,b . The probabilities are such that EX = 0 and EX2 = a 2

b a − b2 = −ab. b−a b−a

We define τ := inf{t ≥ 0 : Bt ∈ (a, b)} = τa,b , i.e. we stop when one of the limits a or b is reached. Using Wald’s equation, it follows: d

Bτ = X, Eτ = −ab = EBτ2 = EX2 , i.e., using the properties of Brownian motion, we get a solution for the first case by a suitable two-sided stopping time. (2) general case In the general case, the distribution of X is a general probability measure:  Q=P

X

∈ M (R , B ) 1

1

1

with EX = 0 =

∞ −∞

x dQ(x).

14

2 Skorohod’s Embedding Theorem and Donsker’s Theorem

We trace this case back to the first case as follows. Let  c :=



0

∞

(−u) dQ(u) =

−∞

v dQ(v). 0

Q can then be written as a mixture of two-point measures μa,b . For the proof let ϕ ∈ Lb be a bounded measurable function with ϕ(0) = 0, then it holds:   c ϕ dQ = 



∞

ϕ(v) dQ(v) 0



∞

=

dQ(v) 0

0

0

 (−u) dQ(u) +

−∞



0

ϕ(u) dQ(u)

−∞

∞

v dQ(v) 0

  dQ(u) vϕ(u) − uϕ(v) .

−∞

Thus, the mixture representation of Q follows  Eϕ(X) = ϕ dQ =

1 c



∞

dQ(v) 0



0

dQ(u)(v − u)

−∞



 −u v ϕ(u) + ϕ(v) v − u   v − u = ϕ dμu,v

(2.2) as a mixture of the two-point measures μu,v   μu,v {u} :=

  v −u and μu,v {v} := . v−u v−u

For this purpose ν be the measure on R+ × R− defined by     ν {0, 0} := Q {0} and for a subset A of the plane with (0, 0) ∈ A   (2.2) 1 dQ = by ν(A) := 1c A (v − u) dQ(u) dQ(v). Then (with ϕ ≡ 1) 1 = ν(R+ × R− ); thus ν is a probability measure. Formula (2.2) can also be read as follows: Let (U , V ) be two random variables on a suitable probability space ( , A, P ) independent of the Brownian motion B = (Bt ) such that P (U,V ) = ν. Because of (2.2) it then holds with ϕ = (ϕ − ϕ(0)) + ϕ(0)   ϕ dQ = E ϕ(x) dμU,V (x), ϕ ∈ Lb . (2.3) Thus, any distribution with zero expectation can be represented as a mixture of two-point measures μa,b .

2.1 Skorohod’s Embedding Theorem

15

τU,V is in general not a stopping time with respect to the σ -algebra AB . But  it is a stopping time with respect to the enlarged σ -algebra At := σ Bs , s ≤  t, U, V . Therefore, τU,V is called an extended stopping time. With τU,V one obtains the assertion. It holds: d

Bτu,v = μu,v .

(2.4)

Because of the independence of (U, V ) from the Brownian motion B it follows because of (2.3): d

d

BτU,V = μU,V = X. Because of P X = Q it follows: The Brownian motion, stopped at τU,V yields the distribution Q. Further, after the first step of the proof, it follows that   (2.4) EτU,V = EE τU,V | U = u, V = v = EEτu,v = EBτ2U,V = EX2 .

 

Remark 2.2 The construction of a (non-extended) stopping time goes back to Azema and Yor (1979). The more elaborate proof can be found, for instance, in Rogers and Williams (2000, pp. 426–430) or in Klenke (2006, pp. 482–485). Random variables can thus be reproduced by stopping a Brownian motion at appropriate times. This procedure can also be undertaken for entire sequences of random variables. Theorem 2.3 (Skorohod’s Embedding Theorem) Let be (Xi ) be an i.i.d. sequence with P Xi = μ, EXi = 0, and finite second moment, EX12 < ∞. Then there exists a sequence of independent, identically distributed random variables (τn )n≥0 withEτn = EX12 such that the partial sum sequence (Tn ) with T0 := 0 and Tn := ni=1 τi is an increasing sequence of stopping times with respect to a Brownian motion B with (1) (2) (3)

P BTn −BTn−1 = μ, ∀ n,   d BTn − BTn−1 = (Xn ), 

 d   Sn = BTn ,

(2.5)

i.e. the sequence (Sn ) is ‘embedded’ in (Bt ). d

Consequence Because of the embedding (Sn ) = (BTn ) a wealth of asymptotic properties of the partial sum sequence can be obtained using a Brownian motion.

16

2 Skorohod’s Embedding Theorem and Donsker’s Theorem

Proof According to Skorohod’s representation theorem, there exists a stopping time d

2 τ1 with Bτ1 = X1 and Eτ1 =  EX1 . The increments of a Brownian motion after the time point τ1 , Bt +τ1 − Bτ1 , are a Brownian motion again according to the strong Markov property. One can apply the previous result to this process: d

∃τ2 : Bτ2 +τ1 − Bτ1 = X2 and Eτ2 = EX22 . Because of the independent increment property of Brownian motion τ2 is independent of Aτ1 and is in particular independent of τ1 . Inductively, we obtain a sequence of stopping times Tn with Tn = Tn−1 + τn ,

d

BTn − BTn−1 = Xn

and Eτn = EXn2 ,

and so that τn is independent of (τi )i≤n−1 . Accordingly, the increments BTi − BTi−1  d  are independent of ATi−1 . Thus it follows that BTn − BTn−1 = (Xi ). From this it follows that 

 d   Sn = BTn

and ETn =

n 

EXk2 .

k=1

  Remark 2.4 The embedding holds in an analogous way for non-identically distributed sum sequences. (Sn ) =

n 

 d   with Xi = BTn

ETn =

i=1

n 

EXk2 .

(2.6)

k=1

A direct corollary of Skorohod’s embedding theorem is the central limit theorem: Theorem 2.5 (Central Limit Theorem) Let be (Xi ) be an i.i.d. sequence with EX1 = 0, EX12 = 1, then it holds: Sn D √ −→ N(0, 1). n

(2.7)

 d Proof By Theorem 2.3 is Sn = BTn with a stopping time Tn = ni=1 τi for an i.i.d. sequence (τi ) with Eτ1 = EX12 = 1 < ∞. Using the standard scaling property of Brownian motion, it follows Sn d BT d √ = √ n = B Tn n n n

2.1 Skorohod’s Embedding Theorem

17

By the strong law of large numbers, it follows: motion has continuous paths, it follows that

Tn n

→ 1 [P]. Since a Brownian

d

B Tn −→ B1 = N(0, 1). n

  In the following theorem, we show that Skorohod’s embedding theorem can be used to transfer the (relatively easy to prove) law of the iterated logarithm for Brownian motion to the proof of Hartmann–Wintner’s theorem for partial sum sequences. Theorem 2.6 (Hartmann–Wintner’s Law of the Iterated Logarithm) Let (Xi ) be an i.i.d. sequence of random variables with EXi = 0 and Var Xi = 1, then it follows lim sup √

Sn = 1 [P ]. 2n log log n

(2.8)

 d Proof By Theorem 2.3 it holds that (Sn ) = (Sn ) with Sn := BTn = nν=1 (BTν − BTν−1 ), BT0 := 0. According to the law of the iterated logarithm for the Brownian motion it holds that (Fig. 2.2)

lim sup  t →∞

Bt 2t log log t

= 1 [P ].

Fig. 2.2 Hartmann–Wintner’s law of the iterated logarithm

18

2 Skorohod’s Embedding Theorem and Donsker’s Theorem

Fig. 2.3 Proof of Hartmann–Wintner’s law

Assertion: lim = 

t →∞

Bt −  S[t ]

= 0 [P ].

2t log log t

Hartmann–Wintner’s theorem then follows from this assertion. The proof uses the strong law of large numbers: Tn −→ 1 [P ]. n

(2.9)

Thus for ε > 0 and t ≥ t0 (ω):  t , t (1 + ε) . ∈ 1+ε 

T[t ]  Let Mt := sup |Bs − Bt |,

t 1+ε

≤ s ≤ t (1 + ε)

 (Fig. 2.3) and consider the

(1 + ε)k

subsequence tk := ↑ ∞. For t ∈ [tk , tk+1 ] it holds: Mt ≤ sup{|Bs − Bt |; tk−1 ≤ s ≤ tk+2 } ≤ 2 sup{|Bs − Btk−1 |; tk−1 ≤ s ≤ tk+2 }. Because of tk+2 −tk−1 = ϑtk−1 , ϑ = (1+ε)3 −1, it holds by the reflection principle d

max Br = |Bt |,

(2.10)

0≤r≤t

and, therefore,  P max

tk−1 ≤s≤tk+2

=P



|Bs − Btk−1 | > (3 ϑ tk−1 log log tk−1 )1/2

max |Br | > (3 log log tk−1 )1/2

0≤r≤1





 tk−1  ≤ 2 κ (log log tk−1 )−1/2 exp − 3 log log , with a constant κ. 2

2.2 Functional Limit Theorem

It follows:

 k

19

P (. . .) < ∞. Therefore, the Borel–Cantelli lemma implies |S˜[t ] − Bt | ≤ (3 ϑ)1/2 [P ]. lim  t log log t

t −→∞

With ϑ −→ 0 the assertion follows.

2.2

 

Functional Limit Theorem

For applications Donsker’s invariance principle is of special importance. It states that continuous functionals of the partial sum sequence converge in distribution to the corresponding functional of a Brownian motion. Idea:  functional of

Sk √ n



D

−→ functional of (Bt ). 0≤k≤n

The space of continuous functions on [0, 1], C = C[0, 1] equipped with the supremum metric is a complete separable metric space. We equip this space with the σ -algebra generated by the projections. This is the same σ -algebra as the one generated by the topology of uniform convergence and also generated by the topology of pointwise convergence on C, the Borel σ -algebra on C. E = Bu (C) = Bp (C) =: B(C).

(2.11)

Definition 2.7 (Convergence in Distribution) A sequence of probability measures μn ∈ M 1 (C, B(C)) converges in distribution to a measure μ ∈ M 1 (C, B(C)), D

μn −→ μ if and only if for all real, continuous, bounded functions on C, ϕ : C → R, it holds: 

 ϕ dμn −→

ϕ dμ.

(2.12)

Also for measures in general metric spaces one introduces the notion of convergence in distribution by the convergence of integrals of continuous bounded functions. Correspondingly, one defines convergence in distribution for stochastic processes with continuous paths   X(n) = Xt(n)

0≤t ≤1

D

−→ X,

20

2 Skorohod’s Embedding Theorem and Donsker’s Theorem

if the associated distributions converge, equivalently, if the expectations of continuous bounded functions  converge, E(X(n) ) −→ E(X). Remark 2.8 D

(1) If X(n) −→ X and if  : C −→ R is P X -almost surely continuous, then it follows  D   X(n) −→ (X), because for ϕ ∈ Cb , ϕ ◦  is a bounded P X -almost surely, continuous function. (2) 

D

μn −→ μ ⇐⇒

 ψdμn −→

ψdμ, ∀ ψ ∈ Cb , ψ uniformly continuous. (2.13)

Now let (Xi ) be an i.i.d. sequence with EXi = 0 and EXi2 = 1. We introduce a suitable scaling of the partial sum process in two steps. (a) Linear interpolation: Let  (n) S

(u) :=

Sn √ , n 1 √ Sk n

u≥n  + (u − k)(Sk+1 − Sk ) , u ∈ [k, k + 1), 0 ≤ k ≤ n − 1,

i.e.  S (n) is the normalized partial sum process defined on [0, n] by linear interpolation. (b) Temporal scaling: The partial sum process S (n) = (St(n) )0≤t ≤1 is obtained from  S (n) by: (n)

St

= S (n) (nt),

0 ≤ t ≤ 1.

By linearly interpolation of the scaled partial sum sequence and by rescaling it to the time interval [0, 1], we have obtained a stochastic process with continuous paths in C[0, 1]. An important corollary of Skorohod’s embedding theorem is Donsker’s invariance principle. Theorem 2.9 (Donsker’s Invariance Principle) Let (Xi ) be an i.i.d. sequence with EXi = 0 and EXi2 = 1, then the partial sum process S (n) converges to a Brownian motion in C[0, 1], D

S (n) −→ B.

(2.14)

2.2 Functional Limit Theorem

21

Meaning Thus, one can simulate a Brownian motion by a partial sum sequence (for example, by a random walk). The theorem is called the invariance principle because the limit of S (n) is independent of the distribution of Xi . The fundamental consequence of the theorem is the possibility of describing the distribution of a functional of a partial sum process approximately by the distribution of the functional of a Brownian motion. This is a generalization of the Central Limit Theorem, which has many fundamental applications in probability theory and mathematical statistics. (n)

Proof of Theorem 2.9 (St ), the scaled and interpolated sum process encodes the Sm sequence ( √ ). It holds: n 

Sm √ n



d

=



BTm √ n



  d = B Tm . n

The normalized stopping time transformation converges a.s., T[ns] −→ s a.s. n From this follows pointwise convergence of

S[ns] √ n

S[ns] d √ = B T[ns] −→ Bs a.s., n n

in distribution 0 ≤ s ≤ 1.

However, even uniform stochastic convergence of B T[ns] holds.

 

n

  Lemma 2.10 The process B T[ns] converges stochastically to B uniformly in n

[0, 1], i.e.

    sup B T[nt] − Bt  −→0. P n

t ∈[0,1]

Proof By a simple monotonicity argument, the following holds  P

     T[ns] − s  < 2δ > 1 − ε sup  n 0≤s≤1

for n ≥ Nδ,ε . As in the proof of Theorem 2.6, using the maximal inequality for Brownian motion, the assertion follows.  

22

2 Skorohod’s Embedding Theorem and Donsker’s Theorem

Proof of Theorem 2.9 (Continued) It is to show: Let ϕ : C[0, 1] −→ R be uniformly continuous and bounded; then it holds Eϕ(S (n) ) −→ Eϕ(B). Because of

(n) St

−

S[nt] √ ∞ n

−→ 0 and P



S[nt] √ n



d

=

B

T

√[nt] n



  = B T[nt] it suffices d

n

with Bt(n) := B T[nt] to show: n

Eϕ(B (n) ) −→ Eϕ(B). But it holds with n := B (n) − B because of the uniform continuity of ϕ :      E ϕ(B (n) ) − ϕ(B)  ≤ E(ϕ(B (n) − ϕ(B))1{n >δ} + |E(ϕ(B (n) − ϕ(B))1{n ≤δ} | ≤ 2 sup |ϕ| P (n > δ) + ε

for δ ≤ δ0 .

By Lemma 2.10 the first term converges to 0. This implies that D

B (n) −→ B and thus the assertion.

 

Remark and Examples (a)  : C −→ R1 , (ω) = ω(1) is a continuous function. Thus it follows from Donsker’s invariance principle:   Sn D d  S (n) = √ −→ (B) = B1 = N (0, 1). n This is precisely the central limit theorem. (b) (ω) := sup{ω(t), 0 ≤ t ≤ 1} is a continuous function. Thus it follows   Sm D  S (n) = max √ −→ M1 = sup Bt . 0≤m≤n n 0≤t ≤1

(2.15)

The maximum of the partial sum process is found at the discrete time points because of the linear interpolation. According to André’s reflection principle   P0 (M1 ≥ a) = P0 (τa ≤ 1) = 2 P0 (B1 ≥ a) = P0 |B1 | ≥ a .

2.2 Functional Limit Theorem

23

d

Thus M1 = |B1 |. M1 is distributed like the absolute value of a standard normally distributed random variable. (c) Let (Sn ) be a symmetric random walk, and let Rn = be range of Sn , i.e. the number of points that are visited by the time n of Sn . Rn = 1 + max Sm − min Sm . m≤n

m≤n

(2.16)

The question is: Which interval will be visited by the time n ? To answer the Rn question, we need to normalize suitably: √ = (S (n) ) is a functional of the n partial sum process with  as in (2.16). Therefore, it follows Rn D √ −→ (B), n

(B) = 1 + sup Bt − inf Bt . t ≤1

t ≤1

(2.17)

(d) An example where one needs the extension to a.s. continuous functionals. We consider (Fig. 2.4) (ω) := sup{t ≤ 1; ω(t) = 0}. (ω) denotes the last time t before 1 at which ω(t) = 0. This functional is not continuous, because (ωε ) = 0, ∀ ε > 0. Further it holds that ωε − ω0  −→ 0 but (ω0 ) = 23 . But for ω ∈ C with (ω) < 1 and such that ω has in each neighborhood Uδ ((ω)) positive and negative values for all δ > 0, it holds that  is continuous in ω. These ω have measure 1 with respect to Brownian motion. So  is P0 almost surely continuous. Let Ln := sup{m ≤ n; Sm−1 Sm ≤ 0}

Fig. 2.4 Example not continuous

24

2 Skorohod’s Embedding Theorem and Donsker’s Theorem

be the index of the last change of sign before n of the random walk. Analogously, let L := sup{0 ≤ t ≤ 1; Bt = 0} be the time of the last zero of the Brownian motion before t = 1. The distribution of L is the arc sine distribution. We now obtain as a corollary. Corollary 2.11 (Arc-Sine Law) For the random walk (Sn ) the normalized time of the last sign change before n, Lnn converges in distribution to the arc-sine distribution Ln D −→ L. n

(2.18)

(e) Positivity range Let (ω := λ\1 ([t ∈ [0, 1]; ω(t) > 0}).  is not continuous on C[0, 1], but  is continuous in ω if λ\1 (]t ∈ [0, 1]; ω(t) = 0}) = 0. The set of these exceptional points has measure 0 with respect to P0 , the distribution of the Brownian motion, because according to Fubini 

1

E0 λ\1 ({t ∈ [0, 1]; Bt = 0}) =

P0 (Bt = 0)dλ\1 (t) = 0.

0

Thus  is P0 -a.s. continuous and it follows from the Skorohod theorem Corollary 2.12 (Convergence of Positivity Ranges) |{m ≤ n; Sm > 0}| D 1 −→ λ\ ({t ∈ [0, 1]; Bt = 0}) n

(2.19)

(f) Erdös and Kac (1946), investigated for k = 2 the following functional for the random walk, based on the function  ω(t)k dλ\1 (t), k ∈ N. (ω) := [0,1]

 is continuous on C[0, 1]. Therefore, from Skorohod’s theorem follows the Erd˝os and Kac theorem: ( S (n) ) = n−1− 2 k

n  m=1

D

k Sm −→



1 0

Btk dt.

(2.20)

2.2 Functional Limit Theorem

25

It is worth noting that also for k ≥ 2 the above approximation in (2.20) only needs the assumption EXi2 = 1, EXi = 0, but not the stronger assumption E|Xi |k < ∞. In the case k = 1 it follows from (2.20) that 3

n− 2

n  m=1

D



1

(n + 1 − m)Xm −→ 0

d

Bt dt = N(0, 1).

3

Stochastic Integration

· The aim of this chapter is to introduce the stochastic integral 0 ϕs dXs for semimartingales X and stochastic integrands ϕ and to describe its properties. If one interprets ϕ as a trading strategy and X as the price process of a security, then the stochastic integral can be understood to be the accumulated gain. The stochastic integral is constructed in a series of steps and extended to general function classes and process classes. In the first part of the chapter, after an introduction to martingales, the stochastic integral for a Brownian motion is introduced as a model case. The construction is then extended in several steps to the integration of general semimartingales. The stochastic integral is a particularly important tool for stochastic model building. In analogy to the ordinary differential equations it allows by means of the corresponding stochastic differential equation e.g. of the form dXt = a(t, Xt ) dt + b(t, Xt ) dBt to construct from local properties, here the drift a and the volatility b, a stochastic model; in the above case, a diffusion process. Semimartingales generalize this principle and can be described by local characteristics, the (local) drift, the continuous diffusion part (martingale part) and a (local) jump part. They form a fundamental class of models and are of particular importance for models in financial mathematics.

3.1

Martingales and Predictable Processes

Let ( , A, P ) be a complete martingale probability space and (At )t ≥0 ⊂ A a filtration in . We postulate in general a regularity property of filtrations:

© Springer-Verlag GmbH Germany, part of Springer Nature 2023 L. Rüschendorf, Stochastic Processes and Financial Mathematics, Mathematics Study Resources 1, https://doi.org/10.1007/978-3-662-64711-0_3

27

28

3 Stochastic Integration

General Condition (Conditions Habituelles) (At ) is right-continuous, i.e. At + =

 s>t

As = At (3.1)

A0 contains all (subsets of) P zero sets. Let X be a d-dimensional stochastic process, X : [0, ∞) × −→ Rd . X is called adapted on the filtration (At ), if Xt ∈ L(At ), for all t, Xt = X(t, ·).  In the standard case based on natural filtration At := ε>0 σ (σ (Xs , s ≤ t + ε) ∪ NP ) =: AX t the general condition is fulfilled. The central concept of this chapter and of stochastic integration theory is the notion of a martingale. Definition 3.1 M = (Mt , At ) is a martingale if (1) M is (At )-adapted (2) E|Mt | < ∞, for all t. (3) E(Mt | As ) = Ms [P ], for all s ≤ t M is called submartingale if (1), (2), and (3’) E(Mt | As ) ≥ Ms [P ] If in (3’) “≤” holds, then M is called a supermartingale. X is called a continuous process if the paths of X are continuous X is called càdlàg process if the paths are right-continuous and have limits on the left-hand side, i.e. Xt = lim Xs , Xt − = lim Xs exists. s↓t,s>t

s↑t,s 0. Then τ + q is a stopping time and τ + q > τn . Now, according to the optional sampling theorem. 1 ≥ E(Xτn 1(τn ≤∞) ) ≥ E(Xτ +q 1(τn ≤∞) ) ≥ 0. n

3.1 Martingales and Predictable Processes

33

Fig. 3.1 Supermartingale

For n → ∞, therefore, by the monotone convergence theorem E(Xτ +q 1(τn 0. The right-continuity of X then implies the assertion   X· (ω) = 0 on [τ, ∞). As a corollary of the optional sampling theorem, we obtain the following characterization of the martingale property by the conservation of expected values under stopping times. Proposition 3.11 Let X be càdlàg and adapted. Then it holds: X is a martingale ⇐⇒ ∀ bounded stopping times τ is Xτ ∈ L1 and EXτ = EX0 . Proof “⇒”: Follows from the optional sampling theorem. “⇐”: For s < t and A ∈ As define the stopping time τ := t1Ac + s1A . Then: EX0 = EXτ = EXt 1Ac + EXs 1A . Further: EX0 = EXt = EXt 1Ac + EXt 1A , because also t is a stopping time. This implies the martingale property : Xs = E(Xt | As ).   The martingale property of a process is preserved under stopping.

34

3 Stochastic Integration

Corollary 3.12 Let M be a (càdlàg) martingale, τ a stopping-time, then it follows M τ = (Mt ∧τ )τ ≥0 is a (càdlàg) martingale. Proof Let M τ be càdlàg and adapted. Let S be a bounded stopping time. Then S ∧τ is a bounded stopping time and hence it follows: EMSτ = EMS∧τ = EM0 = EM0τ . According to Proposition 3.11 then M τ is a martingale with respect to (At ).

 

A standard tool for martingales are Doob’s maximal inequalities. Theorem 3.13 (Doob’s Maximal Inequality) Let X = (Xt )t ∈T be a rightcontinuous martingale or positive submartingale and let X∗ := sup |Xt |. Then it t

follows P (X∗ ≥ λ) ≤ sup t

E|Xt |p for p ≥ 1, λ > 0. λp

Further it holds for 1 < p < ∞: X∗ p ≤

p sup Xt p . p−1 t

Proof The proof follows from the discrete Doob inequality. Let D ⊂ T be countable dense, then because of the right-continuity of X, X∗ = supt ∈D |Xt |. So the discrete Doob inequality can be applied.   We need an essential extension of the martingale notion, the local martingale. Definition 3.14 (a) An adapted process M is called local martingale if there is a sequence of stopping times τn with τn ↑ ∞ such that ∀ n : M τn ∈ M, i.e. M τn is a martingale. (τn ) is called localizing sequence. Mloc := set of local martingales. (b) M ∈ Mloc is called local L2 -martingale ⇐⇒ ∃ a localizing sequence (τn ) such that M τn ∈ M2 . M2loc := set of local L2 -martingales. Remark 3.15 (a) M ⊂ Mloc . Consider the localizing sequence τn = n. (b) Mloc,c ⊂ M2loc ,

3.1 Martingales and Predictable Processes

35

because, e.g. τn = inf{t; |Mt | ≥ n} = inf{t; |Mt | = n} is a localizing sequence with M τn ∈ M2loc . (c) For M ∈ Mloc the following characterization of the martingale property holds M ∈ M ⇐⇒ M is of class (DL), i.e. {MT ; T stopping time, T ≤ a} is uniformly integrable, ∀ a < ∞. Proof “⇒”

According to the optional sampling theorem, it follows 



sup

T ≤a {MT >K}

MT dP = sup

K→∞

T ≤a {MT >K}

Ma dP −→ 0,

so {MT ; T ≤ a} is uniformly integrable. “⇐” Let (Tn ) be a localizing sequence of stopping times and let s < t then M Tn ∧t ∈ M and it holds for A ∈ As    MsTn dP = MsTn ∧t dP = MtTn dP . A

A

A

By the uniform integrability assumption, it follows because of MsTn ∧t −→ Ms and MtTn ∧t −→ Mt 

 Ms dP = A

Mt dP . A

  For a process A define  VtA := sup0=t0 0, ε > 0 holds:

 t    t     ε P sup  f dB  > λ ≤ 4 2 + P fs2 ds > ε λ t ≤T

0

0

50

3 Stochastic Integration

(d) If for f n , f ∈ L0 (B),

T 0

P

|fsn − fs |2 ds −→ 0, then it follows:

 t   t   P n  fs dBs − fs dBs  −→ 0. sup 

0≤t ≤T

0

0

Thus uniform  t stochastic convergence on compacta holds. (e) Let Yt = 0 f dB and σ be a stopping time, then it holds: 

t 0

f 1[0,σ ] dB = Yt ∧σ .

Proof (a), (b) follows from thedefinition with localizing sequence σ n . t (c) Let σ := inf{t ≥ 0; 0 fs2 ds ≥ ε} then f 1[0,σ ] ∈ L2 (B).  t       P sup  f dB  > λ ≤ P (σ < T ) + P ( sup |Zt | > λ, σ ≥ T ) t ≤T t ≤T ∧σ 0  

=:Zt

≤ P (σ < T ) + P (sup |Zt ∧σ | > λ) t ≤T

 ≤ P (σ ≤ T ) + P

 t       f 1[0,σ ] dB  > λ sup 

t ≤T

0

using the Doob inequality, Theorem 3.13  t 2   1  ≤ P (σ ≤ T ) + 2 E sup  f 1[0,σ ] dB  λ t ≤T 0  ∈L2 (B)

with Proposition 3.32 or 3.35 ≤ P (σ ≤ T ) + 4

1 E λ2



T ∧σ 0

fs2 ds

by definition of σ ≤ P (σ ≤ T ) + 4 

T

=P 0

(d) follows from (c).

ε λ2

 ε fs2 ds > ε + 4 2 . λ

3.2 Itô Integral for the Brownian Motion

(e) Let Ytn :=

t 0

51

f 1[0,σ n ] dB be an approximating sequence. Then  ∈L2 (B)

 Ytn∧σ =

t 0

f 1[0,σ ] 1[0,σ n ] dB.

 ∈L2 (B)

t t It holds Ytn∧σ −→ Yt ∧σ and 0 f 1[0,σ ] 1[0,σ n ] dB −→ 0 f 1[0,σ ] dB according to the definition of the integral. But from this follows the assertion.   Lemma 3.40 Let U ∈ L(Au ), u < r. Define: ht := U 1(u,r] (t) (∈ / E, since U is not bounded). Then  t 0 h dB = U (Br∧t − Bu∧t ). h ∈ L (B) and 0

Proof Let hnt := U 1(U ≤n) 1(u,r] (t). Then hn ∈ E and 

t 0

hnt dB = U 1(U ≤n) (Br∧t − Bu∧t ).

t t P It holds: 0 |hns − hs |2 ds = U 2 1(U >n) 0 1(u,r] (s) ds −→ 0. According to Proposition 3.39 the assertion follows.   We can now derive the following approximation result. Theorem 3.41 (Riemann Approximation) Let f be a square-integrable, left continuous, adapted process in the Riemann sense, then it holds with  = {0 = s0 < s1 < · · · < sk = t}, || := max |si+1 − si | 

t 0

f dB = lim

||−→0

k−1 

fsi (Bsi+1 − Bsi )

(stoch.convergence).

i=0

k Proof Let f  := fsi 1(si ,si+1 ] (the fsi are not bounded). Then by i=1  t  Lemma 3.40 0 f dB = i fsi (Bsi+1 − Bsi ). According to the properties of t P the usual Riemann Integrals holds 0 |fs − fs |2 ds −→ 0 and thus according to Proposition 3.39 (d) the assertion follows.  

52

3 Stochastic Integration

Example 3.42 It holds B ∈ L2 (B) and 

T

Bs dBs =

0

1 2 (B − T ). 2 T

Proof Let n = {0 = t0n < · · · < tkn }, k = 2n , tin = via the Riemann approximation holds: n −1 2

i=0



P

n − Bt n ) −→ Btin (Bti+1 i   

=:a

=:a

=:b

i 2n T .

With Proposition 3.41

T

Bs dBs . 0

Because of a(b − a) = 12 (b2 − a 2 − (b − a)2 ) the left side is equal: 2 −1  1  2 n Bt n − Bt2n − (Bti+1 − Btin )2 i+1 i 2 n

i=0

n−1

2 1 n (Bti+1 − Btin )2 −→ T . = (B 2T − QnT ) with QnT := 2 i=0

The right side is just the quadratic variation of B: &B'T = T .  T 1 We thus obtain: Bs dBs = (BT2 − T ). 2 0 As a corollary this implies:  Bt2

=2

t

Bs dBs + t

0

is the decomposition of the submartingale Bt2 into a martingale and a predictable process of finite variation.   We now introduce infinite integrals to conclude this section. Theorem 3.43 Let f ∈ L(P), random variable Z∞ such that 

t

Zt := 0

∞ 0

fs2 ds < ∞ almost surely. Then there exists a

P

f dB −→ Z∞

We define the infinite integral by

f or t −→ ∞.

3.2 Itô Integral for the Brownian Motion

53



∞

(5) Integral for Infinite Intervals 0

f dB := Z∞ .

Proof Let t ≥ u and g := f 1[u,∞) , then 

t



t

g dB = Zt − Zu =

0



u

f dB −

f dB.

0

0

From Proposition 3.39 (c) we get P (|Zt − Zu | > δ) ≤ 4 ε δ2

To ε > 0, δ > 0 be η > 0 such that t0 < u < t: 

t

P u

ε +P δ2



t u

 fs2 ds > ε .

< η8 . Then choose t0 > 0 such that ∀u where

 η fs2 ds > ε ≤ . 2

Then it follows P (|Zt − Zu | > δ) ≤ η,

∀u with t0 < u ≤ t.

Then by the Cauchy criterion for stochastic convergence there exists a limit Z∞ , P

Zt −→ Z∞ for t −→ ∞.   This allows the integral to be introduced for arbitrary stopping boundaries. Remark 3.44 Let τ be a stopping time, f ∈ L(P), 

t 0

P

f 1[0,τ ] dB −→



∞ 0

τ 0

fs2 ds < ∞, then it holds:

f 1[0,τ ] dB.

So for a stopping time τ and for f ∈ L(P) such that 

τ 0

fs2 ds < ∞

(3.6)

we now define 

τ

(6) Integral for Pathwise L2 -Processes 0

 f dB := 0

∞

f 1[0,τ ] dB.

54

3 Stochastic Integration

0 The  τ 2condition (3.6) holds for f ∈ L (B) if τ is a finite stopping time, or if 0 fs ds < ∞ and τ is any stopping time.

We now obtain the optional sampling theorem for stochastic integrals as a consequence of the above considerations. Theorem 3.45 (Optional Sampling Theorem for Local Martingales) Let f ∈ t τ L0 (B), Yt := 0 fs dBs and τ be a stopping time with E 0 fs2 ds < ∞. Then it follows:  EYτ = 0,

EYτ2 = E

τ

fs2 ds < ∞.

0

If in particular Eτ < ∞ then it holds with f ≡ 1: EBτ2 = Eτ.

EBτ = 0, Proof Let Zt :=

t 0

f 1[0,τ ] dB, then by Theorem 3.43 and by the definition in (6)  ∈L2 (B)

the following holds: 

τ

I :=

f dB = lim Zt t −→∞

0

(stochastic limit).

L2

It suffices to show: Zt −→ I , because:  EZt = 0,

EZt2

=1=E

t ∧τ

0

fs2 ds.

With L2 -convergence it follows:  EI = 0,

τ

2

EI = 0

fs2 ds

thus the assertion. It holds with majorized convergence: 

t

E(Zt − Zs )2 = E s

 fu2 1[0,τ ] (u) du = E

t ∧τ s∧τ

s,t →∞

fu2 du −→ 0.

So (Zt ) is a Cauchy net in L2 and, therefore, there exists a limit in I of Zt in L2 , L2

Zt −→ I.  

3.3 Quadratic Variation of Continuous Local Martingales

55

Remark 3.46 (Integral Construction) For the construction of the stochastic integral with respect to Brownian motion, the maximal inequality for martingales and the martingale property of (Bt ) as well as of (Bt2 − t) are used. For continuous martingales M ∈ Mc this process can be carried out analogously. Let M ∈ M2c and let (Mt2 − At ) ∈ Mc with A ∈ V+ then the stochast tic integral 0 f dM can be constructed analogously to that for the BrownT ian motion if E o fs2 dAs < ∞. For this class of processes, the integral was introduced by Kunita and Watanabe (1967). We will generalize the definition of the stochastic integral to this class of integrators in the following section.

3.3

Quadratic Variation of Continuous Local Martingales

For processes A ∈ V and adapted locally bounded progressively measurable t integrands X, the integral (X · A)t := 0 Xs dAs is defined as a stochastic Stieltjes integral and it holds: X · A is adapted and X · A ∈ V. If X is càdlàg, then the above conditions are satisfied. X is locally bounded and progressively measurable. Our goal is to find the integral X · M for continuous martingales M. As for Brownian motion, this requires finiteness of the quadratic variation of M. Definition 3.47 X = (Xt ) is called a process of finite quadratic variation, if ∃ A ∈ V + such that ∀ t > 0 Tt := Tt (X) =



P

(Xti+1 ∧t − Xti ∧t )2 −→ At ,

i

for all decompositions (ti ) of R+ with || −→ 0. At := [X]t is called quadratic variation process of X. Remark 3.48 For the Brownian motion, the quadratic variation is [B]t = t

and it holds

Bt2 − t ∈ M.

B is not of finite variation but of finite quadratic variation. The following basic theorem ensures finite quadratic variation of continuous bounded martingales.

56

3 Stochastic Integration

Theorem 3.49 (a) Every continuous bounded martingale M is of finite quadratic variation and [M] ∈ Vc+ , i.e., the variation process is nonnegative, increasing, and continuous. (b) There exists a unique continuous adapted process A ∈ V + with A0 := 0 such that M 2 − A ∈ Mc . A =: &M' is called the predictable quadratic variation of M. Uniqueness also holds in the class P ∩ V + of predictable and increasing processes. (c) It holds: [M] = &M'. Proof We first prove uniqueness in (b). (b) Uniqueness Let A, B ∈ Vc+ , A0 = B0 = 0 and M 2 − A, M 2 − B ∈ Mc . Then it follows A − B ∈ Mc ∩ V, and A0 = B0 = 0. According to Proposition 3.20 it follows that A = B. Uniqueness also holds in the class of predictable elements in V + . (a) Existence and finiteness of quadratic variation: The proof is divided into several steps. 1st step: Mt2 − Tt (M) ∈ Mc proof  Let ti < s ≤ ti+1, thenit follows :  E (Mti+1 − Mti )2 | As = E (Mti+1 − Ms )2 | As + (Ms − Mti )2 . The mixed term is dropped because of the martingale property. Let s < t, ti < s ≤ ti+1 < · · · < ti+k < t ≤ ti+k+1 . Then it follows   E Tt (M) − Ts (M) | As =

∞    E (Mtj ∧t − Mtj−1 ∧t )2 − (Mtj ∧s − Mtj−1 ∧s )2 | As j =1

=

k    E (Mti+l+1 ∧t − Mti+l ∧t )2 − (Mti+l+1 ∧s − Mti+l ∧s )2 | As l=0

=

k    E (Mti+l+1 ∧t − Mti+l ∧t )2 | As l=1   + E (Mti+1 − Mti )2 − (Ms − Mti )2 | As

= E(Mt2 − Mt2i+1 + (Mt2i+1 − Ms2 ) | As ) because of the martingale property  = E Mt2 − Ms2 | As . 

(3.7)

3.3 Quadratic Variation of Continuous Local Martingales

57

From this follows Mt2 − Tt (M) ∈ Mc . 2. Second step: In the second step, our goal is to show that for a > 0 and a decomposition n of [0, a] with |n | −→ 0 it holds that Tan has a limit in L2 : L2

Tan →[M]a . Assertion: (Tan ) is a Cauchy sequence in L2 . For this let ,  be decompositions. Then according to (3.7)  X := T  − T  ∈ Mc . From this it follows by the first step applied to X 



EXa2 = E(Ta − Ta )2 = ETa (X), where  denotes the joint refinement of  and  . Thus it holds:     EXa2 ≤ 2 · E Ta T  + Ta T  , using (x + y)2 ≤ 2(x 2 + y 2 ). So it suffices to show:  (a) ETa T  −→ 0 for || + | | −→ 0. Let sk ∈ and tl ∈ be the largest point in  before sk such that tl ≤ sk ≤ sk+1 ≤ tl+1 . Then it holds:   ∈Δ

∈Δ

Ts − Tsk = (Msk+1 − Mtl )2 − (Msk − Mtl )2 k+1 = (Msk+1 − Msk )(Msk+1 + Msk − 2Mtl ). It follows that: 



Ta T  ≤ sup |Msk+1 + Msk − 2Mtl |2 Ta (M). k

By Cauchy–Schwarz it follows that:  1  1   ETa T  ≤ E sup |Msk+1 + Msk − 2Mtl |2 2 E(Ta (M))2 2 k

 =:a

It holds: a −→ 0 if || + | | −→ 0. M is, therefore, uniformly continuous on compacta. Now it is to show in the next step: 1  (b) (E(Ta (M)2 ) 2 is bounded independent of ,  .

58

3 Stochastic Integration

W.l.g., let: a := tn , and we write  instead of  . It holds: (Ta )2 =

 n

2 (Mtk − Mtk−1 )2

k=1

=2

n     Ta − Tt k

 k+1 

l>k (Mtl −Mtl−1 )

2

    Ttk − Tt + (Mtk − Mtk−1 )4 k−1

 k (Mtk −Mtk−1 )2

  By the 1st step it follows E(Ta − Tt | Atk ) = E (Ma − Mtk )2 | Ak as M k is a martingale. Thus, by conditioning under Atk (c) E (Ta )2   = 2 nk=1 E[(Ma − Mtk )2 (Tt − Tt )] + nk+1 E(Mtk −Mtk−1 )4 k k−1  ( ' ≤ E 2 supk | Ma − Mtk |2 + sup | Mtk − Mtk−1 |2 Ta  

 ≤4C 2

1.) ETa =

≤4C 2 EMa2 ≤

increasing

≤ ≤ as Thus, (b) follows. Using the Doob inequality for martingales applied to T n − T m it follows: (d) E supt ≤a |Ttn − Ttm |2 ≤ 4E(Tan − Tam )2 −→ 0 for |n | −→ 0 after 2.), 2.a), 2.c). From this assertion 2. follows: (Tan ) is a Cauchy sequence and has a limit in L2 independent of the 12C 2 ETa

12C 4

C2.

L2

decomposition, Tan −→Aa = [M]a . Step 3: Properties of [M]: L2

(a) M 2 − [M] ∈ M, is a martingale, because Ttn − Ttm ∈ M and Ttm −→ [M]t for |m | −→ 0. From this follows for |m | −→ 0 convergence in L1 and thus Ttn − [M]t ∈ M. So after the 1st step M 2 − [M] ∈ M. (b) [M] is continuous. This is because, according to the Doob inequality. E supt ≤a |Ttn − [M]t |2 ≤ 4E|Tan − [M]a |2 −→ 0. nk

So there exists a subsequence (nk ) : supt ≤a |Tt surely (uniform convergence). From this follows that [M]t is continuous. (c) [M]t ⊂ Vc+ is increasing.

−[M]t | −→ 0 almost

3.3 Quadratic Variation of Continuous Local Martingales

59

) For without restriction n+1 is a refinement of n and n , is dense in [0, a]. Then Tsn ≤ Ttn ,)s ≤ t, s, t ∈ n . Thus it follows: [M]t is increasing on n , so also on [0, a], since it is continuous. (3a), (b), (c) ⇒ The quadratic variation process [M] is the unique continuous process A such that M 2 − A ∈ Mc . Thus it holds that [M] is equal to the predictable quadratic variation, [M] = &M'.   For non-continuous martingales, it will be shown later that quadratic variation and predictable quadratic variation are different processes. To extend Theorem 3.49 to non-bounded processes, we need the following connection with stopping times. Proposition 3.50 If τ is a stopping time, and if M ∈ Mc is bounded, then &M τ ' = &M'τ . Proof It holds that M 2 − &M' ∈ Mc . The martingale property is preserved by stopping. Thus, according to Corollary 3.12 Mt2∧τ − &M't ∧τ ∈ Mc thus (M τ )2 − &M'τ ∈ Mc . Because of the uniqueness of the predictable quadratic variation from Theorem 3.49 it follows: &M τ ' = &M'τ .   Remark 3.51 (Localization) (a) It holds: X ∈ Mloc ⇐⇒ ∃(τn ) a sequence of stopping times with τn ↑ ∞ such that Xτn is a uniformly integrable martingale. Let (τn ) be a localizing sequence. Then σn = τn ∧ n defines a localizing sequence such that Xσn is a uniformly integrable martingale. For M ∈ Mloc,c , let be Sn := inf{t : |Mt | ≥ n}. By the transition from τn to τn ∧ Sn , one obtains a localizing sequence (τn ) so that (M τn ) is even a uniformly integrable bounded martingale. So by localizing one can assume w.l.g. that M ∈ Mloc,c is uniformly integrable and bounded.

60

3 Stochastic Integration

(b) There are examples of uniformly integrable local martingales that are not martingales. (c) Let M = (Mn ) be a discrete martingale and ϕ = (ϕn ) be predictable. Then the martingale transformation n    ϕ · M = (ϕ · M)n with (ϕ · M)n = ϕk (Mk − Mk−1 ) k=1

is a local martingale, ϕ · M ∈ Mloc . This is in analogy to the stochastic  t integral in continuous time: If f ∈ L0 (B), then (f · B)t = 0 f dBs ∈ Mloc is a local martingale. The following definition describes the difference between the local martingale property and the martingale property. Definition 3.52 Let X be an adapted process (a) X is a process of class (D) (Dirichlet class), if {Xτ ; τ a finite stopping time} is uniformly integrable. (b) X is called process of class (DL) (local Dirichlet class), if ∀ a > 0 : {Xτ ; τ stopping time, τ ≤ a} is uniformly integrable. Theorem 3.53 If X is a local martingale, then it holds: (a) X ∈ M ⇐⇒ X is of class (DL) (b) X is a uniformly integrable martingale ⇐⇒ X is of the class (D) Proof (a) “⇒”: Be X ∈ M and τ ∈ γ a , i.e. τ is a stopping time and τ ≤ a. Then, according to the optional sampling theorem, it follows Xτ = E(Xa | Aτ ). But this implies the uniform integrability of {Xτ ; τ ∈ γ a }. “⇐”: Let X ∈ DL and (τn ) be a localizing sequence for X such that Xτn is a uniformly integrable martingale. Then it holds Xs∧τn = Xsτn = E(Xtτn | As ) = E(Xt ∧τn | As ). Since τn ↑ ∞ it follows: Xs∧τn −→ Xs a.s. and {Xs∧τn }n is uniformly integrable by condition (DL). So also convergence holds in L1 .

3.3 Quadratic Variation of Continuous Local Martingales

61

Similarly Xt ∧τn −→ Xt a.s. and in L1 . It follows: Xs∧τn = E(Xt ∧τn | As ) −→ E(Xt | As ) and, therefore, Xs = E(Xt | As ) i.e. X ∈ M. (b) “⇐”: If X ∈ Mloc ∩ (D), then the set {Xt ; 0 ≤ t ≤ ∞} ⊂ {Xτ ; τ a finite stopping time}, is, therefore, uniformly integrable. The assertion then follows from a). “⇒”: If X is a uniformly integrable martingale, and τ is a finite stopping time, then the closure theorem yields Xt −→ X∞ almost surely and in L1 and (Xt )t ≤∞ is a martingale. By the optional sampling theorem, it follows that Xτ = E(X∞ | Aτ ). Thus it follows that: {Xτ ; τ a finite stopping time} = {E(X∞ | Aτ ); τ a finite stopping time} is uniformly integrable.   The following theorem shows the existence of predictable quadratic variation &M' for continuous local martingales. As for bounded local martingales, it agrees with the quadratic variation [M]. Theorem 3.54 (Quadratic Variation of Continuous Local Martingales) Let M ∈ Mloc,c then it holds: (a) There exists exactly one process &M' ∈ Vc+ the predictable quadratic variation of M such that M 2 − &M' ∈ Mloc,c . (b) ∀ t, ∀ n decompositions of R+ , with |n | −→ 0, holds P

sups≤t |Tsn − &M't |−→0, i.e. &M' is identical with the quadratic variation [M]. Proof (a) ∃ a sequence of stopping times (Tn ) ↑ ∞ such that with Xn = M Tn ∈ Mc is a bounded martingale. According to Theorem 3.49 there exists a unique process

62

3 Stochastic Integration

(An ) ⊂ Vc+ , An (0) = 0 such that Xn2 − An ∈ Mc . Thus it follows 2 n − An+1 )Tn = Xn2 − ATn+1 ∈ Mc . (Xn+1 n = An almost surely. It follows that the The uniqueness implies that ATn+1 process &M' := An at [0, Tn ] is well-defined. It holds:

(M Tn )2 − &M'Tn ∈ Mc ,

d. h. M 2 − &M' ∈ Mloc,c .

Thus the existence of predictable quadratic variation holds. The uniqueness of &M' follows from the uniqueness on [0, Tn ], ∀ n. (b) Let δ, ε > 0; to t > 0 there exists a stopping time S such that M S is bounded and P (S ≤ t) ≤ δ. To show this let (τn ) be a localizing sequence and define νn := inf{t; |Mtτn | ≥ an }. Then it holds νn ↑ ∞ if an ↑ ∞ sufficiently fast.

E(M τn )2

t −→ 0 if an ↑ ∞ For P (νn ≤ t) = P (sups≤t |Msτn | ≥ an ) ≤ an sufficiently fast. So choose, for example S = νn0 ∧ τn0 . It holds: T  (M) = T  (M S ) and &M' = &M S ' to [0, S]. From this follows   P (sup |Ts (M) − &M S '| > ε) ≤ δ + P sup |Ts (M S ) − &M S ' | > ε . s≤t s≤t



−→0 for ||−→0

  Remark 3.55 From Proposition 3.50 and the proof of Proposition 3.54 it follows for M ∈ Mloc,c and a stopping time τ that &M τ ' = &M'τ . The Doob–Meyer decomposition is an important general version of Theorem 3.54, which also holds for non-continuous processes, which we state without proof. Theorem 3.56 (Doob–Meyer Decomposition) Let X be a submartingale of class (DL). Then it holds:

3.3 Quadratic Variation of Continuous Local Martingales

63

(a) ∃M ∈ M and ∃A ∈ V0+ ∩ P, i.e., there exist a martingale M and an increasing predictable process A with A0 = 0 such that X = M + A. (b) The decomposition is unique with respect to A in the class of increasing and predictable processes starting in 0.  For a proof, compare Karatzas and Shreve (1991, pp. 22–28). A describes the predictable drift of X. Remark 3.57 By localization, the Doob–Meyer decomposition can be extended to other classes of processes. Here are some variants of such extensions: (a) To M ∈ M2loc there exists a predictable process A of finite variation, A ∈ V0+ ∩P such that M 2 − A ∈ Mloc. (b) If X ∈ A+ = {A ∈ V + ; EA∞ < ∞} then X is a submartingale of class (D). It follows that X has a Doob–Meyer decomposition. A consequence of this is: (c) If A ∈ Aloc (A+ loc ) then there exists exactly one local martingale M ∈ Mloc and p Ap ∈ Aloc ∩ P (A+ loc ∩ P) such that A = M + A . Ap is called predictable compensator (predictable compensator) of A or also dual predictable projection. Equivalent to the definition of the predictable compensator Ap is the validity of the projection equations p

EH A∞ = EH A∞ ,

∀ H ∈ L+ (P)

(d) In contrast to the “dual predictable projection” Ap is the predictable projection p (predictable projection) X of a process X ∈ L(A ⊗ B+ ) uniquely defined by the following properties: p (d.1) X ∈ P (d.2) For all predictable stopping times T holds: p XT = E(XT | AT − ) on {T < ∞}. For integrable processes X the predictable projection exists and is unique except for equivalent versions. It holds: p

p

(XT ) = X10,T  + XT 1T ,∞ .

For a Poisson process N with intensity λ > 0 it holds: p

p

Nt = Nt − and Nt = λt.

64

3 Stochastic Integration

(d.3) A process X is called semimartingale, if X has a decomposition of the form M +  A X = X0 +  . ∈Mloc

∈V ∩O

So X is decomposable into a local martingale and an optional process of finite variation. Let S be the set of all semimartingales (cf. Definition 3.18). In general, the decomposition of a semimartingale X is not unique. However, if there exists a decomposition with A ∈ P, i.e. A is predictable, then the decomposition is unique in the class A ∈ V ∩ P. X is called special semimartingale if A is selectable in V ∩ P. The decomposition X = X0 +M +A, A ∈ V ∩P, then is called canonical decomposition of X. Let Sp be the set of special semimartingales. It follows from the Doob–Meyer decomposition: If X is a submartingale in the class (DL), then X ∈ Sp . The property of being a special semimartingale is essentially an integrability condition. It means that the jumps do not become too large. E.g. α-stable processes are special semimartingales for α > 1. The Cauchy process with α = 1 is not a special semimartingale. To describe the relationship between two processes, we define the predictable quadratic covariation. The predictable quadratic variation corresponds to the variance, the predictable quadratic covariation to the covariance. Proposition 3.58 Let M and N be continuous local martingales, M, N ∈ Mloc,c , then it holds: (a) There exists a unique continuous process &M, N' with finite variation such that MN − &M, N' is a continuous local martingale, i.e. ∃! &M, N' ∈ Vc , &M, N'0 = 0, MN − &M, N' ∈ Mloc,c &M, N' is the predictable quadratic covariation resp. bracket process. (b) Let sn := T

   Mtsi+1 − Mtsi Ntsi+1 − Ntsi ; i

then Tn converges uniformly to the predictable covariation process &M, N'. *  * P sup * Ts n − &M, N's * −→ 0, s≤t

if

|n | −→ 0,

3.3 Quadratic Variation of Continuous Local Martingales

65

i.e., the quadratic covariation [M, N] (cf. Proposition 3.78) exists and is identical to the predictable quadratic covariation &M, N'. Proof (a) Existence Consider the local continuous martingales M + N, M − N ∈ Mloc,c . By Theorem 3.54 there exist associated to them unique predictable quadratic variation processes &M + N' ∈ Vc+ , &M − N' ∈ Vc+ . Thus it holds (M + N)2 − &M + N' ∈ Mloc,c and (M − N)2 − &M − N' ∈ Mloc,c . By polarization MN = 12 ((M + N)2 − (M − N)2 ) it follows MN − 12 (&M + N' − &M − N') ∈ Mloc,c .

 =:&M,N'

Uniqueness follows as in Theorems 3.54 and 3.49. Localization reduces the problem to the case of bounded martingales. For M ∈ Mloc,c it holds &M τ ' = &M'τ . This provides consistency in the above definition of covariation and, as a consequence, uniqueness. (b) follows by polarization as in (a) of the corresponding quadratic increments.   Remark 3.59 It follows from (b): The (predictable) quadratic covariation is a bilinear form on the set of continuous local martingales &αM1 + βM2 , N' = α&M1 , N' + β&M2 , N'. That quadratic covariation has similar properties to the covariance is also seen in the following proposition; part (c). Proposition 3.60 Let M be a continuous local martingale M ∈ Mloc,c . (a) Let τ be a stopping time, then &M τ , N τ ' = &M, N τ ' = &M, N'τ (b) &M'a = 0 a.s. ⇐⇒ Mt = M0 a.s. ∀ t ≤ a (c) In general for a < b it holds: {ω; Mt (ω) = Ma (ω), ∀ t ∈ [a, b]} = {ω; &M'b (ω) = &M'a (ω)} a.s., i.e., if the quadratic variation on an interval is constant, then the process is also constant there.

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3 Stochastic Integration

Proof (a) follows from the construction of the quadratic covariation and the corresponding property of the quadratic variation. (b), (c) By (a), it suffices to show (b) for the case of bounded martingales M. Then Mt2 − &M't ∈ Mc . It follows for all t ≤ a: E(Mt −M0 )2 = E&M't and analogously E(Mb −Ma )2 = E&M'b −E&M'a = 0. From this, (b) and (c) follow.   The predictable quadratic variation describes the growth of the process. Like covariance, quadratic covariation describes the dependence behavior. The quadratic covariation process induces a measure d|&M, N'| := dV &M,N' ( d&M'. This measure is absolutely continuous with respect to the variation process &M' or also with respect to &N'. This continuity is a consequence of the following inequality. Proposition 3.61 Let M, N ∈ Mloc,c , H, K : ( × R+ , A ⊗ B+ ) −→ (R, B)  0

t

|Hs Ks | d|&M, N'|s ≤



t 0

Hs2 d&M's

1/2  t 1/2 · Ks2 d&N's . 0

Proof It suffices to consider the case where the integrands are bounded and ≥ 0. We first show the statement for elementary processes. Then the statement follows for bounded positive integrands with the monotone class theorem. To this end let K, H be processes of the form K = K0 1{0} + K1 1[0,t1 ] + · · · + Kn 1[tn−1 ,tn ] , H = H0 1{0} + H1 1[0,t1 ] + · · · + Hn 1[tn−1 ,tn ] with Hi , Ki ∈ L(A) and Hi , Ki bounded. With &M, N'ts := &M, N't − &M, N's is then for r ∈ Q: &M + rN, M + rN'ts = &M'ts +2r &M, N'ts +r 2 &N'ts ≥ 0.    =:c

=:b

=:a

Therefore, the above quadratic form is also positive for r ∈ R and as a consequence

3.3 Quadratic Variation of Continuous Local Martingales

67

the discriminant D = b2 − ac ≤ 0, i.e. |&M, N'ts | ≤ (&M'ts )1/2(&N'ts )1/2

a.s.

But from this, according to Cauchy–Schwarz,it follows.  t  t |Hs Ks | d|&M, N'|s ≤ |Hi Ki ||&M, N'ti+1 | i 0

i

≤



t

i

≤

t

|Hi ||Ki |(&M'ti+1 )1/2 (&N'ti+1 )1/2 i i



t

Hi2 &M'ti+1 i

i



0

t

Ki2 &N'ti+1 i

i

t

=

1/2  

1/2   Hs2 d&M's

0

t

1/2

1/2 Ks2 d&N's

In particular V &M,N' ( ds) is continuous with respect to the measure d&M's . The statement of the theorem now follows from monotone convergence.   The variation measure of the covariation is, therefore, continuous with respect to the variation measure of each component. The proposition is central to the extension of the stochastic integral to L2 -martingales. With the help of Hölders inequality it follows from Proposition 3.61 Corollary 3.62 (Kunita–Watanabe Inequality) Let be p, q ≥ 1, H , K, M, N be as in Proposition 3.61. Then it holds: 

∞

E 0

*  * |Hs Ks | d|&M, N'|s * *

0

∞

Hs2 d&M's

1/2 * *  * * * ·* * * p

∞ 0

1 p

+

1 q

= 1, and

Ks2 d&N's

1/2 * * * *

q

The Kunita–Watanabe inequality induces an important bilinear form on the set of L2 -bounded martingales. Definition 3.63 Let H2 := {M ∈ M; sup EMt2 < ∞} t

be the class of L2 -restricted martingale. Remark 3.64 By Doob’s convergence theorem, the following holds for M ∈ H2 : Mt −→ M∞ , a.s. and in L2 .

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3 Stochastic Integration

Moreover the closure theorem holds: If we add the limit to the martingale, the martingale property is preserved, (Mt )t ≤∞ ∈ M2 . In particular, the set (Mt )t ≤∞ is uniformly integrable and it holds Mt = E(M∞ | At ),

M∞ ∈ L2 .

The Brownian motion is not in H2 because it is not uniformly bounded in L2 , but only after localization. Two important norms on H2 are the following: Norms for the Class H2 ∗ := sup |M | ∈ L2 for M ∈ H2 . Let Define M∞ t t   ∗  = E sup M 2 1/2 the M∗2 -norm MM∗2 := M∞ 2 t t     1/2 1/2 2 MH2 := M∞  = EM∞ = lim EMt2 the H2 -norm ∗ . The H2 -norm, on the other The M∗2 -norm is the L2 -norm of the supremum M∞   1/2 hand, is the L2 -norm of M∞ , i.e. the limit of EMt2 . It holds according to the maximal inequality of Doob Doob

∗ MH2 ≤ M∞ 2 ≤ 2 · MH2 ,

i.e. the norms are equivalent. H2 provided with  · H2 is a Hilbert space. H2 with  · M∗2 it is not a Hilbert space. However, this norm is useful for approximation arguments. Proposition 3.65   (a) H2 ,  · H2 is a Hilbert space. (b) The mapping H2 −→ L2 ( , A∞ , P ), M −→ M∞ is an isometry. (c) Hc2 ⊂ H2 is a closed subspace. Proof (a), (b) M −→ M∞ is bijective, because one can specify the inverse directly using the representation Mt = E[M∞ | At ]. The isometry property follows from the definition, because the H2 -norm is defined via the L2 -norm of M∞ . Each Y ∈ L2 ( , A∞ , P ) generates an L2 -martingale Mt = E(Y | At ) such that Mt −→ Y . The L2 -bounded martingales can, therefore, be identified with the L2 -space.

3.3 Quadratic Variation of Continuous Local Martingales

69

(c) Closedness of the set of elements in H2 : We consider a sequence of continuous L2 -bounded martingales which converges in H2 : Let {M n } ⊂ Hc2 , M n −→ M in H2 . Then it follows: n − M∞ 22 −→ 0. E sup |Mtn − Mt |2 ≤ 4 · M n − M2H2 = 4 · M∞ t

n

Thus there exists a subsequence such that supt |Mt k − Mt | −→ 0 almost surely. Thus M ∈ Hc2 , because the limit of continuous functions under uniform convergence is continuous. The closure property for continuous L2 -bounded martingales is thus obtained using the equivalence of the two norms introduced above.   For continuous local martingales, the property to be an element in Hc2 can be written via the quadratic variation. Proposition 3.66 Let M ∈ Mloc,c . Then it holds: M ∈ Hc2 ⇐⇒ M0 ∈ L2

and

&M' ∈ A+ ,

&M' ∈ A+ means: the variation process is an integrable increasing process, i.e. E&M'∞ < ∞. Further it holds for M ∈ Hc2 : (a) M 2 − &M' is a uniformly integrable martingale (b) ∀ stopping times S ≤ T it holds: E(MT2 − MS2 | AS ) = E((MT − MS )2 | AS ) = E(&M'TS | AS ), i.e., the expected conditional quadratic increment is described by the conditional variation process between stopping times. Proof “⇒”: Let Tn ↑ ∞ be stopping times such that M Tn is a bounded martingale. Then, according to the optional sampling theorem,  E

M Tn

2 t

 − &M Tn 't

= EMT2n ∧t − E&M'Tn ∧t =

EM02

(3.8)

< ∞ so M0 ∈ L . 2

∗ ∈ L2 and E&M' Further M∞ Tn ∧t −→ E&M'∞ (monotone convergence). It follows that the right-hand side in (3.8) for n −→ ∞, t → ∞ converges to 2 − E&M' . EM∞ ∞

70

3 Stochastic Integration

2 − E&M' and thus E&M' < ∞. So it holds: 0 ≤ EM02 = EM∞ ∞ ∞ “⇐”: By (3.8) it is EMT2n ∧t ≤ E&M'∞ + EM02 =: K < ∞. It follows by the Lemma of Fatou:

EMt2 ≤ limEMT2n ∧t ≤ K. So {Mt } ⊂ L2 is bounded. In particular, therefore, L1 -convergence holds. To prove the martingale property: With (Tn ) as introduced above,it holds for s ≤ t: E(Mt ∧Tn | As ) = Ms∧Tn . From this it follows by means of uniform integrability E(Mt | As ) = Ms . 2 ≤ limEM 2 ≤ K < ∞ (M ) is a L2 -bounded martingale and Because EM∞ t t 2 thus M ∈ Hc . For the proof of (a) and (b): M 2 − &M' is a uniformly integrable martingale, because ∗ 2 sup |Mt2 − &M't | ≤ (M∞ ) + &M'∞ ∈ L1 as &M' ∈ A+ . t

Therefore, M 2 − &M' is uniformly majorized by a L1 -function and, therefore, uniformly integrable. By Theorem 3.53 is M 2 − &M' of the class (D), i.e., the set {Mτ2 − &M'τ ; τ finite stopping time} is uniformly integrable. Hence, by the optional sampling theorem, (b) it follows.   A consequence of the martingale property of M 2 − &M' is the following corollary: Corollary 3.67 Let M ∈ Hc2 , M0 = 0. Then it follows: 1/2  1/2 MH2 = &M'∞ 2 = E&M'∞ . 1/2

The Hilbert space norm on H2 , is identical to the L2 -norm of &M'∞ i.e. with 1/2  . E&M'∞ The following proposition gives an application of the variation process to the characterization of the convergence set of continuous local martingales. This is an extension of Doob’s convergence theorems to non L1 -bounded local martingales. Proposition 3.68 (Convergence Sets) Let M be a continuous local martingale M ∈ Mloc,c , then it holds: {&M'∞ < ∞} ⊂



lim Mt exists P −almost surely

t →∞

 almost surely.

3.4 Stochastic Integral of Continuous Local Martingales

71

Proof Let w.l.g. M0 = 0, and let τn := inf{t; &M't ≥ n}. Then &M τn ' = &M'τn ≤ 2  n. It follows that (M τn ) is bounded in L2 since M τn − &M τn ' is a martingale. Thus, according to the martingale convergence theorem, there exists limt →∞ Mtτn a.s. and in L2 . On {&M'∞ < ∞} it holds τn = ∞ for n ≥ n0 (ω). But this implies lim Mtτn = lim Mt ,

t →∞

t →∞

n ≥ n0 .  

On the set where the variation is finite at infinity, the martingale converges. Even the inverse holds, i.e. a.s. equality of the sets holds.

3.4

Stochastic Integral of Continuous Local Martingales

We construct the stochastic integral in a series of steps.

3.4.1

Stochastic Integral for Continuous L2 -Martingales

We define the stochastic integral first for the class M2c = {M ∈ Mc ; EMt2 < ∞, ∀ t} of continuous L2 martingales. Important examples for continuous L2 martingales can be found in the class of diffusion processes. These processes are of importance in stochastic analysis. They are the stochastic analogue of elliptic partial differential equations. First, as before we introduce the integral for elementary predictable processes.  Let f ∈ E, ft (ω) = U0 1{0} + m j =0 Uj · 1(sj ,sj+1 ] , 0 = s0 < s1 < · · · < sm , Uj ∈ B(Asj−1 ). For these integrands, we define the integral as for the Brownian motion: 

t

m−1 

f dM := U0 M0 +

0

  Uj Mstj+1 − Mstj =: (f · M)t .

j =0

By analogy with the stochastic integral for Brownian motion, and using predictable quadratic variation, we obtain the basic properties. Proposition 3.69 Let f , g ∈ E, M, N ∈ M2c , then it holds:  (a) (b)



 (f + g) dM = f d(M + N) =



 f dM + f dM +



g dM f dN

72

3 Stochastic Integration



t   (c) f · M ∈ and &f · M, N't = fs d&M, N's = f · &M, N' t 0  t 2  t   2 (d) E sup  f dM  ≤ 4 · E g d&M', ∀ T < ∞.

M2c

t ≤T

0

0

Proof (a), (b) analogous to Proposition 3.32 (c) The space of L2 -martingales M2c is a vector space. The mapping X → &X, N' is linear. So it suffices to prove (c) for the case f := U 1(s,u] (t), U ∈ B(As ). In this case:   (f · M)t = U Mu∧t − Ms∧t = U Mtu − Mts ∈ M2c . Next it is   &f · M, N't = &U M.u − M.s , N't   = U &M u , N't − &M s , N't   = U &M, N'ut − &M, N'st connection between quadratic variation and stopping times  = f · &M, N' t 

(d) follows from the maximal inequality of Doob.   The predictable covariation of (f · M) with another process N is given by the Lebesgue–Stieltjes integral of f with respect to the covariation process &M, N'. In particular it holds: &f · M, f · M' = &f · M' = f 2 · &M' For the extension of the stochastic integral, a measure-theoretic argument for the Doléans measure introduced in the following definition is important. Definition 3.70 (a) Let M be a continuous L2 -martingale, M ∈ M2c , then we define " L2 (M) := f ∈ L(P);



T

E 0

# f 2 d&M' < ∞,

∀T < ∞

3.4 Stochastic Integral of Continuous Local Martingales

73

(b) We define the restricted Doléans measure μT on the σ -algebra P in = [0, ∞) ×   μT (C) =

T



T

1C d&M't ⊗ P = E

0

1C (t, ω) d&M't .

0

(c) As Doléans measure we denote the measure μ at ( , P) defined by 

∞

μ(C) = E

1C (t, ω) d&M't . 0

It holds: L2 (M) =



L2 (μT , P).

T >0

μT is a finite measure on since μT ( ) = EMT2 < ∞, M ∈ M2c . For M ∈ Hc2 the Doléans measure μ is finite on and it holds that L2 (μ, P) ⊆ L2 (M). For L2 -martingales M, the restricted Doléans measures are finite measures. Because of the finiteness of the measures, the elementary predictable processes are dense in L2 ( , P, μT ), ∀ T > 0. Using this measure-theoretic denseness statement, the extension of the stochastic integral to integrands in L2 (M) is as follows: To g ∈ L2 (M) there exists a sequence of elementary processes (g N ) ∈ E such that: 



N

(g N − g)2 dμN = E

(g N − g)2 d&M' ≤ 2−2N .

0

For elementary integrands we had already introduced the stochastic integral. We now define  t YtN := g N dM ∈ M2c . 0

This sequence is a Cauchy sequence. As in Chap. 3.2 for the Brownian motion, there exists a limiting process Y ∈ M2c so that  2 E sup YtN − Yt −→ 0. t ≤T

For each further elementary process f ∈ E holds:  E sup |YtN t ≤T

T

− (f · M)t | ≤ 4 · E 2

0

|gsN − fs |2 d&M's .

74

3 Stochastic Integration

For N −→ ∞ it follows  2  t    f dM  ≤ 4E E sup Yt − t ≤T

0

T

 (gs − fs )2 d&M's = 4

(gs − fs )2 dμT .

0

So the limit Y is defined independently of the approximating sequence (g N ) and we get Theorem 3.71 Let M be a continuous L2 -martingale and let g ∈ L2 (M). Then the associated process Y = (Yt ) is called stochastic integral of g with respect to M 

t

Yt :=

g dM = (g · M)t .

0

The properties (a)–(d) from Proposition 3.69 hold true. This part of the definition process is as for Brownian motion. The following characterization of the stochastic integral using the covariation process gives an alternative and somewhat more abstract way to introduce the stochastic integral. This construction route is taken in Revuz and Yor (2005). Theorem 3.72 (Characterization of the Stochastic Integral) Let be M ∈ M2c and g ∈ L2 (M) then it holds: g · M is the is the uniquely determined element  ∈ M2c which solves the following system of equations: &, N' = g · &M, N',

∀ N ∈ M2c .

(3.9)

Proof By Theorem 3.71 it holds: &g · M, N' = g · &M, N',

∀ N ∈ M2c ,

i.e. the stochastic integral g · M satisfies the system of equations (3.9). To prove uniqueness, we assume that (3.9) holds for  ∈ M2c . Then it follows: & − g · M, N' = 0 [P ]-a.s. ∀ N ∈ M2c . If one chooses N specifically N :=  − g · M, then it follows: &N, N' = &N' = 0,

N0 = 0.

With Proposition 3.60 then follows N = 0, i.e.  = g · M.

 

3.4 Stochastic Integral of Continuous Local Martingales

75

Remark 3.73 (Stochastic Integral and Theorem of Riesz) (a) Equation (3.9) leads to a an alternative construction possibility for the stochastic 2 2 integral: For M ∈ Hc2 ⊂ M2c and  f2 ∈ L (μ) ⊂ L (M), μ the Doléans measure, we define on the Hilbert space Hc ,  · H2c the functional T

Hc2 −→ R,

  N −→ E f · &M, N' ∞ .

(3.10)

The mapping T is linear and continuous in the Hilbert space norm  · H2c . This follows from the Kunita–Watanabe inequality. The norm of the image can be estimated by the norm of the preimage. By the Theorem of Riesz it follows that: ∃! ∈ Hc2 so that   (, N)H2c = E∞ N∞ = E f · &M, N' ∞ = T (N).

(3.11)

We now define the stochastic integral: f · M as this solution  = f :  = f =: f · M. By Theorem 3.72 it remains to show that  is a solution of (3.9) in order to obtain as a consequence the equality of both constructions of the stochastic integral. Note that for all stopping times τ : E(f N)τ = EE(f | Aτ )Nτ = E(f )∞ Nτ τ = E(f )∞ N∞ = E(f · &M, N τ ')∞

= E(f · &M, N'τ )∞ = E(f · &M, N')τ . It follows that f N − f · &M, N' ∈ M. This implies: &f , N' = f · &M, N'.

(3.12)

Thus, according to (3.9) the stochastic integral as defined by Riesz’s representation theorem f is identical to the earlier introduced stochastic integral. (a) Stochastic Integral as an Isometry The stochastic integral for M ∈ Hc2 is an isometry of L2 (μ) to Hc2 , i.e. L2 (μ) −→ Hc2 ,

f −→ f = f · M is an isometry.

(3.13)

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3 Stochastic Integration

For by definition of the Doléans measure it holds:  f 2L2 (μ) = E

f 2 d&M' = E2∞ = 2H2

(3.14)

c

by the characterization in (3.9). Because of this isometry, the stochastic integral for Hc2 and integrands in L2 (μ) is most simply introduced as an isometric continuation of the stochastic integral for elementary integrands E ⊂ L2 (μ), since these lie dense in L2 (μ).

3.4.2

Extension to the Set of Continuous Local Martingales

In this section we extend the construction of the stochastic integral to the set of continuous local martingales. Analogous to the Brownian motion case, we denote by Mloc,c , the class of continuous local martingales. For M ∈ Mloc,c we introduce as class of integrands similar to Brownian motion the class L0 (M) as a class of integrands: " L (M) := f ∈ L(P);



T

0

0

t) +  ∈V



0

t

f (y) dP τn (y) .



For all T > 0 and t ≤ T it holds for n ≥ n0 , P (τn > t) > 0. The above representation then implies that f ∈ V.

∈V

 

We now treat some decomposition theorems for general semimartingales and describe their variation and covariation processes.

3.5.1

Decomposition Theorems

Let M ∈ M2loc with M0 = 0, i.e. there exists a localization τn ↑ ∞ such that M τn ∈ M2 is an L2 -martingale. Then (M τn )2 is a submartingale of class (DL). By the Doob–Meyer’s theorem, it follows: There exists exactly one predictable increasing process A ∈ V+ ∩ L(P) = (Vp )+ , A0 = 0, such that M 2 − A ∈ Mloc . (Definition on [0, τn ], consistent continuation on [0, ∞).) The process A =: &M' is called the predictable quadratic variation of M. Using the polarization formula, we can then introduce for M, N ∈ M2loc the predictable quadratic covariation of M, N &N, M' ∈ L(P) and characterize it by the property MN − &M, N' ∈ Mloc .

(3.20)

Since Mloc,c ⊂ M2loc this generalizes the predictable variation process of continuous local martingales.

3.5 Integration of Semimartingales

85

As in the continuous case, we get an inequality for N, M ∈ M2loc given by:  1/2 |&M, N't | ≤ &M't &N't

a.s.

∀ t,

i.e., the root of the product of the variations is a majorant for the absolute value of the covariation. The reason that the square-integrable processes play a special role is the following decomposition theorem: Proposition 3.87 (Special Decomposition Theorem) Let be X ∈ S, then there exists a decomposition of X of the form X = X0 + M + A

with M ∈ M2loc and A ∈ V.

The above decomposition is for A ∈ V unique. The argument for the above decomposition is to transfer the large jumps of M in a decomposition to the process A without losing the finite variation. Applying this special decomposition theorem to a local martingale, we obtain as corollary Corollary 3.88 Let M ∈ Mloc , M0 = 0, then there exists a process M  ∈ M2loc and A ∈ V with M = M  + A. As a consequence of this, one can restrict the definition of the stochastic integral for semimartingales to the case where the martingale component lies in M2loc . A useful property of predictable increasing processes (as in the definition of special semimartingales) is formulated by the following lemma. Lemma 3.89 Let A be a predictable process A ∈ Vp of finite variation, A0 = 0, then A ∈ Aloc , A has locally integrable variation, i.e., there exists a localization (τn ) such that Aτn has integrable variation ∀ n ∈ N. As a corollary of this, we have the following integrability statement. Lemma 3.90 Let f ∈ L(P) be a predictable  t process, and let A ∈ A+ ∩ L(P) be predictable and increasing, and further let 0 |fs | dAs < ∞, ∀ t a.s. Then it holds: 

t

Bt :=

fs dAs ∈ L(P) ∩ Aloc ,

0

i.e., the integral B = f · A is predictable and locally integrable.

86

3 Stochastic Integration

$ t % Proof Let f ∈ E and let H := X càdlàg; 0 f dX ∈ L(P) , then H ⊂ {continuous processes}, because f · X is continuous. For all càdlàg processes f the integral f · X is defined. By the monotone class theorem, it follows: H contains all constrained predictable processes. Then by localization it follows: B = f · A ∈ L(P) ∩ Aloc ,

∀ A ∈ V+ ∩ L(P), f ∈ E.

From this, again using the monotone class theorem, the assertion for f ∈ L(P), f bounded, follows. For general predictable functions, the statement is reduced to the bounded case by localization: Let f k := (f ∧k)∨(−k) then the sequence f k converges pointwise: f k −→ f . By assumption the sequence |f k | is majorized by an integrable function: 

t

|f | ≤ |f | and k

|f |s dAs < ∞.

0

The assertion now follows with the theorem of majorized (dominated) convergence.  

3.5.2

Stochastic Integral for M2loc

The stochastic integral is now introduced in the next step in a similar way as for M2c .   t Let M ∈ M2loc , and L0 (M) := f ∈ L(P); 0 fs2 d&M's < ∞, a.s. ∀ t < ∞ . t Then, by Lemma 3.90 Bt := 0 fs2 d&M's ∈ P ∩ Aloc , for f ∈ L0 (M). So there exists a localizing sequence σn ↑ ∞ : EBtσn < ∞. t For M ∈ M2 the stochastic integral Xt := 0 f dM is defined as for martingales 2 M ∈ Mc . Then one extends the class of integrators to M2loc by localization as before:  Let (σn ) be a localizing sequence such that M σn ∈ M2 and let τn := σn ∧ σn . We define on [0, τn ] the stochastic integral by 

t 0



t

f dM := 0

f 1[0,τn ] dM τn on [0, τn ].

To identify the predictable quadratic covariation, we use the Doob–Meyer decomposition:

3.5 Integration of Semimartingales

87

Let N ∈ Mloc ∩ P, N0 = 0 and N ∈ V, then P (Nt = 0, ∀ t) = 1 because of the uniqueness in Doob–Meyer’s decomposition. This implies the uniqueness of &X, N'. We get X = f · M ∈ M2loc and the integral properties hold (cf. Proposition 3.69) as well as the theorem on dominated convergence for the stochastic integral. In particular, the covariation formula also holds.

3.5.3

Stochastic Integral for Semimartingales

Let X ∈ S be a semimartingale. Then according to the decomposition theorem one can decompose X: X = M + A,

M ∈ M2loc , A ∈ V.

This decomposition is not unique. Our goal is to construct an integral that is independent of the decomposition. Suppose M, N ∈ M2loc , A, B ∈ V

X = M + A = N + B,

are two decompositions of X. For an elementary functions f ∈ E then, according to the definition of the integral 

t 0



t

f dM +



t

f dA =

0



t

f dN +

0

f dB

(3.21)

0

The set of f satisfying this condition is closed with respect to pointwise convergence. By the monotone class theorem and the dominated convergence theorem, it follows: Equation (3.21) holds also for bounded predictable processes f ∈ P. With a truncation argument, it follows: (3.21) holds for f ∈ P under the conditions  t  t fs2 d&M's < ∞ a.s., |fs | d|A|s < ∞ a.s.,  0t 0 t (3.22) fs2 d&N's < ∞ a.s., |f |s d|B|s < ∞ a.s. ∀ t, and 0

0

i.e. equality holds in the class of all f with (3.22).

88

3 Stochastic Integration

We, therefore, define the integration class L0 (X) := {f ∈ L(P); ∃ decomposition X = M + A,

 with M ∈ M2loc , A ∈ V, such that (3.22) holds .

For this class the definition of the integral is unambiguous and the integral properties hold as in Chap. 3.4.2, since essentially only the theorem of majorized convergence is used. The statement on the quadratic variation for continuous semimartingales carries over as follows: Proposition 3.91 (Quadratic Variation) Let be X ∈ S, be a càdlàg semimartingale and let (τin ) be a sequence of decompositions with |n = | −→ 0. Then Qnt :=

 2 P n ∧t − Xτ n ∧t Xτi+1 −→ [X]t i

and the representation formula holds  Xt2

=

X02

t

+2· 0

Xs− dXs + [X]t .

(3.23)

The convergence is uniform on [0, t ]. The proof of this proposition is similar to the proof in Proposition 3.78 using the theorem of dominated convergence. The essential step is the proof of the following Proposition 3.92 Let X ∈ S be càdlàg, Y càdlàg, and let (τin ) be a sequence of n n ∧t − Xτ n ∧t ). Then decompositions with n −→ 0. Let further Zt := Yτin (Xτi+1 i it holds    t  n  P  sup Zt − Y− dX −→ 0. t ≤T

0

With the quadratic variation, we also obtain the quadratic covariation by polarization from quadratic variation. Definition 3.93 (Quadratic Covariation) For X, Y ∈ S we define the quadratic covariation [X, Y ] :=

 1 [X + Y ] − [X − Y ] 4

The partial integration formula also carries over.

3.5 Integration of Semimartingales

89

Proposition 3.94 (Partial Integration Formula) For X, Y ∈ S it holds: 

t

Xt Yt = X0 Y0 + 0

 X− dY +

t 0

Y− dX + [X, Y ]t .

In particular, Proposition 3.94 yields in the case of deterministic functions of finite variation, the classical partial integration formula as a special case (cf. Proposition 3.81). Proposition 3.95 Let M, N ∈ Mloc , and let f ∈ L(P) be locally bounded, then it holds:  t (a) Yt := f dM = (f · M)t ∈ Mloc 0

(b) MN − [M, N] ∈ Mloc . Proof (1) From the decomposition theorem, it follows that there exists a decomposition M = M ∗ + A with M ∗ ∈ M2loc and A ∈ V. A = M − M ∗ is as the difference of two local martingales itself a local martingale, A ∈ Mloc . By construction, the stochastic integral is f · M ∗ ∈ M2loc . It remains to show: f · A ∈ Mloc . 2 Let τn1 := inf{t; |A|t ≥ n} and (τn2 ) be a localizing sequence with Aτn ∈ M and define τn := inf{τn1 , τn2 }. Then it holds for t < τn that |A|t ≤ n. Thus it follows |A|τn ≤ 2n + |Aτn | and |A|τn ∈ A. Let g = f 1[0,τn ] , B = Aτn then (f · A)τn = g · B and |B| ∈ A. Define H := {h ∈ L(P); h bounded, h · B ∈ M}, then it holds: H ⊃ E, H is closed with respect to bounded pointwise convergence; so H is a monotone class. Thus it follows: H ⊃ B(P) and, therefore, f · A ∈ Mloc . (2) follows from the characterization property a) and Proposition 3.91.

 

Remark 3.96 For M, N ∈ M2loc it holds MN − &M, N' ∈ Mloc , &M, N' ∈ Vp is of finite variation and predictable. The quadratic covariation [M, N] is in general not predictable. By Proposition 3.95 it follows [M, N] − &M, N' ∈ Mloc

90

3 Stochastic Integration

i.e. &M, N', the predictable covariation from M and N, is the predictable quadratic variation of the covariation process [M, N]. The predictable variation process of the covariation is identical to the predictable covariation of M and N. We now describe some properties of the jumps of semimartingales. Let (X)t := Xt − Xt − be the jump process at t. If the process X is càdlàg, then there are at most countably many jumps. Proposition 3.97 Let X ∈ S, Y ∈ P be locally bounded, Z := Y · X, then it holds: (a) The jump process of Z is the integral Y · X, Z = Y · X (b)   [X] = (X)2 (c) 0≤s≤t (Xs )2 ≤ [X]t . Proof For the case that Y = Y− is left-continuous: (a) Let Y = Y− and Ytn := Ztn :=

 

n ] (t), Yτin 1(τin ,τi+1   Yτin Xτt n − Xτt n the integral of the elementary process i+1

i

Ytn w.r.t. X. Then, because of the left-continuity of Y P

Ytn −→ Yt . By the dominated convergence theorem for semimartingales it follows analogously to Proposition 3.78:    t   P sup Ztn − Y dX −→ 0.

t ≤T

0

It holds Z n −→ Z and by the dominated convergence theorem Y n · X −→ Y · X because of the uniform convergence on compact intervals.

3.5 Integration of Semimartingales

91

Because of Z n = Y n · X, Z = Y · X, therefore, it follows Z = Y · X. (b) Using the representation formula from Proposition 3.91 for quadratic variation, it follows:  t 2 2 Xt = X0 + 2 · X− dX + [X]t . 0

If we form the increment on both sides, then we get from (a): X2 = 2X− · X +  [X]. On the other hand it holds: (X)2t = (Xt − Xt − )2 = Xt2 − Xt2− − 2 · Xt − (Xt − Xt − )(X2 )t − 2Xt − (X)t . Thus it follows  [X] = (X)2 . (c) follows by definition of [X] and according to b), because the continuous component also contributes to the quadratic variation.   Remark 3.98 The sum of the quadratic jumps of X is smaller than the variation process at t. In particular, this sum is finite. Typically, equality does not hold. In addition to the part of the quadratic variation that comes from the jumps, there is also a part that comes from the continuity regions of the process. The quadratic variation grows due to the jumps, but also due to the continuity regions. A useful computational rule for quadratic covariation follows: Lemma 3.99 Let X ∈ S, A ∈ V, i.e. X is a semimartingale and A is of finite variation, then (a) If A or X is continuous, then [X, A] = 0. (b) The covariation of X and A is the sum of the products of the jumps of the increments  (X)s (A)s . [X, A] = 0 0, i.e. f is constant in each component ξi on [0, T ]. Then the stochastic exponential is given by " #   1 if Et = exp i&ξ, Xt ∧T ' + |ξ |2 (t ∧ T ) ∈ Mc . 2 It follows with the martingale property from (3).   1 1 2 2 E ei&ξ,Xt '+ 2 |ξ | t | As = ei&ξ,Xs '+ 2 |ξ | ·s ,

0 ≤ s ≤ t ≤ T.

(4.9)

This implies   1 2 E ei&ξ,Xt −Xs ' | As = e− 2 |ξ | (t −s). So the conditional expectation is independent of As and, therefore, coincides with the expected value.   1 2 E ei&ξ,Xt −Xs ' | As = e− 2 |ξ | (t −s)   = E E ei&ξ,Xt −Xs ' | As  = E ei&ξ,Xt −Xs ' ) = ϕXt −Xs (ξ ). But this is the characteristic function of the normal distribution. So the increments are Xt −Xs are independent of As and the increments of X are independent

114

4 Elements of Stochastic Analysis

and normally distributed. But these are exactly the defining properties of d

Brownian motion: Xt − Xs is independent of As and Xt − Xs = N(0, t − s). Thus X is a Brownian motion.   Because of the special importance we formulate the theorem for d = 1 separately. Corollary 4.20 Let d = 1 and B = (Bt ) be a continuous process with B0 = 0, then it follows: B is a Brownian motion ⇐⇒ B ∈ Mloc,c and &B't = t, ⇐⇒ (Bt ),

∀t

(Bt2 − t) ∈ Mloc,c .

Thus a Brownian motion is the unique continuous local martingale such that &B't = t. For the class of jump processes, this property (local martingale and &N't = t) characterizes the compensated Poisson processes. Thus the Poisson process and Brownian motion occupy a prominent position as prototypes of these classes of processes. A fundamental problem concerning Brownian motion B and more general classes of processes is the description of more general L2 -functionals Y = F (B) ∈ L2 or equivalently of functionals F on the infinite-dimensional Wiener space, (C, W ), W the Wiener measure. In financial mathematics, associated completeness theorems are fundamental. One way to describe such functionals is based on the construction of a basis using Hermitian polynomials, another way of description is based on stochastic integrals. Here we first treat the method of Hermitian polynomials. The Hermitian polynomials hn are defined as normalized solutions of the equations  x2  x2 dn = (−1)n hn (x)e− 2 , exp − n dx 2

hn (0) = 1.

hn is called a Hermitian polynomial of degree n. The Hermitian polynomials are x2

orthogonal polynomials to the weight function e− 2 , i.e., they are closely coupled to the normal distribution density: 

x2

hn (x)hm (x)e− 2 dx = 0,

n = m.

4.1 Itô Formula

115

The functions (hn ) form an orthogonal basis in the space L2 (exp(−x 2 /2) dx), i.e. in the L2 -space of the standard normal distribution with the weight function exp(−x 2 /2). One can also introduce the Hermitian polynomials in a related way.  u2  hn (x) = exp ux − . n! 2

 un n≥0

(4.10)

Expanding and rearranging the exponential function yields the above representation. The hn (x) then occur as weights of un , i.e., (4.10) provides a representation of the generating function of the polynomial sequence (hn ). Coded in the exponential function are the hn (x) as coefficients in x and on the other hand all hn (x) in the expansion by the second variable u. One can normalize this representation in the following way with another parameter a:  √  x 2 (u√a)2   au2  = exp u a √ − exp ux − 2 2 a   n  x u (4.10) · a n/2 · hn √ , Hn (x, 0) := x n . = n! a n≥0



(4.11)

=:Hn (x,a)

The Hn retain the orthogonality properties of the Hermitian polynomials. With respect to the variables x they are polynomials of degree n in x. The first standardized Hermitian polynomials Hn = Hn (x, a) are n 0 1

2

3

4

5

Hn 1 x x 2 − a x 3 − 3ax x 4 − 6ax 2 + 3a 2 x 5 − 10ax 3 + 15a 2 x

The space of square-integrable functions of the Brownian motion can be described in terms of the standardized Hermitian polynomials. These form a basis in the L2 -space when one inserts the Brownian motion and its quadratic variation. Moreover, they allow the construction of a large class of continuous local martingales as functionals of given local martingales M and their predictable quadratic variations &M'. Proposition 4.21 Let M ∈ Mloc,c , M0 = 0, then it follows (n)

Lt

  := Hn Mt , &M't ∈ Mloc,c

116 (n)

Lt

4 Elements of Stochastic Analysis

has a representation as a multiple stochastic integral L(n) t = n!





t

dMs1

0



s1

dMs2 · · ·

0

sn−1 0

· · · dMsn

 .

The stochastic exponential of M has the expansion E (M)t = λ

∞ n  λ n=0

n!

(n)

Lt .

Proof The exponential martingale E λ (M) is a continuous local martingale and it is a solution of the stochastic differential equation (cf. Corollary 4.8): 

t

E (M)t = 1 + λ λ

E λ (M)s dMs .

0

  With E λ (M)s = exp λMs − 12 λ2 &M's it follows by (4.11) with u = λ, a = &M's , x = Ms . E λ (M)t = 1 + λ

 t ∞ λn (n) L dMs . n! s 0 n=0

Now let τ be a stopping time such that M τ , &M'τ are bounded. Then it follows by the theorem of dominated convergence that the integral and the sum can be interchanged and thus it holds E λ (M)t = 1 +

 ∞  λn+1 n=0

n!

t 0

L(n) s dMs =

∞  λn n=0

n!

L(n) s .

By comparison of coefficients we get (0)

L

≡ 1,

L(n+1) t

 = (n + 1) 0

t

L(n) s dMs .

So L(n) ∈ Mloc,c . Thus the normalized Hermitian polynomials define a system of local martingales.  

4.2 Martingale Representation

4.2

117

Martingale Representation

Let B be a Brownian motion with Brownian filtration (At ) = (AB t ), i.e., the completed right continuous filtration generated by the Brownian motion. The goal of this section is to represent martingales with respect to this filtration as stochastic integrals 

t

Mt = M0 +

Hs dBs . 0

It will be shown below that any martingale admits such a representation. In financial mathematics, the integrand H from such a representation can be interpreted as a trading strategy. The proof of the above integral representation is based on the special class of integrands, # " n   := f = λj 1(tj−1 ,tj ] , j =1

of the set of staircase functions with a bounded support. Let 

f

Mt := (f · B)t =

t

f (s) dBs ∈ M

0

be the corresponding stochastic integral with respect to the Brownian motion. The integrands f ∈  are deterministic and predictable. The predictable variation of M f is  t &M f 't = f 2 (s) ds. 0

Let E f be the associated stochastic exponential generated by M f   1 E f := exp M f − &M f ' . 2 A preliminary remark from Hilbert space theory follows: A subset A of a Hilbert space H is called total in H , ⇐⇒ ∀ h ∈ H :

h ⊥ A ⇒ h = 0.

118

4 Elements of Stochastic Analysis

Total subsets are suitable for approximating elements of a Hilbert space. It holds A ⊂ H is total ⇐⇒ the linear hull of A, linA, is dense in H. This equivalence follows from an application of Hahn–Banach’s theorem. The following lemma shows that the set of stochastic exponentials of staircase functions is total in L2 (A∞ , P ). Lemma 4.22 Let K := staircase functions. Then

  f E∞ ; f ∈  be the set of stochastic exponentials of K is total in L2 (A∞ , P ),

equivalently, linK is dense in L2 (A∞ , P ). Proof Let Y ∈ L2 (A∞ , P ) and let Y ⊥ K, then it is to show that: Y ≡ 0. The assertion is equivalent to Y · P |A∞ = 0, i.e., the measure with density Y with respect to P restricted to A∞ (a signed measure) is the zero measure. It suffices to show that the measure is the zero measure on a generator of A∞ . We define for all n-tuples of complex numbers z = (z1 , . . . , zn ) ∈ Cn #   " n   ϕ(z) := E exp zj Btj − Btj−1 Y . j =1

ϕ is analytic, that is, complex differentiable in each variable. One can easily specify a majorant here and swap integral and expectation value (similar to exponential families). For real arguments λj holds ϕ(λ1 , . . . , λn ) = E exp

" n

#   λj Btj − Btj−1 Y = 0,

j =1

since Y ⊥ K and with λj = 0 it follows EY = 0. On the real axis, i.e., in Rn the function ϕ = 0. Then, using the uniqueness theorem for analytic functions, it follows that ϕ = 0 on Cn . Specifically, it follows from this # "    λj Btj − Btj−1 Y. 0 = ϕ(iλ1 , . . . , iλn ) = E exp i

4.2 Martingale Representation

119

This can be written as the integral of the e-function with respect to the measure with density Y with respect to P . Then, by the uniqueness theorem for characteristic functions, it follows that the image measure of the random variable under the measure Y · P is the zero measure, (Y · P )(Bt1 , . . . , Btn − Btn−1 ) = 0. The successive differences of Bti produce the same σ algebra as Bt1 , . . . , Btn . So it follows Y · P |σ (Bt , . . . , Bt ) = 0 n 1 and thus Y · P |A∞ = 0.

 

Lemma 4.22 is the key to the following completeness theorem. Theorem 4.23 (Completeness Theorem) Let F ∈ L2 (AB ∞ , P ), be a squareintegrable Brownian motion functional. Then there exists exactly one integrand H ∈ L2 (μ) = L2 (B, μ), μ the Doléans measure on P, such that F can be represented as a stochastic integral 

∞

F = EF +

Hs dBs . 0

Any L2 -functional can be represented uniquely as a stochastic integral. Proof Let F be the set of functionals for which such a representation exists, "  F := F ∈ L2 (A∞ , P ); ∃H ∈ L2 (μ), F = EF +

# Hs dBs ⊂ L2 (A∞ , P ).

∞ 0

  F is a linear subspace of L2 A∞ , P (1) F is closed. ∞ To the proof let F ∈ F . Then it holds with E 0 Hs dBs = 0 and 

2

∞

Hs dBs

E

 =E

0 2

 2

EF = (EF ) + E 0

0 ∞

∞

Hs2 ds

(isometry property) (4.12)

 Hs2 ds

2

= (EF ) +

2

H dμ.

120

4 Elements of Stochastic Analysis

% $ ∞ Let F n = EF n + 0 Hsn dBs be a Cauchy sequence in F with respect to L2 convergence L2 (A∞ , P ). Then it follows in particular: (EF n ) is also a Cauchy sequence, because |EF n − EF m | ≤ F n − F m 2 , i.e., also the first moments form a Cauchy sequence and (   is a Cauchy sequence in L2 A∞ , P . The mappings   −→ L2 A∞ , P L2 (μ) −→ H2 H −→ H · B −→ (H · B)∞

∞ 0

H n dB) = (H · B)

are isometries.

  So H n ⊂ L2 (μ) a Cauchy sequence in the complete space L2 (μ). Therefore, there exists a limit H ∈ L2 (μ) such that H n −→ H ∈ L2 (μ). Because of the isometry property, convergence follows in L2 (A∞ , P ) 

∞

F −→ (lim EF ) + n

n

H dB. 0

Thus every Cauchy sequence in F has a limit in F , i.e. F is closed.   f (2) K = E∞ ; f ∈  ⊂ F . By the Itô formula it holds for f ∈  and M f := f · B 

f

Et = E(M f )t = 1 + 

t

f

f

Es dMs

0 t

=1+

f Es f (s) dBs

0 f



t

=1+

Hs dBs 0

with Hs := Es f (s). H ∈ L2 (μ), because f is a bounded staircase function. f Therefore, E f has a representation as a stochastic integral, i.e. E∞ ∈ F . As a corollary to (2), we now get (3) F ⊃ lin K and F is closed. According to Lemma 4.22 it thus follows F = L2 (A∞ , P ).

4.2 Martingale Representation

121

(4) Uniqueness ∞ ∞ Let H, G ∈ L2 (μ) and F = EF + 0 Hs dBs = EF + 0 Gs dBs be two ∞ representations of F . Then it follows 0 = 0 (Hs − Gs ) dBs . So it also follows ∞ 0 = E 0 (Hs − Gs ) dBs and thus 

∞

0=E

2 (Hs − Gs ) dBs



∞

=E

0

(Hs − Gs )2 ds = H − G2

0

the norm with respect to the Doléans measure. From this follows H = G [μ].

 

Remark 4.24 For F ∈ L2 (μ∞ , P ) thus there exists an integral representation with ∞ an integrand H ∈ L2 (μ, P) = L2 (B) := {f ∈ L(P); E 0 fs2 ds < ∞}. Analogously, for square-integrable functionals of (Bs ; s ≤ T ) representations with integrands H ∈ L2 (μT , P) are obtained. As a consequence, we now have the following representation theorem for martingales. Theorem 4.25 (Martingale Representation Theorem) Let M ∈ H2 = H2 (AB , P ) (or M ∈ M2 ). Then there exists exactly one H ∈ L2 (μ) (or H ∈ L2 (B)), such that 

t

Mt = M0 +

Hs dBs ,

∀ t ≤ ∞ (∀ t < ∞).

0

Remark 4.26 In particular, every martingale in H2 (resp. M2 ) has a continuous version. The integral representation is such a continuous version. Proof Let M ∈ H2 , then M∞ ∈ L2 (AB ∞ , P ). According  ∞ to Theorem 4.23 there exists exactly one H ∈ L2 (μ) such that M∞ = M0 + 0 Hs dBs and M0 = EM0 is constant. But it follows from this Mt = E(M∞ | At )  ∞  = M0 + E Hs dBs | At 0



t

= M0 + 0



∞

Hs dBs + E t

 Hs dBs | At .

122

E

4 Elements of Stochastic Analysis



∞ t Hs

dBs | At



= 0 since the Brownian motion has independent increments

is predictable. To prove this, first consider the class E and then and H ∈ apply the monotone class theorem. For M ∈ M2 we obtain the above integral representation on [0, Tn ], Tn ↑ ∞. This then can be consistently extended to [0, ∞).   L2 (B)

The stochastic integral representation can now be extended to the class Mloc . Theorem 4.27 Let M ∈ Mloc = Mloc (AB ), then there exists a process H ∈ L2loc (B) such that 

t

Mt = M0 +

Hs dBs .

(4.13)

0

In particular Mloc (AB ) = Mloc,c (AB ), i.e., any local martingale with respect to the Brownian filtration is continuous. Proof (1) For M ∈ H2 the above representation follows by Theorem 4.25. (2) If M is uniformly integrable, then M∞ ∈ L1 . L2 (A∞ ) ⊂ L1 (A∞ ) is dense in L1 (A∞ ). So there exists a sequence (M n ) ⊂ 2 n | −→ 0. H such that E|M∞ − M∞ From this, using Doob’s maximal inequality, it follows   1 n P sup |Mt − Mtn | > λ ≤ E|M∞ − M∞ | −→ 0. λ t Therefore, according to Borel–Cantelli, there exists a P -almost surely convergent subsequence (nk ) such that M nk −→ M almost surely, uniformly in t. It follows that M has a continuous version. (3) M ∈ Mloc . Then there exists a continuous version of M (with suitable stopping after (2). Therefore, there further exists a sequence of stopping times τn ↑ ∞ such that M τn is bounded and L2 is bounded. M τn is in H2 and, therefore, has a stochastic integral representation Mtτn = cn +



t 0

Hsn dBs .

4.2 Martingale Representation

123

But it follows cn = c, and Htn+1 = Htn on [0, τn ]. We define Ht := Htn at [0, τn ]; then  Hs ∈ L2loc (B)

and

t

M =c+

Hs dBs . 0

  The stochastic integral representation is also valid for the d-dimensional Brownian motion. Theorem 4.28 Let B be a d-dimensional Brownian motion and let M ∈ Mloc (AB ). Then it follows: d  t  ∃ H i ∈ L2loc (B), 1 ≤ i ≤ d, ∃ c ∈ R1 such that Mt = c + Hsi dBsi . i=1

0

Proof The proof uses an analogous argument as in d = 1 with the multivariate stochastic exponential.   Remark 4.29 How is the integrand of the stochastic integral representation obtained? (1) Assuming that  t the predictable covariation is Lebesgue continuous, i.e., let &M, B i 't = 0 Hsi ds. Because of &B i , B j 's = δij · s it follows according to the Radon–Nikodým theorem Hti =

d&M, B i 't d1

is equal to the Radon–Nikodým derivation. The Clark–Ocone formula (cf. Sect. 4.3.2) provides such a representation formula under more general conditions. (2) An example of an explicit form for the integrand H : If f is harmonic, i.e. f = 0, then Mt := f (Bt ) is a martingale. From the Itô formula it follows the explicit integral representation formula 

t

Mt = f (Bt ) = M0 + 0

∇f (Bs ) dBs .

124

4 Elements of Stochastic Analysis

(3) Non-uniqueness of representation for M ∈ Mloc : For M ∈ Mloc as in Theorem 4.28, the integral representation is in general not unique. Example Let 0 < a < T and τ := inf{t ≥ 0;

t

1 a T −u

Then it follows from the strong oscillation of T : P (τ < T ) = 1. Let ψ(ω, s) :=

⎧ ⎨1 ⎩

1 T −s

0

t

dBu = −Ba }.

1 a T −u

dBu in the vicinity of

0 ≤ s < a, a ≤ s < τ, τ ≤ s ≤ T.

Then ψ ∈ L2loc and 

a

 ψ(ω, s) dBs = Ba = −

(ω, s) dBs .

0

Thus ψ ∈ L2loc (B) and is a local martingale.

T

a

T 0

ψ(ω, s) dBs = 0 but ψ ≡ 0 and M = ψ · B ∈ Mloc

Alternative Representation of L2 (AB ∞ ) by Multiple Stochastic Integrals Let dimension d = 1. Let n = {(s1 , . . . , sn ) ∈ Rn+ ; s1 > s2 > · · · > sn } and let L2 (n ) = L2 (n , λ\n+ |n ) be the corresponding L2 -space. We consider the subspace En of L2 (n ) defined by " # . En := f ∈ L2 (n ); f (s) = ni=1 fi (si ), s = (s1 , . . . , sn ), fi ∈ L2 (R+ ) . 2 En ⊂ L2 ( . n ) is total in L (n ). For f = i fi ∈ En define the multiple stochastic integral





∞

In f := 0

f1 (s1 ) dBs1

s1 0

 f2 (s2 ) dBs2 · · ·



sn−1 0

fn (sn ) dBsn ∈ L2 (A∞ )



∈Asn−1

(cf.: Hermitian functionals in Proposition 4.21 for indicator functions f ). Contrary to the notation used so far, here the integrands are to the right of the integrator. Assertion: In is norm-preserving, In f L2 (A∞ ) = f L2 (n ) ,

f ∈ En .

4.2 Martingale Representation

Proof Let n = 2, f =

125

.2

i=1 fi

∈ E2 , then

 I2 f L2 (A∞ ) = E

∞



f1 (s1 ) 0

∞

=E 0 ∞

=

2

f2 (s2 ) dBs2 dBs1



=:Hs

1

 

0





s1

Hs2 ds1 1





ds1 f1 (s1 )

s1

2 0

0

 f22 (s2 ) ds2 = f 2L2 ( ) . 2

 

Analogously for n > 2 with induction. The closure of the linear hull of multiple stochastic integrals Cn := clin{In (En )} ⊂ L2 (A∞ ) is called n-th Wiener chaos.

(1) The mapping In : En → L2 (A∞ , P ) is linear and continuous. There exists a unique continuous and linear extension In : L2 (n ) → Cn . The extension In : L2 (n ) → Cn is an isometry. In particular Cn = In (L2 (n )); therefore, it holds Cn ∼ = L2 (n ). Alterna2 tively, for f ∈ L (n ) we can directly define the multiple integral  In (f ) =

∞



sn−1

···

f (s1 , . . . , sn ) dBs1 . . . dBsn .

0

0

(2) Cn ⊥ Cm , m = n. We first show: C1 ⊥ C2 . For this consider 

 

∞

E 0

f1 (s1 ) dBs1

 ∈C1



=E

0

= 0

g1 (s1 ) dBs1



 ∞ f1 (s1 )g1 (s1 )

0 ∞



 

∞

0

 f1 (s1 )g1 (s1 ) E

0

0

∈C2

s1

s1



s1

g2 (s2 ) dBs2

 g2 (s2 ) dBs2 dBs1  g2 (s2 ) dBs2 dBs1 = 0.

 =0

The general case follows by induction with a similar argument.

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4 Elements of Stochastic Analysis

Remark 4.30 The elements of C1 = c lin{Bt ; 0 ≤ t} ⊂ L2 (A∞ ) are Gaussian random variables. C2 is the orthogonal complement of C1 in the set of quadratic functions. Theorem 4.31 (Wiener Chaos Representation) The space of L2 -functionals of the Brownian motion has a representation as a direct topological sum. L2 (AB ∞) =

∞ 5

C0 / R1 ,

Cn ,

n=0 n 2 i.e. ∀ Y ∈ L2 (AB ∞ ) there exists a sequence f ∈ L (n ), n ∈ N, with

Y =

∞ 

in L2 (P ).

In (f n )

(4.14)

n=0

Proof According to Proposition 4.21 it holds f

E∞ = E∞ (M f ) =  =f ·B

∞ 

In (f n ),

n=0

.  with f n (s1 , . . . , sn ) = ni=1 f (si ), f = i αi 1(si−1 ,si ) . In general, for f with bounded compact support and M f = f · B: E(M )∞ f

 t  sn−1  s1 ∞  1 (n) Lt = = f (s1 ) dBs1 f (s2 ) dBs2 · · · f (sn ) dBsn . 0 0 0 n!  n=0

=In (f n )

It follows: f The representation in (4.14) holds for lin({E∞ ; f has a bounded, compact support}). f According to Lemma 4.22 it follows that lin({E∞ ; f bounded, compact sup2 B port}) ⊂ L (A∞ ) is dense. So there exists to Y ∈ L2 (A∞ , P ) a sequence f (Yn ) ⊂ lin({E ∞ }) with Yn → Y in L2 ,

and Yn =

∞  m=0

L2

Im (f n ) −→Y. m ∈Cm

Ym is for every m an orthogonal series. It follows that (Im (fmn ))n is for each m a Cauchy sequence, and hence has a limit: Im (fmn ) → Gm ∈ Cm .

4.2 Martingale Representation

127

Since Im is an isometry, it follows that also the sequence (fmn )n ) converges, fmn → fm in L2 (m ). It follows Gm = Im (fm ), and thus the representation of Y : Y =

∞ 

Im (fm ).

m=0

  The basic martingale representation theorem states that the Brownian motion has the representation property for local martingales, i.e., one can represent martingales with respect to Brownian filtration AB as an integral with respect to the Brownian motion. In general, this representation property does not hold for martingales X ∈ M2 or H2 with respect to the filtration AX generated by those martingales. Let for X ∈ H2 " #  ∞ 2 2 L (X) := f ∈ L(P); E fs d&X's < ∞ = L2 (μ) 0

be the set of L2 -integrands on [0, ∞). The Kunita–Watanabe decomposition identifies for a general functional F ∈ L2 (A∞ , P ) a uniquely representable part. Theorem 4.32 (Kunita–Watanabe Decomposition) Let be X = (Xt ) ∈ H2 , F ∈ L2 (A∞ , P ), A∞ = AX ∞ , then it follows: There exist unique processes (Hs ) ∈ L2 (μ) and L ∈ L2 (A∞ , P ) such that 

∞

(1) F = EF + Hs dXs + L   ∞ 0  fs dXs = 0, ∀ f ∈ L2 (μ) or equivalently &L, X' = 0, for the (2) E L 0

martingale (Lt ) = (E(L | At )) generated by L. $ %  = 0 and be H the set of functionals that  ∈ L2 (A∞ , P ); E F Proof Let L20 = F can be represented as a stochastic integral "  ∈ L20 ; H = F

= ∃ f ∈ L (μ) : F



∞

2

# fs dXs .

0

(1) H is stable under stopping, i.e., for each element F ∈ H and each stopping time τ holds Fτ ∈ H , where Ft = E(F | At ) is the martingale generated by F .

128

4 Elements of Stochastic Analysis

∞ For the proof let F = 0 fs dXs ∈ H . Then, according to the optional sampling theorem, it follows 

∞

Fτ = E(F | Aτ ) = 0

f 1[0,τ ] dXs

and  EFτ2

∞

=E 

0

≤E

∞

(f 1[0,τ ] )2 d&X' f 2 d&X' < ∞.

0

(2) The orthogonal space H ⊥ is stable, i.e. with N ∈ H ⊥ and with Nt = E(N | At ), the martingale generated by N, satisfies Nτ ∈ H ⊥ for each stopping time τ . It needs to be shown that for F ∈ H it holds: EF Nτ = 0. However, this follows using the stability of H as follows: EF Nτ = EE(F Nτ | Aτ ) = EE(F | Aτ )Nτ = ENτ Fτ = EFτ E(N | Aτ ) = EE(Fτ N | Aτ ) = EFτ N = 0, as N ∈ H ⊥ and Fτ ∈ H . So Nτ ∈ H ⊥ and, therefore, H ⊥ is stable. (3) Let M ∈ H2 and be w.l.g. M0 = 0 (otherwise transition to M − M0 ), then M∞ ∈ L20 . We define & ∞ | H ) the orthogonal projection in L2 (A∞ , P ), Y∞ := E(M L∞ := M∞ − Y∞ , Yt := E(Y∞ | At )

and

Lt := E(L∞ | At ).

t Then it holds: Mt = Yt + Lt , Yt = 0 fs dXs since Y∞ ∈ H and (Lt , At ) ∈ H2 with L∞ ∈ H ⊥ . H, H ⊥ are stable. From this it follows ∀ F ∈ H and ∀ stopping times τ : Lτ ∈ H ⊥ , Fτ ∈ H and, therefore, ELτ Fτ = 0. From this, by the characterization of the martingale property by stopping times, it follows: (Lt Ft ) ∈ M. This implies: &F, L' = 0, ∀ F = f ·X ∈ H , since &F, L' is the unambiguous predictable process A such that F L − A ∈ M. As consequence we obtain &f · X, L' = f · &X, L' = 0, ∀ f ∈ L2 (μ), or, equivalently: &X, L' = 0.

4.2 Martingale Representation

129

(4) Uniqueness Suppose there are two different representations of F F = EF + H 1 · X + L1 = EF + H 2 · X + L2 Then it follows: L1 − L2 = (H 2 − H 1 ) · X. Since L1 − L2 ∈ H ⊥ it follows &L1 − L2 , (H 2 − H 1 ) · X' = 0. This implies (H 2 − H 1 )2 · &X' = 0. In consequence it follows H 1 = H 2 in L2 (μ) and L1 = L2 .  

So F has a unique decomposition. Remark 4.33

(a) Decomposition theorem for martingales A direct corollary of Kunita– Watanabe’s theorem is a decomposition theorem for martingales. Theorem 4.32 gives the decomposition of L2 -functionals. 2 2 Let X = (Xt , At ) ∈ H2 , At = AX t and M ∈ H = H (X) be a martingale with respect to this filtration. Then there exists a unique decomposition  Mt = M0 +

t

ϕs dXs + Lt

0

with L ∈ H2 , ϕ ∈ L2 (μ) and &L, X' = 0, i.e. L is perpendicular to X. L is the non-hedgeable part of M. (b) There is also a vector-valued version of the decomposition theorem. Let X = (X1 , . . . , Xd ), M ∈ H2 (AX ), then M has a decomposition Mt = M0 +

d   i=1

t 0

ϕsi dXsi + Zt

  with Z ∈ H2 , and &Z, X' = 0 and it holds E di=1 (ϕsi )2 d&Xi 's < ∞. This decomposition is unique. (c) Locally square-integrable martingales One can get away from the assumption of square integrability by localization. Let X ∈ M2loc, N ∈ Mloc (AX ) then it follows: There is exactly one decomposition Nt = N0 + Ht + Lt ,

t 0[λ\+ ⊗ P ] and there exists a Brownian motion B with respect to AM such that  t 1 d&M't = ft P a.s. and Mt = M0 + fs2 dBs dλ\t 0 Proof According to Lebesgue’s differentiation theorem there exists ft = lim n(&M't − &M't − 1 ) n

λ\ ⊗ P a.s.

and ft =

d&M't ∈ L(P). dλ\t

1

f − 2 ∈ L2loc (M), because 

t



− 12 2

0

Now define Bt :=

t

) d&M'Ms =

(fs

0

t 0

− 12

fs

fs−1 fs dλ\(s) = t < ∞.

dMs . Then B ∈ Mloc,c and it holds 

t

&B't = 0

fs−1 d&M's = t.

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4 Elements of Stochastic Analysis

According to Lévy’s theorem B is a Brownian motion and it is 

t

Mt = M0 +

1

− 12

fs2 (fs 

dMs )

0 t

= M0 +

1

fs2 dBs . 0

  In the case where the variation process is only Lebesgue continuous, &M' ( λ\+ there is an analogous statement formulated with an extension of the basic space. Proposition 4.35 Let M = (M 1 , . . . , M d ) ∈ Mloc,c and let &M i ' ( λ\t , 1 ≤ i ≤ d, then there exists a d-dimensional Brownian motion B on an extension  P) ⊃ ( , A, P ) of the basic space and there exists f ∈ L2 (B) with values  A, ( , loc d×d such that in R  t Mt = M0 + fs · dBs 0

with (f · B)i =

4.3

 j

f i,j · B j .

Measure Change, Theorem of Girsanov

Topic of this section is the study of properties of processes when the measure changes. Let X be a semimartingale with respect to an underlying measure P , (X, P ) ∈ S and let Q be a P -continuous probability measure, Q ( P . Then a fundamental question is whether also (X, Q) is a semimartingale and how the drift and martingale components of X change by the transition from P to Q. We want to decompose the process (X, Q) into drift and martingale components. This is of great importance in financial mathematics, because it turns out that the prices of derivatives and the form of optimal trading strategies can be computed simply by introducing suitable new measures Q. To motivate the general form of Girsanov’s theorem, we consider a simple example in discrete time. Example 4.36 Let Z1 , . . . , Zn be independent, normally distributed, Zi ∼ N(0, 1), Z = (Z1 , . . . , Zn ) random variables on ( , A, P ). We define a new measure P using the Radon–Nikodým derivative. dP := exp dP

 n i=1

μi Z i −

 n 1 2 μi . 2 i=1

4.3 Measure Change, Theorem of Girsanov

133

This density has a similar shape to that in the exponential martingale. It holds: Eeμ·Z = e 2 μ 1

2

is the Laplace transform of the one dimensional normally distributed random variable μ · Z. It follows that the new measure P is a probability measure,  ∈ M 1 ( , A). P How is the sequence (Zi ) distributed with respect to the new measure P? It holds: dP˜ Z dP˜ Z dP Z (z) = (z) · (z) dλ\n dP Z dλ\n     1 2 1 2 1 = exp μi zi − μi · · exp − zi 2 (2π)n/2 2  2  1 1  = zi − μi . · exp − 2 (2π)n/2 i := Zi − μi , 1 ≤ i ≤ n, is an i.i.d. sequence with respect to P. This It follows: Z implies that the sequence (Zi ) with respect to P has the following representation: i + μi with an i.i.d. sequence (Z i ) ∼ N(0, 1) and with shif t (μi ). Zi = Z  the sequence (Zi ) is, therefore, an i.i.d. With respect to the new measure P normally distributed sequence with shifts μi . Girsanov’s theorem shows that a similar behavior holds in a more general form. Denotation Let P , Q ∈ M 1 ( , A) and be (At ) ⊂ A is a filtration, without restriction one can choose A = A∞ . Then we define • Q ( P , Q is absolutely continuous with respect to P ⇐⇒ NP ⊂ NQ • Q ∼ P , Q is equivalent to P ⇐⇒ NP = NQ loc

• Q ( P , Q is locally continuous with respect to P ⇐⇒ Qt ( Pt , ∀ t > 0, where Qt := Q|At and Pt := P |At are the measures restricted to At . loc

For stopping times τ be Pτ := P |Aτ , Pτ− := P |Aτ− . The statement Q ( P is equivalent to the condition that a localizing sequence (τn ) of stopping times exists with Qτn ( Pτn , n ∈ N. loc

• Q ∼ P , Q is locally equivalent with respect to P loc

loc

⇐⇒ Q ( P and P ( Q.

134

4 Elements of Stochastic Analysis

Proposition 4.37 (a) Let Q ( P and let Dt := E density quotient

dQ dP .



dQ dP

 | At be the martingale generated by the

Then it follows Dt =

dQt [P ], dPt

∀ t,

i.e., Dt coincides with the density quotient of Q to P on the restricted σ -algebra At . loc

(b) Let Q ( P be locally P -continuous, then the density process D is a P martingale Dt =

dQt ∈ M(P ) and it holds dPt

(Dt ) is uniformly integrable ⇐⇒ Q ( P and then it holds  Dt := E

 dQ | At . dP

(4.18)

Proof (a) follows directly by verifying the Radon–Nikodým equation. (b) According to (a) it follows that (Dt ) is a P -martingale. The rest of the assertion follows by the closure theorem for martingales: (Dt ) is uniformly integrable if and only if Dt −→ D∞ in L1 and P a.s. From the L1 -convergence it then follows: Q = D∞ P

i.e. 

Dt = E

D∞ =

dQ . dP

 dQ | At . dP   loc

Remark 4.38 Local continuity of measures Q(P does not in general imply continuity Q ( P (at infinity). There are a number of 0–1 laws (e.g., for Gaussian measures) which state that if there is local continuity in the limit, then the measures are either orthogonal or equivalent. In preparation for Girsanov’s theorem for the Brownian motion, we need a martingale characterization of Brownian motion that is related to one of the three

4.3 Measure Change, Theorem of Girsanov

135

equivalences from the Lévy characterization of Brownian motion: If the stochastic exponential "  d  if Et := exp i

d

k=1 0

1 2 d

fk (s) dXsk +



k=1 0

t

# fk2 (s) ds

is a complex martingale, then X is a Brownian motion. The following is a variant of this characterization: Proposition 4.39 Let (Xt , At ) be a continuous process such that "

Zt =

Zt(u)

# 1 2 := exp uXt − u t ∈ Mloc , 2

u ∈ (−ε, ε),

then (Xt , At ) is a Brownian motion.



Proof There is a localizing sequence τn ↑ ∞ such that for all 0 ≤ s < t, and all A ∈ As it holds:   (u) (u) Zs∧τn dP = Zt ∧τn dP , (4.19) A

A

i.e., the martingale property holds when localizing Z with a sequence τn . In the proof of the characterization theorem this has already been used. In an exponential family with a real parameter, it holds by the theorem of majorized convergence, that one can swap differentiation (derivative w.r.t. u) and integration: ∂ ∂u



 ··· =

∂ . . . , u ∈ Uε (0). ∂u

This property is needed for a small interval Uε (0) around zero. Derivation of both sides in (4.19) results in    (u)  (u)   dP , A ∈ As . Xt ∧τn − u(t ∧ τn ) Zt ∧τn dP = Xs∧τn − u(s ∧ τn ) Zs∧τ n A

A

Differentiation again w.r.t. u provides    A

(Xt ∧τn − u(t ∧ τn )

=

   A

2

 (u) − (t ∧ τn ) Zt ∧τn dP

Xs∧τn − u(s ∧ τn )

2

 (u) − (s ∧ τn ) Zs∧τn dP .

We now have two equations that hold in an neighborhood of zero. For u = 0 the first equation says Xt is a continuous local martingale, X ∈ Mloc,c . The second

136

4 Elements of Stochastic Analysis

equation says, Xt2 − t ∈ Mloc,c . By Lévy’s theorem it follows that X is a Brownian motion.   The following theorem of Girsanov describes changes in measure for the Brownian motion. Theorem 4.40 (Girsanov’s Theorem for the Brownian Motion) Let (Xt , At ) be a Brownian motion with respect to P , let ϕ ∈ L0 (X) and let 

t

Lt := exp 0

1 ϕs dXs − 2



t 0

 ϕs2 ds

be the exponential martingale of ϕ · X. Then it holds: (a) (Lt , At ) is a local martingale and a supermartingale, and it holds:  Lt = 1 +

t

Ls ϕs dXs 0

(b) If ELT = 1, then (Lt , At )0≤t ≤T is a martingale. Let dP t := Xt − := LT and X dP



t

ϕs ds, 0

t , At , P) is a Brownian then P is a probability measure on ( , A) and (X motion. Concerning the new measure P, therefore, X is a Brownian motion with a random drift. Remark 4.41 We will deal with a reverse of Girsanov’s theorem in the following. Continuous measure changes in Brownian motion correspond to the addition of a random drift and the densities of continuous measure changes are described by stochastic exponentials. Proof (a) Part (a) and the first part of (b) follow from Proposition 4.12 via exponential martingales. (b) According to Proposition 4.39, it is to show that (u) Zt

"   #  t 1 2 := exp u Xt − ϕs ds − u t ∈ Mloc (P). 2 0

4.3 Measure Change, Theorem of Girsanov

137

Let, therefore, Dt be the density quotient process. Dt :=

 dP    dPt | At = EP LT | At = Lt as (Lt ) ∈ M. = EP dPt dP

Then it holds: (Zt(u) Dt ) is a local martingale w.r.t. P , since (u)

(u)

Zt Dt = Zt Lt " t #  1 t = exp (ϕs + u) dXs − (ϕs + u)2 ds ∈ Mloc (P ), 2 0 0 using that (X, τ P ) is a Brownian motion. Let τn be a localizing sequence such that Z (u) D n ∈ M and let σ is a bounded stopping time. According to the 

P optional sampling theorem Dτ = E ddP | Aτ for all bounded stopping times τ . Further, again according to the optional sampling theorem

EPZτ(u) = EP Zτ(u) Dτn ∧σ n ∧σ n ∧σ (u) = EP Z0 D0 as Zτn ∧t Dτn ∧t M(P ) (u)

= EPZ0 ,

∀ bounded stopping times σ.

With the characterization of the martingale property by stopping times it follows Z (u) ∈ Mloc (P) and thus according to Proposition 4.37 the assertion.   Remark 4.42 (a) Sufficient for the requirement ELT = 1 is the Novikov condition  ∞ E exp 12 0 ϕs2 ds < ∞ as well as the Kazamaki condition. (b) There is also an analogous multivariate version of Girsanov’s theorem as well as a version on [0, ∞). For d ≥ 1 the density quotient process is of the form "

1 Lt = exp (ϕ · X)t − 2 d

i=1

 0

t

# (ϕsi )2 ds

.

If X is a d-dimensional Brownian motion with respect to P , then    t i i i  ϕs ds (Xt ) = Xt − 0

 with density is a Brownian motion with respect to the continuous measure P process L bzgl. P .

138

4 Elements of Stochastic Analysis

In the above version of Girsanov’s theorem, we are concerned with the question how a Brownian motion behaves under a measure transformation. For the generalization to the case of general martingales or semimartingales we need the following statement about the density process. loc

Proposition 4.43 Let Q ( P then the density process is Dt > 0 Q-a.s. ∀ t and it holds, inf {t ∈ R+ ; Dt = 0 or Dt − = 0} = ∞ Q-a.s. if Q ( P . Proof It is sufficient to consider the case Q ( P otherwise one can reduce the problem to the case [0, t], t > 0. Let τ := inf{t; Dt = 0 or Dt − = 0}. (Dt ) is a nonnegative local martingale and hence a nonnegative supermartingale. According to Corollary 3.10 it follows D = 0 on [τ, ∞). So D∞ − dQ dP = 0 on {τ < ∞}. Because of Q = D∞ P , therefore, follows Q({τ < ∞}) = 0, i.e. inf{t ∈ R+ ; Dt = 0 or Dt − = 0} = ∞. Thus the assertion follows.   Proposition 4.44 loc

(a) If Q ( P , then for all stopping times τ on Aτ ∩ {τ < ∞},

Q = Dτ P

i.e. Q has the stopped density process as density with respect to P on Aτ on τ < ∞. (b) If Q ( P , then Q = Dτ P on Aτ . Proof First, we prove (b). (b) According to Proposition 4.37, the density process in the continuous case is given by Dt = E

 dQ dP

 | At .

1 This is a uniformly integrable martingale, and Dt −→ D∞ = dQ dP in L and a.s. By the optional sampling theorem, for all stopping times τ it follows :

Dτ = E

 dQ|Aτ | Aτ = . dP dP |Aτ

 dQ

4.3 Measure Change, Theorem of Girsanov

139

loc

(a) For Q ( P D is uniformly integrable on finite intervals [0, t]. Let A ∈ Aτ , then it follows for all t: A ∩ {τ ≤ t} ∈ Aτ ∧t . From this follows with the optional sampling theorem:   Q A ∩ {τ ≤ t} =

 A∩{τ ≤t }

  E Dt | Aτ ∧t dP 

 =

A∩{τ ≤t }

Dτ ∧t dP =

A∩{τ ≤t }

Dτ dP ;

so the assertion holds on [0, t]. For t ↑ ∞ it follows:   Q A ∩ {τ < ∞} =

 Dτ dP . A∩{τ 0 Q a.s. Let X ∈ L1 (At , Q), and be Mu := EQ (X | Au ). Then it follows M ∈ M(Q), and, therefore, according to Proposition 4.45 DM ∈ M(P ). From this it follows Ds EQ (X | As ) = Ds Ms = EP (Dt Mt | As ) = EP (XDt | As ), therefore, the proposition. Case X ∈ L+ (At ) is analogous.

 

The following theorem extends the statement of Girsanov’s theorem to continuous local martingales under the assumption of a continuous density process. Theorem 4.47 (Girsanov’s Theorem for Continuous Local Martingales) Let loc

Q ( P and let the density process D be continuous, then it holds: (a) If X is a continuous semimartingale with respect to P , then X is also a continuous semimartingale with respect to Q: X ∈ Sc (P ) ⇒ X ∈ Sc (Q). (b) If M ∈ Mloc,c (P ), then it holds:  := M − D −1 · &M, D' ∈ Mloc,c (Q) M

and

 Q = &M'P &M'

 is called Girsanov The mapping  · : Mloc,c (P ) → Mloc,c (Q), M → M transform.

4.3 Measure Change, Theorem of Girsanov

141

loc

(c) If D > 0 [P ] (i.e. P ∼Q), then it follows:   (1) ∃! L ∈ Mloc,c (P ) : D = E(L) = exp L − 12 &L' , D is the stochastic exponential of L. · (2) L = log D0 + 0 Ds−1 dDs = L(D), L is the stochastic logarithm of D. (3) P = E(− L)Q Proof (a) follows from (b) since X ∈ S(P ) has a decomposition X = M + A with M ∈ Mloc,c and A ∈ Vc .  = M − D −1 · &M, D' ∈ Mloc,c (Q). Then by (b) it follows M From this then follows the decomposition  + (A + D −1 · &M, D') ∈ Sc (Q); X=M

 ∈V c

Thus X is a continuous Q-semimartingale. (b) If XD ∈ Mloc (P ) then it follows by Proposition 4.45 X ∈ Mloc (Q). So to  ∈ Mloc,c (P ). show is: MD Using the partial integration formula for continuous semimartingales it holds: t Dt = M0 D0 + M

 

t

s dDs + M

0 t

= M0 D0 +

s dDs + M

0

 

t

s + &M,  D't Ds dM

0 t

Ds dMs − &M, D't

0

+ &M, D't ∈ Mloc,c (P ). To this purpose, we use the following relation:  + &M,  D' = D · M − D · (D −1 · &M, D') +&M, D' D·M 

 ∈Mloc,c

=&M,D'

− D −1 · &&M, D', D'. The last term is zero, since &M, D' is predictable and of finite variation.

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4 Elements of Stochastic Analysis

 Q = &M'Q = [M]Q = [M]P = &M'P Finally, it holds that &M' (c) According to the Itô formula 

t

log Dt = log D0 +

0

Ds−1 dDs

=Lt



1 − 2



t 0

1 Ds−2 d&D's = Lt − &L't . 2

But it follows that D is the stochastic exponential of L is, Dt = E(L)t and, therefore, also L = L(D), L is the stochastic logarithm of D. To prove (c): By definition and the first part of (b).  = M − D −1 · &M, D' M = M − &M, D −1 · D' = M − &M, L' ∈ Mloc,c (Q)  = L − &L' ∈ Mloc,c (Q). is a locally continuous Q-martingale. Further : L It follows that: 1 E(− L) = exp(−L + &L' − &L') = E(L)−1 . 2 Thus it follows that L)Q. P = E(L)−1 Q = E(−   Thus, continuous locally non-negative martingales, in particular continuous density quotients of equivalent measures, are characterized by having a representation as a stochastic exponential of a continuous local martingale. Remark 4.48 In particular, for the case of the Brownian motion M = B it holds for  the Girsanov transform M:  ∈ Mloc,c (Q) M

and

P Q &M' t = &M't = t.

 Q) is a Brownian motion. The density process By Lévy’s theorem it follows: (M, D has the form   1 D = E(L) = exp L − &L' with L ∈ Mloc,c (P ). 2

4.3 Measure Change, Theorem of Girsanov

143

According to the martingale representation theorem L is of the form L = L0 + ϕ · B with ϕ ∈ L0 (B). The Girsanov transform t = Bt − &B, L't = Bt − B



t

ϕs ds 0

thus yields a stochastic shift and we obtain Girsanov’s theorem on Brownian motion as a special case of the general theorem. There is also a version of Girsanov’s theorem for non-continuous local martingales. We use in the proof some statements from Jacod and Shiryaev (1987, I, Sections 3.5 and 4.49). loc

Theorem 4.49 (General Measure Transformation) Let Q ( P with density process D and let M ∈ Mloc (P ) with M0 = 0 and [M, D] ∈ Aloc (P ). Then it holds:  := M − 1 · &M, D' = M − &M, L(D)' ∈ Mloc (Q), L(D) the stochastic (a) M D− logarithm c 'Q ; the predictable quadratic variations of the continuous (b) &M c 'P = &M  with respect to P and Q are identical. martingale components of M, M Proof (a)

 is wellis locally bounded according to Proposition 4.43 such that M defined. A := D1− · &M, D' ∈ V ∩ P is predictable, since in Aloc (P ) (cf. Jacod and Shiryaev (1987, I, Sect. 3.5)). It follows by the partial integration formula for semimartingales 1 D−

AD = A · D + D− · A = A · D + &M, D' (cf. Jacod and Shiryaev (1987, I, Sect. 4.4)). Again with partial integration we get  = MD − AD = M− · D + D− · M +[M, D] − A · D −&M, D' MD    ∈Mloc (P )

=

∈Mloc (P )

∈Mloc (P )

+ [M, D] − &M, D' ∈ Mloc (P ) Y 



∈Mloc (P )

∈Mloc (P )

With Y := M− · D + D− · M − A · D ∈ Mloc (P ) this martingale property of  with respect to P implies according to Proposition 4.45 MD  ∈ Mloc (Q). M

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(b) M =

 M 

∈Mloc (Q)

+  A ∈ SQ

is a special semimartingale with respect to Q.

∈V P

Equality of the quadratic variation of M under P and Q hods. For because of loc

P ( Q the Pt stochastic limit implies the same Qt stochastic limit (transition to a.s. convergent subsequences), i.e. [M]Q = [M]P . For A ∈ V the continuous martingale part of A equals zero, Ac = 0 since c A ∈ V ∩ Mloc,c . c = M c . Thus it follows: M Therefore, as a result we get: c 'Q = &M c 'Q = [M]Q − &M



(Ms )2

s≤·

 c 'P . = [M]P − (Ms )2 = &M s≤·

  Similarly, a generalization of Proposition 4.49 to a version of Girsanov’s theorem for general semimartingales holds. loc

Proposition 4.50 Let Q ( P , X ∈ S P , ϕ ∈ L0P (X), then it follows: X ∈ S Q , ϕ ∈ L0Q (X) and it holds: [X]Q = [X]P ,

(ϕ · X)Q = (ϕ · X)P

and

&Xc 'Q = &Xc 'P .

Proof For the proof, we refer the reader to Jacod and Shiryaev (1987, III, Sect. 3.13).   Remark 4.51  (a) Let ϕ ∈ L0loc (M), then it follows ϕ ∈ L0loc (M). For the Girsanov transformation of ϕ · M it holds:  ϕ · M = ϕ · M.   it holds L0 (M) = L0 (M). Proof Since &M' = &M' loc loc ϕ is locally bounded and, therefore, it holds  = ϕ ·M − (ϕD −1 ) ·&M, D' = ϕ ·M − D −1 ·&ϕ ·M, D' = ϕ ϕ ·M ·M.

 

4.3 Measure Change, Theorem of Girsanov

145

(b) (P , Q) is called Girsanov pair, if P ∼ Q on A∞ and D is continuous. Then it follows: S(P ) = S(Q)  and the Girsanov transformation GQ P : Sc (P ) −→ Sc (Q), M → M is bijective.

4.3.1

Applications of the Girsanov Theorem

The Girsanov transform is one of the important operations that allow to perform analysis on infinite-dimensional spaces. We outline two applications of Girsanov’s theorem.

4.3.1.1 Theorem of Cameron–Martin Let (Cd [0, T ],  · ∞ , W ) be the Wiener space, that is W is the Wiener measure on Cd [0, T ] the distribution of a d-dimensional Brownian motion on [0, T ]. The standard construction of the Brownian motion B on = Cd [0, T ] then is given by the evaluation mappings βt : Bt (ω) = βt (ω) = ω(t),

w ∈ Cd [0, T ].

For each continuous function h ∈ Cd [0, T ], h : [0, T ] −→ Rd , let τh (ω) be the h-shift (the translation by h), i.e.   (τh ω)(t) = βt τh (ω) = βt (ω) + h(t) We consider the image measure of the Wiener measure under the h-shift Wh := W τh = P B+h . Wh is the distribution of the Brownian motion shifted by h (Fig. 4.3). A fundamental question is: for which functions h ∈ Cd [0, T ] does it hold, that Wh ≈ W , i.e. the translated measure Wh is equivalent to W and how to determine h in this case the density dW dW . This is a basic problem for the statistical analysis of stochastic processes resp. measures on the infinite-dimensional space of continuous functions. For two Gaussian measures on Cd [0, T ] a 0–1 law holds. Either the measures are orthogonal, that is, they can be perfectly distinguished statistically, or they are equivalent. If B σ is a Brownian motion with variance var(Btσ ) = σ 2 t then B σ1 and B σ2 for σ1 = σ2 are orthogonal. Since the quadratic variation of B σ is [B σ ]t = σ 2 t it follows that B σ1 and B σ2 are concentrated on disjoint path sets implying orthogonality of the measures.

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Fig. 4.3 Brownian motion, normal and shifted, B = Brownian motion; h(y) = cos(4πy), B + h = addition of B and h

Cameron-Martin’s theorem answers the question for which continuous functions h ∈ Cd [0, T ] Wh ≈ W resp. Wh ⊥W holds and gives a formula for the density dWh / dW . Obviously, for the equivalence of Wh and W necessary that hi (0) = 0 for h = (h1 , . . . , hd ). Definition 4.52 Let " H : = h ∈ Cd [0, T ]; h = (h1 , . . . hd ), hi absolutely continuous, hi (0) = 0, 

T

and 0

# |hi (s)|2 ds < ∞ .

The subspace H ⊂ Cd [0, T ] is called Cameron–Martin space. Remark 4.53 The Cameron–Martin space H is a Hilbert space with scalar product  T &h, g' := i 0 hi (s)gi (s) ds and hH := &h, h'1/2 . H ⊂ Cd,0 [0, T ] = {h ∈ Cd [0, T ]; h(0) = 0} and H is dense with respect to the ·  · supremum norm ·∞ . For h ∈ H , h ◦ β = 0 h ·dβ = di=1 0 hi dβ i satisfies the Novikov condition. Therefore, the stochastic exponential E(h ◦ β) is a martingale. The following theorem was proved by Cameron–Martin in 1949 by functional analytic methods. The proof given here is based on an inventive combination of Girsanov’s theorem with the martingale representation theorem.

4.3 Measure Change, Theorem of Girsanov

147

Theorem 4.54 (Theorem of Cameron–Martin) The shifts Wh of the Wiener measure are either equivalent or orthogonal to W and it holds: (a) Wh ≈ W ⇐⇒ h ∈ H (b) For h ∈ H the Cameron–Martin formula holds   dWh = E (h ◦ β)T . dW Proof (a) “⇒”: Let Wh ≈ W , then according to Girsanov’s theorem (in multivariate version) it follows: There is a continuous martingale L with respect to the Wiener measure W such that dWh = E(L)T dW

 := β − &β, L', and B

i := β i − &β i , L') is a Brownian motion with respect to Wh . (i.e. B By definition of Wh with respect to Wh also the representation β = B + h, with a Brownian motion B with respect to Wh holds. So it holds with respect to Wh : h=β−B  − B) + &β, L't ∈ Sc (Wh ), = (B h is a deterministic continuous semimartingale with respect to Wh . It follows that h has finite variation, V h (T ) < ∞. This gives us two decompositions of β with regard to Wh ,  + &β, L', β =B +h=B  Since h, &β, L' are predictable, it follows that with Brownian motions B, B. β is a special semimartingale. Because of the uniqueness property of the decomposition of special semimartingales, it thus follows that h = &β, L' a.s. L is a martingale in the filtration generated by β. By the martingale representation theorem there exists a predictable process  ∈ L0 (B),

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4 Elements of Stochastic Analysis

T 0

|is |2 ds < ∞, 1 ≤ i ≤ d, such that with respect to the Wiener measure W L = L0 +  · β.

From this it follows: hi (t) = &β i , L't i i i = (  t · i&β , β ')t = 0 s ds, for all t ∈ [0, T ]. Therefore, by Lebesgue’s differentiation theorem hi is absolutely continuous and it holds is = hi (s) almost surely. So (independently of ω ∈ ) h ∈ H since  ∈ L0 (B) and it holds dWh = E(L)T dW = E( · β)T = E(h · β)T . The Radon–Nikodým density of Wh is thus given by the stochastic exponential of h · β. “⇐”: For h ∈ H let Q be the probability measure Q := E((h · β))T · W According to Theorem 4.40 of Girsanov it holds: Q ∼ W and with respect to Q it holds:    · β = (β i ) = B i + hi (s) ds 0

= (B i + hi ) = B + h,

because hi (0) = 0

with a Brownian motion (B, Q). β is, therefore, a Brownian motion with shift and, therefore, it holds: Q = Wh . Let h ∈ Cd [0, T ], then β is with respect to Wh a Gaussian process. The 0–1 law for Gaussian measures holds: W, Wh are equivalent Wh ∼ W

or W, Wh are orthogonal Wh ⊥ W.

Either Wh and W can be distinguished on the basis of one observation; this is exactly the case if W ⊥ Wh ⇐⇒ h ∈ H . Or W , Wh are equivalent and W ∼ Wh ⇐⇒ h ∈ H .  

4.3 Measure Change, Theorem of Girsanov

149

Denote by supp μ the support of a measure μ, i.e. the smallest closed set A such that μ(Ac ) = 0. As a consequence of the above consideration one obtains also a simple argument for the determination of the support of the Wiener measure W . Corollary 4.55   supp W = Cd,0 [0, T ] = {f ∈ Cd ([0, T ]); f (0) = 0}. Proof There exists an element x ∈ suppW = ∅. Then it follows: x + H ⊂ suppW , for Wh ≈ W for h ∈ H . H is a dense subset of Cd,0 [0, T ]. Thus the assertion follows.  

4.3.2

The Clark Formula

Another application of Girsanov’s theorem is the derivation of a formula for the integrand of the martingale representation theorem for the case of the Brownian motion: the Clark formula. β Let (At ) = (At ) be the Brownian filtration in Cd [0, T ] and let β be the standard construction of the Brownian motion. Let X be an L2 -functional of the Brownian motion X := F (β) ∈ L2 (AT ). According to the martingale representation theorem X has a representation of the form 

T

X = EX +

s · dβs .

0

How can one determine  as a functional of F . For the special case of harmonic functions F we had obtained a solution to this problem using the Itô formula. We make the following assumption. Assumption F is Lipschitz, i.e. 1. ∃K : |F (β + ) − F (β)| ≤ K. 2. There is a kernel F  from Cd [0, T ] to [0, T ] such that ∀  ∈ H  1 lim F (β + ε) − F (β) = ε→0 ε



T

(t)F  (β, dt) a.s.

0

The directional derivative of F is given by a measure F  (β, dt).

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4 Elements of Stochastic Analysis

Remark 4.56 (a) If F is Fréchet differentiable with bounded derivative, then conditions (1) and (2) hold. The continuous linear forms on Cd [0, T ] are the bounded measures on [0,T]. (b) In the following theorem we need the predictable projection. Let X ∈ L(A ⊗ B+ ) be a measurable process, then there exists exactly one p predictable process X ∈ L(P) such that for all predictable stopping times τ p p

  Xτ = E Xτ | Aτ − on {τ < ∞}. p

X is the predictable projection of X, X := & πP (X). p X is characterized by the equations 



T

T

Xs dAs = E

E 0

p

∀ As ↑∈ L(P).

Xs dAs

(4.20)

0 p

In many cases, the predictable projection X of X is given by the left-hand limit Xt = Xt − or also by EP (Xt | At − ). The following theorem determines the integrand  of the martingale representation of F (β) as the predictable projection of the derivative measure F  (β, (t, T ]).

p

Theorem 4.57 (Clark Formula) Let be X = F (β) ∈ L2 (AT , W ) and let the assumptions (1), (2) hold. Then the integrand  in the martingale representation T F (β) = EX + 0 s dBs of X is given by:   =& πP F  (β, (t, T ]) .  is the predictable projection of F  (β, (t, F ]) in L2 (P, μ). Proof Let u ∈ B(P) be bounded, P-measurable and be 

t

ψt :=

us ds,

t := ψ

0



t

us dβs

)T W. and Q := E(εψ

0

By Girsanov’s theorem, it follows: β − εψ is a Brownian motion with respect to Q. Thus the expected value of F (β) can also be calculated by a change of measure: EW F (β) = EQ F (β − εψ) )T . = EW F (β − εψ)E(εψ

4.3 Measure Change, Theorem of Girsanov

151

Equivalent to this is the following equation ' )T − 1) + EW (F (β − εψ) − F (β)) EW (F (β − εψ) − F (β))(E(εψ ( )T − 1) = 0. + EW F (β)(E(εψ In this form, the differentiability assumptions on F can now be applied. Multiplication by 1ε leads to: 1 (. . .) = 0 = I1 (ε) + I2 (ε) + I3 (ε), ε

∀ ε > 0,

(4.21)

where Ij (ε) denote the three resulting terms. For ε → 0 converges I1 (ε) → 0 because F satisfies the Lipschitz condition. 1  ε |F (β − εψ) − F (β)| ≤ ψ and E(ε ψ)T → 1. To: I2 (ε) + I3 (ε) → 0: Using the differential equation for the stochastic exponential, the following holds. ε

−1



  T −1 = E(εψ)



T

)s us dβs . E(εψ

0

The integrand is majorized and converges pointwise to u. According to the theorem on majorized convergence for stochastic integrals, it follows ε

−1

 T − 1) −→ (E(εψ)



T

in L2 (P , AT ).

us dβs 0

Thus it holds: I2 (ε) + I3 (ε) −→ 0 and, therefore, by (4.21) and Assumption 2:  EW F (β)

T



T

us dβs = − lim I2 (ε) = EW F (β) ε↓0

0

ψt F  (β, dt).

0

With the properties of the stochastic integral, it follows from (4.22)  EW

T

 s us ds = EW

0



T

T

s dβs 0



= EW F (β)  = EW 0

us dβs 0

T

us dβs 0

T

ψt F  (β, dt)



t

with ψt =

us ds 0

(4.22)

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4 Elements of Stochastic Analysis

 = EW

T

ut F  (β, (t, T ]) dt

by partial integration

0

 = EW

0

T

  ut & πP F  (β, (t, T ]) dt,

as u ∈ L(P).

The last equality follows from the projection equations for the predictable projection in (4.20). However, this equation implies that the predictable projection & πP (F  (β, (t, T ))) of F  (β, (t, T ]) is equal to the integrand ,   =& πP F  (β, (t, T ]) in L2 (P, μ).  

4.4

Stochastic Differential Equations

Stochastic differential equations are an important tool for modeling stochastic timedependent processes. The local behavior of the processes can often be well described and then leads to stochastic differential equations as suitable models. Diffusion processes governed by local drift and diffusion coefficients are an important class of examples. Two examples of stochastic differential equations have already occurred so far. Remark 4.58 (1) For a continuous local martingale M ∈ Mloc,c the stochastic exponential   E(M)t = exp Mt − &M't is the unique solution of the stochastic integral equation 

t

Xt = 1 +

Xs dMs . 0

Equivalent to this integral equation is the stochastic differential equation "

dXt = Xt dMt , X0 = 1.

(2) The geometric Brownian motion is described by the stochastic differential equation dXt = μXt dt + σ Xt dBt ,

X0 = x0 .

4.4 Stochastic Differential Equations

153

Here we have a local drift and a local variation term. The drift term is proportional to Xt and has the value μXt . The variation term σ Xt is also proportional to Xt . The solution to this stochastic differential equation is the geometric Brownian motion Xt = x0 exp



 1  μ − σ 2 t + σ Bt . 2

The geometric Brownian motion is a standard model in financial mathematics. One can see some properties of X directly from the explicit solution, e.g., the evolution of the expected value: EXt = x0 eμt . This follows from the fact that the stochastic exponential E(σ B) is a martingale,   1 2 exp σ Bt − σ t ∈ M. 2 For t → ∞ it holds Bt /t → 0. Therefore, it follows: Xt −→ 0 if If μ > σ 2 /2 then Xt ∼ e

  2 μ− σ2 t

σ2 2

> μ > 0.

.

We now want to study more generally the following type of stochastic differential equations: Given a Brownian motion B on ( , A, P ) with natural filtration A = AB and let X be a solution of the stochastic differential equation "

dXt = b(t, Xt ) dt + σ (t, Xt )dBt X0 = Z,

(4.23)

t t i.e. Xt = Z + 0 b(s, Xs ) ds + 0 σ (s, Xs )dBs . b is a local drift term and σ is a local diffusion term. One can also consider the above stochastic differential equation on an interval [s, t]. The initial value Z should in this case lie in L(As ). Solutions X of (4.23) are called generalized diffusion processes, since they are generated by the diffusion of a Brownian motion. More precisely, we speak of strong solutions. One can introduce an additional degree of freedom, namely the construction of a suitable Brownian motion, and then obtain an extended class of solutions—the weak solutions. The stochastic differential equations in (4.23) describe the Markovian case of diffusion equations, where b = b(t, Xt ), σ = σ (t, Xt ) depends only on the current state Xt of the process. Non-Markovian diffusion equations are a generalization of (4.23) with general predictable drift and diffusion coefficients bt , σt .

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Remark 4.59 The differential equation (4.23) can be understood as a generalization of the deterministic differential equations in which the stochastic diffusion term is absent, or equivalently in which σ = 0, 

t

Xt = X0 +

b(s, Xs ) ds. 0

For such deterministic differential equations, the solution behavior is well studied. Under a local Lipschitz condition with respect to x for the coefficient b = b(x, t) in x, boundedness of b on compacta in [0, ∞) × Rd , i.e., in space and time, one can apply the iteration procedure of Picard Lindelöf and obtain: The iteration (0)

Xt

= x,

Xt(n+1) = x +



t 0

b(s, Xs(n) ) ds,

n ≥ 0,

converges for sufficiently small times t to a solution in [0, t]. The solution is unique. However, one cannot expect a global solution. For example, consider the differential equation 

t

Xt = 1 + 0

Xs2 ds,

then for the coefficient b(s, x) = x 2 the local Lipschitz condition holds. The unique local solution of the differential equation with x = 1 is Xt =

1 . 1−t

This solution explodes for t ↑ 1. There is no global solution. A fundamental question for stochastic differential equations, as in the case of deterministic equations, is the existence and uniqueness of solutions. There are examples of coefficients where a deterministic solution does not exist, or exists only on compact intervals, but where there is a global solution with the diffusion term. The first general existence and uniqueness statement is an analogue of Picard Lindelöf’s theorem for deterministic differential equations. Theorem 4.60 Let b, σ be two measurable coefficient functions which are locally Lipschitz in x, i.e. ∀ T > 0, ∃K = KT < ∞ with |b(t, x) − b(t, y)| + |σ (t, x) − σ (t, y)| ≤ K|x − y|,

0 ≤ t ≤ T.

4.4 Stochastic Differential Equations

155

Further, the coefficients b, σ have an at most linear growth in x, i.e. |b(t, x)| + |σ (t, x)| ≤ K(1 + |x|), 0 ≤ t ≤ T , ∀ x and the initial value has a finite second moment, EZ 2 < ∞, then it holds: There exists a unique solution to the stochastic differential equation (4.23) in [0,T] i.e. if X and Y are two solutions, then it holds: Xt = Yt ∀ t ∈ [0, T ] [P ]. The solution has bounded second moments: E sups≤T |Xs |2 < ∞. Proof The proof relies on an iteration and fixed point argument similar to the deterministic case. Let E be the class of all (At )-adapted continuous processes X = (Xt )0≤t ≤T so that X2 := E sup0≤t ≤T |Xt |2 < ∞. This ground space (E,  · ) is a Banach space, that is, a complete, separable, normed vector space. Now we make use of a fixed point argument: For a process X ∈ E we define a new process 

t

(X)t := Z +



t

b(s, Xs ) ds +

σ (s, Xs )dBs .

0

0

The above assumptions ensure the well-definedness of these integrals;  is a welldefined mapping  : E → E. We are looking for a fixed point of . To apply Banach’s fixed point theorem, we show that  is a contraction with respect to the introduced norm. With the inequality (a + b)2 ≤ 2(a 2 + b 2 ) it holds:   2  t |(X)t − (Y )t |2 ≤ 2 sup b(s, Xs ) − b(s, Ys ) ds 0

s≤T



t

+ sup s≤T

2 (σ (s, Xs ) − σ (s, Ys )dBs )

.

0

According to the Doob inequality and using the Lipschitz condition, it follows  t  2       E sup  σ (s, Xs ) − σ (s, Ys ) dBs  ≤ 4E t ≤T

0



T 0

T

= 4E

(σ (s, Xs ) − σ (s, Ys ))dBs

(σ (s, Xs ) − σ (s, Ys ))2 ds

0

≤ 4K 2 E sup |Xs − Ys |2 T . s≤T

As a consequence, one obtains: (X) − (Y ) ≤ (4K 2 T 2 + 4K 2 T )1/2X − Y .

2

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4 Elements of Stochastic Analysis

Furthermore, using the inequality (a + b + c)2 ≤ 3a 2 + 3b2 + 3c2 one obtains for the initial value (0) the estimate  t t 2 2       2    |(0)t | ≤3 Z + sup  b(s, 0) ds  + sup  σ (s, 0) ds  2

t ≤T

t ≤T 0

0

and it follows with the triangle inequality as above   E sup |(0)t |2 ≤ 3 EZ 2 + K 2 T 2 + 4K 2 T < ∞. t ≤T

As a result we obtain:  : E → E is a Lipschitz mapping with the Lipschitz constant. 1/2  . k(T ) := 2(K 2 T 2 + 4K 2 T ) If one chooses T sufficiently small (T ≤ T0 ), then k(T ) < 1, i.e. the mapping is a contraction on E. By Banach’s fixed point theorem it follows:  has exactly one fixed point X ∈ E for T ≤ T0 , i.e. the differential equation (4.23) restricted to [0, T0 ] has a solution in E, and in E the solution is unique. It remains to show: Every solution of (4.23) lies in E. Let X be a solution of (4.23) and let τn := inf{s ≥ 0; |Xs | ≥ n} and let f (n) (t) := E sup |Xs |2 .. s≤t ∧τn

Then it holds as above with the Lipschitz and linear growth condition   2 E sup |Xs | ≤ 3 EZ + E

t ∧τn

2

s≤t ∧τn

K 2 (1 + |Xs |2 ) ds

0



t ∧τn

+ 4E

 2

2

K (1 + |Xs |) ds

0

    t ≤ 3 EZ 2 + 10K 2 1 + E sup |Xu |2 ds 0

u≤s∧τn



t

≤ 3(EZ 2 + 10K 2 T ) + 30K 2

f (n) (s) ds.

0

So with suitable constants a, b:  f

(n)

t

(t) ≤ a + b 0

f (n) (s) ds,

0 ≤ t ≤ T.

4.4 Stochastic Differential Equations

157

Continuous functions f satisfying such a recursive inequality can grow exponentially at most. This is the content of the following lemma of Gronwall from the theory of differential equations. Lemma 4.61 (Gronwall’s Lemma) Let f be continuous on [0, T ] and for t ≤ T let  t (4.24) f (t) ≤ a + b f (s) ds. 0

Then it follows   f (T ) ≤ a 1 + ebT . Proof Let u(t) := e−bt

t 0

f (s) ds. Then u(0) = 0 and by assumption it holds:

   t u (t) = e−bt f (t) − b f (s) ds ≤ ae−bt . 0

It follows: 

T

u(T ) ≤ a 0

e−bt dt =

 a a − e−bT + 1 ≤ . b b

Thus we obtain as a result the desired bound 

T

f (T ) ≤ a + b

  f (s) ds ≤ a + bu(T )ebT ≤ a 1 + ebT .

0

  Thus it follows in the proof of Theorem 4.60: f (n) (T ) < K < ∞, K = KT independent of n. By Fatou’s lemma, it follows: E sup |Xs |2 = E lim sup |Xs |2 ≤ K < ∞. s≤T

n s≤T ∧τn

Thus X satisfies the integrability condition and it follows: X ∈ E. This implies the existence and uniqueness of solutions for t ≤ T0 . n−1 For arbitrary times T we consider the intervals [0, Tn ], [ Tn , 2T n ], . . . , [ n T, T ]. T For n ≥ n0 sufficiently large so that for n ≤ T0 it follows the existence of the (1) solution with initial value X0 = Z in the first interval [0, Tn ]. The bound given in the proof is independent of the initial value. In the second interval we consider as (2) new initial value X0 = X T from the solution on the first interval. This procedure n can be iterated to go through all intervals. In each case, we use the value on the

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4 Elements of Stochastic Analysis

right edge as the new initial value. The solution can be combined consistently, since the bound only depends on EZ 2 . For EZ 2 we had found a universal upper bound. Therefore, the solution intervals do not decrease. So as a result we get a unique solution on [0, T ].  Two examples of the determination of solutions of stochastic differential equations follow. Example 4.62 (Brownian Bridge) A continuous Gaussian process with normally distributed, finite-dimensional marginal distributions B 0 = (Bt0 )0≤t ≤1, on the time interval [0, 1] is called Brownian bridge if EBt0 = 0,

0 ≤ t ≤ 1 and

Cov(Bs0 , Bt0 ) = s(1 − t),

0 ≤ s ≤ t ≤ 1.

Compared to the Brownian bridge, the Brownian motion has the covariance function min(s, t). In particular, (Fig. 4.4) VarBs0 = s(1 − s) and B00 = B10 = 0, i.e., a Brownian bridge starts and ends at zero. The special importance of the Brownian bridge B 0 results from the fact that the empirical process Vn has the Brownian bridge as a limit process, Vn (s) =

 √  n Fn (s) − s , s ∈ [0, 1]

D

Vn −→B 0 . Therefore, as with Donsker’s invariance principle, all continuous functionals of Vn converge to the corresponding functional of the Brownian bridge. In statistics, many

Fig. 4.4 Brownian bridge

4.4 Stochastic Differential Equations

159

empirical quantities of interest are given or approximated by functionals of the empirical process. Thus, one gets the possibility to approximate the distribution of statistical functionals by the distribution of functionals of Brownian bridge. One way to construct a Brownian bridge is as follows: Take a Brownian motion under the condition that it has value zero at the point one (i.e., at the end): d

B 0 = B|B1 = 0. So one considers the paths of Brownian motion that end up in the ‘neighborhood of zero’ at time 1. A second way to construct a Brownian bridge is as follows: Define Bt0 := Bt − tB1 . B 0 is a Brownian bridge, because for this process EBt0 = 0 and EBs0 Bt0 = E(Bs − sB1 )(Bt − tB1 ) = (s ∧ t) − ts − st + st = s(1 − t),

for s ≤ t.

B 0 is a Gaussian process with the covariance of Brownian bridge. Thus B 0 is a Brownian bridge. The Brownian bridge constructed above B 0 is not a process adapted to the filtration of Brownian motion. Indeed, for the definition one uses the value of the Brownian motion at the point t = 1. The question is: Can one also generate a Brownian bridge on the Brownian filtration from the Brownian motion itself? The idea is to construct a Brownian bridge using a differential equation where a drift in the zero direction is introduced. Let the drift b(t, Xt ) be given at time t by b(t, x) := −

x . 1−t

This drift drives the process in the residual time 1 − t in the direction of zero. The corresponding stochastic differential equation is given by dXt = −

Xt dt + dBt . 1−t

(4.25)

To solve this differential equation, we use the ‘variation of coefficients’ method familiar from the theory of differential equations. We consider the solution ansatz    t Xt = a(t) x0 + b(s)dBs , 0

(4.26)

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4 Elements of Stochastic Analysis

with a, b differentiable. With partial integration it follows    t dXt = a  (t) x0 + b(s)dBs dt + a(t)b(t)dBt . 0

If a(0) = 1, a(t) > 0, ∀ t then X solves the following stochastic differential equation 



dXt = aa(t(t)) Xt dt + a(t)b(t)dBt X0 = x0

(4.27)

To solve the differential equation (4.25), we obtain by comparing coefficients 1 a  (t) =− , a(t) 1−t

a(t)b(t) = 1.

1 . Using x0 = 0 it These equations have the solution a(t) = 1 − t and b(t) = 1−t follows:  t 1 Xt = (1 − t) dBs , 0 ≤ t ≤ 1, 0 1−s

is a solution of the differential equation (4.25). Assertion: X is a Brownian bridge. By construction X is a continuous Gaussian process and EXt = 0. For s ≤ t it follows with the independence of the increments of the Brownian motion  s   t  1 1 dBu dBv EXs Xt = (1 − s)(1 − t)E 0 1−u 0 1−v  s 2 1 dBu = (1 − s)(1 − t)E 1 − u 0  s 1 = (1 − s)(1 − t) du = s(1 − t) = Cov(Bs0 , Bt0 ) 2 0 (1 − u) The solution X of the stochastic differential equation (4.25) thus has the covariance of a Brownian bridge Cov(Xs , Xt ) = s(1 − t) = Cov(Bs0 , Bt0 ). Thus X is a Brownian bridge and X is adapted to the filtration of Brownian motion B.

4.4 Stochastic Differential Equations

161

Example 4.63 (Ornstein–Uhlenbeck Process) The Ornstein–Uhlenbeck model dates back to Langevin in 1908 and Ornstein and Uhlenbeck in 1930. The Ornstein– Uhlenbeck process is a model for the velocity of molecular particles and is defined as the solution of the stochastic differential equation "

dXt = − αXt dt + σ dBt , X0 = x0 ,

σ and α are positive constants. The Ornstein–Uhlenbeck model is a modification of the Brownian motion model. Since the paths of Brownian motion are nowhere differentiable, one cannot define a velocity for Brownian motion. The Ornstein–Uhlenbeck process is ‘mean reverting’. For Xt > 0 there is a negative, for Xt < 0 a positive drift proportional to Xt which drives the process towards zero. This ‘mean reverting’ property causes a damping of the paths of the Brownian motion and leads to a stationary limit distribution. However, the paths remain non-differentiable. To solve the differential equation, we use the method for variation of the coefficients in (4.27) with a  (t) = −α, a(t)

a(t)b(t) = σ,

a(0) = 1.

It then follows: a(t) = e−αt ,

b(t) = σ eαt

and we obtain    t Xt = e−αt x0 + σ eαs dBs = x0 e−αt + σ

0



t

e−α(t −s)dBs .

0

X is a Gaussian process with (t → ∞) EXt = x0 e−αt −→ 0  t VarXt = σ 2 e−2α(t −s) ds 0

=

 σ2 1 − e−2αt −→ 2α 2α σ2 

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4 Elements of Stochastic Analysis 2

The variance does not vanish but converges in the limit towards σ2α . The stronger the drift α is, the smaller becomes the variance. The process Xt converges in distribution to the normal distribution  σ2  D , Xt −→ N 0, 2α the stationary distribution of this process. If one starts the Ornstein–Uhlenbeck process in the stationary distribution, then one obtains a stationary process. By analogy with Markov chains in discrete time, the Ornstein–Uhlenbeck process is a Markov process in continuous time and converges for t → ∞ to the stationary distribution. The existence and uniqueness theorems can be dealt with in the same way for the multidimensional case. Let B = (B 1 , . . . , B r ) be an r-dimensional Brownian motion, i.e., a vector of r independent Brownian motions which drives the differential equation. We call the generated filtration (At ). Let b : R+ × Rd −→ Rd (s, x) −→ b(s, x) be a drift vector, σ be a matrix of diffusion coefficients given by a d × r-matrix σ : R+ × Rd −→ Rd×r   (s, x) −→ σij (s, x) and Z ∈ Ld (A0 ) an initial vector. The diffusion process X is then defined by the following stochastic differential equation: ⎧  t  t ⎪ ⎪ X = Z + b(s, X ) ds + σ (s, Xs ) dBs ⎪ t s ⎨ 0 0 t r  t  j i i i ⎪ ⎪ Xt = Z + b (s, Xs ) ds + σij (s, Xs ) dBs . ⎪ ⎩ 0

j =1 0

Theorem 4.64 Let the Lipschitz condition: |b(t, x) − b(t, y)| + |σ (t, x) − σ (t, y)| ≤ K|x − y| and the linear growth condition:   |b(t, x)| + |σ (t, x)| ≤ K 1 + |x| ,

E|Z|2 < ∞,

0 ≤ t ≤ T hold.

(4.28)

4.4 Stochastic Differential Equations

163

Then there exists exactly one solution to the stochastic differential equation (4.28), such that E sup |Xs |2 < ∞. s≤T

Example 4.65 (Linear Differential Equations) For the special class of linear stochastic differential equations of the form dX =  t d×1



 A(t) Xt + a(t) dt + σ (t) dBt ,      d×d d×1

X0 = ξ,

(4.29)

d×r r×1

d×1

with locally bounded, deterministic coefficients A, a, σ , solutions can be given in explicit form. Such linear differential equations arise, for example, in economics, in models with many factors (multi-factor models). To find the solution of (4.29), we consider the corresponding inhomogeneous deterministic linear differential equation where the stochastic term is missing ζ˙ (t) = A(t)ζ (t) + a(t),

ζ (0) = ξ ∈ Rd .

(4.30)

By standard existence and uniqueness statements, there exists a unique solution to (4.30). To determine this solution, we consider the associated homogeneous matrix differential equation: ˙ (t) = A(t) (t), 

(0) = I.

d×d

(t) is a d × d-matrix. The matrix system has a unique solution  the fundamental solution of the associated homogeneous differential equation ζ˙ (t) = A(t)ζ (t). (t) is nonsingular for all t. Because otherwise there exists a λ ∈ Rd , λ = 0, and t0 such that (t0 )λ = 0. But (t)λ is unique solution of the homogeneous equation with ζ (t0 ) = 0. It follows: (t)λ = 0, ∀ t in contradiction to (0) = I . The solution of the deterministic equation (4.30) is now available via the fundamental solution    t −1 (4.31) ζ(t) = (t) ζ(0) +  (s) · a(s) ds . 0

For the stochastic differential equation we get a solution ansatz in a similar way    t  t −1 (s) · a(s) ds + −1 (s) · σ (s)dBs . Xt = (t) X0 + 0

0

(4.32)

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4 Elements of Stochastic Analysis

By means of partial integration (product formula) we can verify with the Itô formula that X is a solution of the linear stochastic differential equation (4.29). Uniqueness follows according to the existence and uniqueness theorems. Remark 4.66 Solutions of the SDE (4.23) are Markov processes, i.e. it holds: E(f (Xt ) | As ) = E(f (Xt )|Xs ) = ψ(Xs ),

for s < t,

with ψ(x) := E(f (Xt ) | Xs = x), as in the case of Brownian motion. The Markov property is a consequence of the flow property. Let (Xst,x )s≥t be a solution of (4.23), starting in x at time t,  Xst,x = x +

s t

 b(u, Xut,x ) du +

s t

σ (u, Xut,x )dBu ,

t ≤ s.

Then the flow property holds: t,Xt0,x

Xs0,x = Xs

,

(4.33)

s > t.

(4.33) is a corollary of the uniqueness of solutions and the continuity of solutions in (s, t, x). In d = 1 there is an existence and uniqueness statement under weakened assumptions on the diffusion coefficient, cf. Karatzas and Shreve (1991, Prop. 5.2.13). Theorem 4.67 (Yamada, Watanabe) d = 1: Let there exist K < ∞ such that |b(t, x) − b(t, y)| ≤ K|x − y| and |σ (t, x) − σ (t, y)| ≤ h(|x − y|), ε for a function h ↑, h(0) = 0 and 0 h−2 (u) du = ∞, ∀ ε > 0 (e.g. h(u) = uα ). Then there exists exactly one solution of the SDE (4.23).

1 2

≤ α ≤ 1 and

Example 4.68 The solution X of the stochastic differential equation dXt = a(b − Xt+ ) dt +



γ Xt+ dBt

with X0 = x ≥ 0, γ > 0, a, b ≥ 0

is called Feller’s branching diffusion with immigration, or in financial mathematics Cox–Ingersoll–Ross model for the time evolution of interest rates. With h(x) = Cx α , α ∈ [ 12 , 1] the condition of rate Theorem 4.67 is satisfied with α = 12 , K = √ γ + α. It holds Xt ≥ 0, ∀ t and Xt > 0, ∀ t > 0 if 2ab γ ≥ 1, Xt = 0 infinitely often with probability 1 if

2ab γ

< 1.

4.4 Stochastic Differential Equations

165

In the special case a = b = 0 (no drift) holds with the help of the Itô formula e

−λXt

−e

−λx

λ2 −γ 2



t

e

−λXs



t

Xs ds = λ

0

 e−λXs γ Xs dBs ∈ M.

0

The Laplace transform ϕ(t, λ, x) = Ex e−λXt of Xt solves the differential equation γ λ2 d λ2 d ϕ(t, λ, x) = γ EXt e−λXt = − ϕ(t, λ, x), dt 2 2 dλ

ϕ(0, λ, x) = e−λx .

The unique solution of this equation is:  ϕ(t, λ, x) = exp

 λ − x . γ /2t + 1

For γ = 2 this is identical to the limit of rescaled Galton–Watson branching processes.

4.4.1

Strong Solution—Weak Solution of Stochastic Differential Equations

The problem of strong solutions of stochastic differential equations concerns the question: Given a probability space ( , A, P ), a Brownian motion B with filtration AB on ( , A, P), we are looking for a solution of (4.23), adapted to AB . By extension of the notion of strong solution, the weak solution has an additional degree of freedom: We are looking for a probability space ( , A, P ), a filtration and a Brownian motion. B with respect to this filtration and a solution X of (4.23) with respect to this Brownian motion B, i.e., (X, B) is a solution of 

t

Xt = X0 + 0 d

X0 = μ,



t

σ (s, Xs ) dBs +

b(s, Xs ) ds 0

(4.34)

the initial distribution of X0 is given and equal to μ.

This additional degree of freedom allows the construction of solutions even in some cases where no strong solution exists. An example of such a situation is the following. Example 4.69 We consider the differential equation dXt = sgn(Xt )dBt ,

X0 = 0,

(4.35)

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4 Elements of Stochastic Analysis

where sgnx := 1(0,∞) (x) − 1(−∞,0) (x) denotes the sign function. (4.35) is equivalent to 

t

Xt = X0 +

(4.36)

sgn(Xs ) dBs 0



t

⇐⇒ Bt =



t

dBs =

0

∀ t ≥ 0.

sgn(Xs )dXs ,

(4.37)

0

(4.37) follows from (4.35) by multiplication by sgnXt . Assertion: There exists a weak solution of (4.36). For let X be a Brownian motion on ( , A, P ), A = AX , then define B by (4.37). Then it follows: t B is a continuous martingale and &B't = 0 (sgn(Xs ))2 ds = t. It follows that B is a Brownian motion. So (B, X) is a weak solution of the differential equation in (4.36). But there is no strong solution ! For let (X, B) be any weak solution. Then as above X is a continuous martingale with &X't = t, that is, a Brownian motion. By an approximation argument (cf. Karatzas and Shreve (1991, p. 562)), one obtains σ (B) ⊂ σ (|Xs |; s ≤ t) ⊂ σ (Xs ; s ≤ t). The second inclusion is proper. So X is not adapted to σ (B). An example of the extended possibilities of weak solutions is provided by the following theorem. In the case σ ≡ 1 only a boundedness, a linear growth and measurability assumption is posed on the drift term b but no Lipschitz condition. Theorem 4.70 The stochastic differential equation dXt = b(t, Xt ) dt + dBt ,

0 ≤ t ≤ T,

(4.38)

  b(t, x) bounded and measurable, and |b(t, x)| ≤ K 1 + |x| , has a weak solution with initial distribution P X0 = μ,

μ ∈ M 1 (Rd , B d ).

Proof Let X = (Xt , At , (Px )x , ( , A)) be a d-dimensional Brownian motion with start in x with respect to Px . Then Zt : = exp

 d  j =1 0

t

j

bj (s, Xs )dXs −

  1 = exp (b · X)t − &b · X't 2

1 2



t 0

 |b(s, Xs )|2 ds

4.4 Stochastic Differential Equations

167

is the exponential martingale. Z is a martingale with respect to Px , because by the linear growth condition and the boundedness of b the Novikov condition is satisfied. We define dQx := ZT . dPx According to the Theorem of Girsanov it follows  Bt := Xt − X0 −

t

b(s, Xs ) ds,

0≤t ≤T

0

is a Brownian motionwith respect to Qx , with Qx (B0 = 0) = 1 ∀ x. Define Qμ (A) := Qx (A)dμ(x), then it follows  Xt = X0 +

t

b(s, Xs ) ds + Bt with respect to Qμ .

0

With respect to Qμ B is a Brownian motion and  (Qμ )

=

X0

 0 QX x dμ(x)

=

εx dμ(x) = μ

0 QX x (A) = 1A (x).

Thus it follows that (X, B, ( , A, Qμ ), (At )) is a weak solution of (4.38).

 

Remark 4.71 (a) The statement of Theorem 4.70 can be generalized as follows. Let X be a solution of the stochastic differential equation dXt = μt dt + σt · dBt . Let νt is another drift such that ϑt :=  Mt = exp With

dQ dP

− ϑt · Bt −

μt −νt σt

1 2



is bounded, then it follows that t

 ϑs 2 ds

∈ Mc (P ).

0

:= MT it follows by Girsanov’s Theorem t = Bt + B



t

ϑs ds is a Brownian motion with respect to Q 0

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4 Elements of Stochastic Analysis

and it holds t with respect to Q. dXt = νt dt + σt dB

(4.39)

Proof: dXt = μt dt + σt · dBt = νt dt + σt · dBt + (μt − νt ) dt = νt dt + σt ( dBt + νt dt ).  t dB

 is thus on the new probability space with measure Q a weak The pair (X, B) solution of (4.39).  (b) Girsanov can also be used to prove the weak uniqueness of solutions, i.e., the uniqueness of the distributions of weak solutions can be proved (cf. Karatzas and Shreve (1991, p. 304)). Definition 4.72 (Pathwise Uniqueness) The solution of the stochastic differential equation (4.23) with initial distribution μ ∈ M 1 (Rd , B d ) is called pathwise unique if for every two weak solutions (X, B1 ), (Y, B2 ) on ( , A, P ) with respect to filtrations (At ) and (Bt ) and initial distribution μ it holds: P (Xt = Yt , ∀ t ≥ 0) = 1. Remark 4.73 For the stochastic differential equation dXt = sgn(Xt ) dBt from Example 4.69 with X also −X is a solution, i.e., pathwise uniqueness does not hold. The solution is however weakly unique, i.e., for two solutions X, Y with the same initial distribution μ it holds P X = P Y . Proposition 4.74 The following statements are equivalent : (a) The stochastic differential equation (4.23) has a unique strong solution. (b) There exists a weak solution and pathwise uniqueness holds (for the initial distribution μ = P X0 ). Under (a), (b) it holds: the solution is weakly unique (Karatzas and Shreve 1991). Definition 4.75 (a) X ∈ C d is called the solution of the local martingale problem with drift b and diffusion a, initial distribution μ (term: X ∈ LMP (a, b, μ)) if:  · b(s, Xs ) ds ∈ Mloc,c P X0 = μ and M := X − 0  t with &M i , M j 't = 0 aij (s, Xs ) ds ∀ i, j, t ≥ 0

(4.40)

4.4 Stochastic Differential Equations

169

(b) The solution of LMP (a, b, μ) is unique, if for two solutions X and Y it holds: PX = PY. The local martingale problem is defined by the martingale property of M. There is an important connection between the notion of weak solution and the local martingale problem. Theorem 4.76 (Weak Solution—Local Martingale Problem) (a) X is exactly then a solution of the local martingale problem LMP (σ σ T , b, μ)  =:a

 ⊃ such if there exists a Brownian motion B, if necessary, on an extension that (X,B) is a weak solution of (4.23). (b) There exists a unique weak solution of (4.23) with initial distribution μ ⇐⇒ The local martingale problem LMP (σ σ T , b, μ) is uniquely solvable. Proof We give the proof for m = d = 1; d is the dimension and m is the number of Brownian motions. We consider the stochastic differential equation dX = σ (t, Xt ) dBt + b(t, Xt ) dt.  t    d×1

d×m

m×1

d×1

“⇐” If (X, B) is a weak solution of the stochastic differential equation in (4.23), then it follows  t  t Xt − b(s, Xs ) ds = σ (s, Xs )dBs ∈ Mloc,c . 0

0

So X solves the local martingale problem LMP (σ σ T , b, μ). t “⇒” Now be X be a solution of LMP (σ 2 , b, μ) and Mt := Xt − 0 b(s, Xs ) ds ∈  on  ⊃ and a Brownian motion B Mloc,c . Then there exists an extension    ( , A, P ), such that (cf. e.g. Klenke (2006, p. 542)) 

t

Mt =

s . |σ (s, Xs )|dB

0

With Bt :=

t 0

s then the following representation holds sgn(σ (s, Xs ))dB 

t

Mt =

σ (s, Xs )dBs . 0

So (X, B) is a weak solution of the stochastic differential equation (4.23).

 

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4 Elements of Stochastic Analysis

The connection between weak solutions and the local martingale problem can now be used to prove general existence theorems for weak solutions and for the local martingale problem, respectively. The formulation of the local martingale problem is particularly suitable for the application of approximation arguments and convergence in distribution. Theorem 4.77 Let b, σij R+ × Rd → R continuous and bounded. Then for all initial distributions μ ∈ M 1 (Rd , B d ) with x2m dμ(x) < ∞ for a m > 1 there exists a solution of the local martingale problem LMP (a, b, μ) with a = σ σ T , and a weak solution of the stochastic differential equation in (4.23). Proof Sketch We first introduce a discretized version of the stochastic differential equation. Let tjn := 2δn and n (t) = tjn for t ∈ (tjn , tjn+1 ] the dyadic rational approximation of g. Then for y ∈ Cd ([0, ∞)) b(n) (t, y) := b(y(n (t))),

σ (n) (t, y) := σ (y(( t))))

is progressively measurable. Let further be (Bt , AB t ) be a d-dimensional Brownian motion on ( , A, P ) and ξ a random variable independent of B with P ξ = μ. We define the sequence (X(n) ) = (n) (Xt , At ) recursively by an Euler scheme: X0(n) := ξ (n)

Xt

     (n) (n) (n) (n)  := Xtj + b X (n) (t − tj ) + σ X (n) Bt − Bt (n) , tj

tj

j

(n)

tj

(n)

< t ≤ tj +1 .

Then, by definition of b(n), σ (n) : Xt(n) = ξ +



t 0



t

b(n) (s, X(n) ) ds +

σ (n) (s, X(n) )dBs .

(4.41)

0

X(n) thus solves the modified stochastic differential equation with approximate coefficients. Now, as in the proof of the general existence and uniqueness theorem (Theorem 4.60), the following estimate is also valid for moments of order 2m > 2 (cf. Karatzas and Shreve (1991, Problem 3.15)). The proof uses the BurkholderDavis inequality instead of Doob’s inequality. If b(t, y)2 + σ (t, y)2 ≤ K(1 + maxs≤t y(s)2 ) and if (X, B) is a weak solution of the diffusion equation (4.23) with EX0 2m < ∞ then for all T < ∞

and

E maxs≤t Xs 2 m ≤ C(1 + EX0 2 m )eCt , t ≤ T EXt − Xs 2 m ≤ C(1 + EX0 2 m )(t − s)m , s < t ≤ T . (4.42)

4.4 Stochastic Differential Equations

171

For the solution X(n) of the discretized stochastic differential equation in (4.41) thus it holds: (n)

sup E|Xt n

− Xs(n) |2 m ≤ C(1 + EEξ 2 m )(t − s)m ,

s ≤ t < T.

(n)

be the associated probability measure on Cd = Let P (n) := P X C([0, ∞), Rd ). Then it follows that the sequence (P (n) ) is tight, because it satisfies the stochastic Arzela–Ascoli criterion: (n)

∃ ν, α, β > 0 : ∀ n ∈ N : EX0 ν ≤ M and (n)

EXt

− Xs(n) α ≤ CT |t − s|1+β ,

∀ s, t ≤ T , ∀ T . D

exists a convergent subsequence. W.l.g. it holds P (n) −→P ∗ , i.e.  So there  (n) f dP → f dP ∗ for all continuous, bounded functions f on Cd . We assert: (1) P ∗ ({y ∈ Cd ; y(0) ∈ }) = μ(), i.e. P ∗ has the initial distribution μ. For f ∈ Cb (Rd ) it holds 

∗

E f (y(0)) = lim E n

D

(n)

f (y(0)) =

f dμ,

D

since P (n) −→ P ∗ implies that μ = (P (n) )π0 −→ (P ∗ )π0 . So P ∗ has the initial distribution μ. (2) The standard construction on Cd (πt ), P ∗ ) is a solution of the associated martingale problem, i.e. ∗



t

E (f (y(t)) − f (y(s)) −

Au f (y(u)) du | As ) = 0[P ∗ ]

(4.43)

s 2 (Rd ) and with the associated differential operator for 0 ≤ s < t < ∞, f ∈ CK

 2  1 ∂ f (y(t)) At f (y) = b(t, y) · ∇f (y(t)) + σ T (t, y) σ (t, y). 2 ∂xi ∂xj With Itô’s formula, it follows that the formulation in (4.43) is equivalent to the formulation in the definition of the local martingale problem.

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4 Elements of Stochastic Analysis

X(n) solves the stochastic differential equation with discretized coefficients. Therefore, by Theorem 4.76 X(n) is a solution of the associated local martingale 2 (Rd ) it holds: problem, i.e., for f ∈ CK    t f,n A(n) f (y) du, A ∈ Mc (P (n) ), Mt := f (y(t)) − f (y(0)) − t u 0

2 f ∈ CK

(n)

with At f (y) := b(n) (t, y) · ∇f (y(t)) + 12 (σ (n) )T (t, y)D 2 f (y(t))σ (n) (t, y). It follows for As -measurable g ∈ Cb (Cd ) : " E

(n)

 f (y(t)) − f (y(s)) −



t s

# A(n) u f (y(u))du

g(y) = 0.



=Fn (y)=Fns,t (y)

t Let F (y) := f (y(t))−f (y(s))− s Au f (y)(u) du, then it holds: Fn (y)−F (y) → 0 uniformly on compacta of Rd (cf. Karatzas and Shreve (1991, p. 325)). For K ⊂ Cd compact it holds M := sup y(s) < ∞ and limn supy∈K ωt (y, 2−n ) = y ∈ K, 0≤s≤t 0, ∀ t < ∞ with the modulus of continuity ωt on [0, t]. b, σ are uniformly continuous on {y; y ≤ M}. It follows that sup {b(n) (s, y) − b(y(s)) + σ (n) (s, y) − σ (y(s))} ≤ ε,

∀ n ≥ nε .

0≤s≤t

Thus it follows that E ∗ F (y)g(y) = 0, ∀ g. Thus ((πt ), P ∗ ) is a solution of the local martingale problem (4.43). By Theorem 4.76, it, therefore, follows that there exists a weak solution to the stochastic differential equation. Remark 4.78 (a) An analogous existence statement for weak solutions holds also in the timeinhomogeneous case with bounded continuous coefficients b = b(t, x), σ = σ (t, x). (b) From the solvability of the Cauchy problem ∂u ∂t = Au, u(0, ·) = f , u bounded [0, T ] × Rd weak uniqueness of solutions follows (Stroock and Varadhan 1979; Karatzas and Shreve 1991). Sufficient for the solvability of the Cauchy problem is that the coefficients are bounded and Hölder continuous and that the diffusion matrix a is uniformly positive definite.

4.5 Semigroup, PDE, and SDE Approach to Diffusion Processes

4.5

173

Semigroup, PDE, and SDE Approach to Diffusion Processes

The aim of this section is to describe the construction of diffusion processes by means of semigroup theory, partial differential equations (PDE)—in particular the Kolmogorov equations—and by means of stochastic differential equations (SDE), and to discuss the interactions of these approaches. Let X = (Xt )t ≥0 be a Markov process in Rd with transition probabilities Ps,x (t, A) := P (Xt ∈ A|Xs = x),

A ∈ B d , 0 ≤ s < t ≤ T , x ∈ Rd .

Then the Chapman–Kolmogorov equation holds  Ps,x (t, A) =

Rd

Pu,z (t, A)Ps,x (u, dz)

∀ s < u < t, x ∈ Rd .

Examples are the Brownian motion B with transition kernel Ps,x (t, A) = (2π(t − s))

− d2



e−|y−x|

2 /2(t −s)

dy

A

or the multidimensional Ornstein–Uhlenbeck process with transition kernel 

Ps,x (t, A) = π(1 − e

−2(t −s)

)

"

− d  2

exp A

2 # y − e−(t −s)x dy. 1 − e−2(t −s)

Definition 4.79 An Rd -valued Markov process (Xt )0≤t ≤T is called diffusion process, if ∀ t ≤ T , x ∈ Rd , c > 0 it holds: 

1 ε {|y−x|≥c} Pt,x (t + ε, dy) = 0 1 limε0 ε {|y−x| y. The above argument can be carried out exactly and yields the following existence statement. −

Theorem 4.89 Let , a be continuous and , a satisfy the Lipschitz condition and the linear growth condition in x. Further let there exist a c > 0 such that a(t, x) ≥ c, t ≤ T , x ∈ R. Then it follows: The Kolmogorov backward equation (4.47) has a unique transition function F as a solution. There exists a continuous Markov process X with Ps,x ∼ Fs,x , and X is a diffusion 1 process with coefficients , σ = |a| 2 .

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181

Remark 4.90 (a) An analogous result holds in dimension d ≥ 1 under the condition: ∃c > 0, ∀ t and ∀ v ∈ Rn \ {0} the ‘uniform ellipticity inequality’ holds v 2 a(t, x)v ≥ c|v|2 . (b) If X is stationary, then for all s ≥ 0: Ft (x, y) = Fs,x (s + t, y). Therefore, it follows: ∂ Fs,x (s + t, y) = 0. ∂s Thus, the stationarity equation holds ∂ ∂ Ft (x, y) = − Fs,x (u, y)|u=s+t . ∂t ∂s From this follows the Kolmogorov equation in the stationary case: ⎧ ∂ ∂ 1 ∂2 ⎪ ⎨ ∂t Ft (x, y) = (x) ∂x Ft (x, y) + 2 a(x) ∂x 2 Ft (x, y) " 1, x < y, ⎪ ⎩ limt 0 Ft (x, y) = 0, x > y.   (c) Let u(t, x) := f (y)Pt (x, dy) = f (y)Ft (x, dy) where f ∈ Cb(2) . From the above equation it then follows 

= Au d2 u(0, x) = f (x), A := (x) dx + 12 a(x) dx 2, ∂u ∂t

(4.48)

i.e., u is a solution of the Cauchy problem to the differential operator A. A is the infinitesimal generator of the Markov process X. (d) Kolmogorov forward equation Let ps,x (t, dy) = ps,x (t, y) dy be the transition density of a Markov process X. Then, according to Chapman–Kolmogorov, it follows for s ≤ u ≤ t  ps,x (t, y) =

pu,z (t, y)ps,x (u, z) dz

The Kolmogorov forward equation, also called Fokker–Planck equation, for p is obtained by variation at a time point in the future. Theorem 4.91 (Kolmogorov Forward Equation) a,  satisfy the conditions of ∂a ∂ 2 a Theorem 4.89 and in addition let ∂ ∂x , ∂x , ∂x 2 satisfy the Lipschitz and the linear growth condition.

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Then the Kolmogorov forward equation for the transition density p  1 ∂2   ∂ ∂  ps,x (t, y) = − (t, y)ps,x (t, y) + a(t, y)p (t, y) s,x ∂t ∂y 2 ∂y 2

(4.49)

lim ps,x (t, y) = δx (y)

t s

has a unique solution ps,x . There exists exactly one diffusion process X with transition density p. Proof (Idea) Let X  be a diffusion process with transition density p to the above data. Let ϑ(t) := ξ(y) ps,x (t, y) dy, ξ ∈ Cb2 , and let ε > 0. The Chapman–Kolmogorov equation applied to ϑ(t + ε) yields: ϑ(t + ε) =





 ξ(y)

pt,z (t + ε, y)ps,x (t, z) dz dy

  =

 ξ(y)pt,z (t + ε, y) dy ps,x (t, z) dz

  ≈

 ξ(z) + ξ  (z)(y − z) 1 + ξ  (z)(y − z)2 pt,z [t + ε, y] dy 2

 ps,x (t, z) dz

   1   = ξ(z) + ξ (z)ε(t, z) + ξ (z)εa(t, z) ps,x (t, z) dz 2 R    1 = ϑ(t) + ε ξ  (z)(t, z) + ξ  (z)a(t, z) ps,x (t, z) dz. 2 It follows with partial integration    1 ξ  (z)(t, z) + ξ  (z)a(t, z) ps,x (t, z) dz 2    1 ∂2 ∂ (a(t, z)p (t, z)) dz. = ξ(z) − ((t, z)ps,x (t, z)) + s,x ∂z 2 ∂z2

ϑ  (t) =

On the other hand, by definition 

ϑ (t) =





∂ ps,x (t, z) ξ(z) ∂t

 dz,

∀ ϑ ∈ Cb2 .

From this, by comparing the integrands, the Kolmogorov forward equation follows in (4.48).  

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In comparison to the backward differential equation in Theorem 4.89, the forward differential equation in Theorem 4.91 is stated for the transition densities p (instead of the transition distribution function F ). This leads to the relatively stronger assumptions in Theorem 4.91. An analogous result to Theorem 4.91 also holds in the multivariate case d ≥ 1. 3. Approach Through Stochastic Differential Equations (Itô Theory) We assume that the coefficients √ , a satisfy the conditions of Theorem 4.89. Let d = 1 and let σ (t, x) = a(t, x). Then |a(t, x) − a(t, y)| 1 |σ (t, x) − σ (t, y)| = √ ≤ √ |a(t, x) − a(t, y)|. √ 2 c a(t, x) + a(t, y) Thus σ satisfies the Lipschitz condition and the linear growth condition holds. Thus, by the existence and uniqueness theorem, there exists a unique solution of the stochastic differential equation (4.23). Theorem 4.92 Under the above assumptions, the unique solution X of the stochastic integral equation 

t

Xt = ξ + 0



t

σ (s, Xs )dBs +

(s, Xs ) ds 0

is a diffusion process with coefficients , a. The transition distribution function Fs,x (t, y) from Xt for t, y is fixed, is the unique solution of the Kolmogorov backward equation. Under the additional conditions in Theorem 4.91, there exists a transition density ps,x (t, y). p is the unique solution of the Kolmogorov forward equation. Remark 4.93 If d ≥ 1 then the solution σ (t, x) of σ (t, x)σ (t, y)T = a(t, x) is not unique in general. If e.g. U is an orthogonal matrix, U U T = I, then σ U is also a solution of this equation and U B is also a Brownian motion. The transition distribution function of the solution X of the stochastic differential equation is independent of the choice of σ . We compare the presented methods for constructing a diffusion process X by means of a special case: Comparison of Methods Let d = 1, a(x) = 2 and (x) = −x. Thus, the problem to solve is to construct a diffusion process in d = 1 with coefficients a(x) = 2 and (x) = −x and the infinitesimal generator Af (x) = f  (x) − xf  (x).

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(1) Semigroup theory To apply the semigroup theory, one has to check that (I − n1 A)−1 exists, and whether (I − n1 A)−1  ≤ 1 for a n ∈ N; a task which is already not easy in this example; the spectrum of the operator has to be investigated. (2) PDE method We analyze the Kolmogorov forward equation ∂ ∂t pt (x, y)

=

∂ ∂2 ∂y (ypt (x, y)) + ∂y 2 pt (x, y)

limt ↓0 pt (x, y) = δx (y).

(4.50)

We assume that the stationary limit p(y) = lim pt (x, y) exists, and is independent of x. For the density of the invariant measure for the diffusion X we obtain the stationarity condition: ∂t∂ p(y) = 0. Then it follows from (4.50) in the limit the equation for the stationary density: p(y) + yp (y) + p (y) = 0. This equation has as solution the density p(y) = √1 e−y /2 . 2π These preliminary considerations lead us to the approach with undetermined 2

coefficients λ(t), ϑ(t): pt (x, y) = For this we get ∂ pt (x, y) = pt (x, y) · ∂t



√ 1 2πϑ(t )

·e

−(y−λ(t)x)2 2ϑ(t)

.

 ϑ  (t)(y − λ(t)x)2 λ (t)x(y − λ(t)x) ϑ  (t) + + . 2ϑ(t) 2ϑ 2 (t) ϑ(t)

Further it holds ∂ ∂2 (ypt (x, y)) + 2 pt (x, y) ∂y ∂y   1 y(y − λ(t)x) (y − λ(t)x)2 + − = pt (x, y) 1 − . ϑ(t) ϑ 2 (t) ϑ(t) Substituting in (4.50) results in −

ϑ  (t) (y − λ(t)x)2 λ (t)x(y − λ(t)x) + ϑ  (t) + 2ϑ(t) ϑ(t) 2ϑ 2 (t) =1−

1 y(y − λ(t)x) (y − λ(t)x)2 + − . ϑ(t) ϑ(t) ϑ(t)

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185

Coefficient comparison yields the equivalence of this differential equation with: ϑ  (t) = −2ϑ(t) + 2 and ϑ  (t)λ(t) + 2ϑ(t)λ (t) = −2λ(t). From the initial conditions it follows: ϑ(0) = 0, λ(0) = 1. This implies the solution ϑ(t) = 1 − e−2t ,

λ(t) = e−t

and thus the transition density pt (x, y) = 

  (y − e−t x)2 . exp − 2(1 − e−2t ) 2π(1 − e−2t ) 1

p is the transition density of the Ornstein–Uhlenbeck process. (3) SDE Method To determine the diffusion process by means of stochastic differential equations √ with. a(x) = 2, (x) = −x, and thus σ (x) = 2. The stochastic differential equation to be solved is : dXt =

√ 2dBt − Xt dt where X0 = 0,

or the equivalent integral equation  t√  t Xt = x0 + Xs ds. 2dBs − 0

0

Its solution results directly from the Itô formula (cf. Example 4.63) Xt = e−t x0 +

√  t −(t −u) e dBu . 2 0

It follows that Xt is normally distributed:    t   e−(t −u) du = N e−t x0 , 1 − e−2t Xt ∼ N e−t x0 , 2 0

and thus the transition density is given in explicit form. The SDE method is the easiest to carry out for this example.

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PDEs, Infinitesimal Generator, and SDEs In the following, we deal with some statements on the relation between PDEs, infinitesimal producer and SDEs. From the semigroup property in (4.48) or the t martingale property of f (Xt , t) − f (x, 0) − 0 Af (Xs , s) ds, where Af =

  ∂f ∂ 2f ∂f + As f, As f (x) = bi (x) + aik (x) , ∂s ∂xi ∂xi ∂xk i

i,k

it follows that u(t, x) = Ex f (t, Xt ) is the solution of a partial differential equation. Corollary 4.94 If X is a weak solution of the stochastic differential equation (4.28), 1,2 and if σij is bounded on the support supp f , then it holds: f ∈ CK Ex f (t, Xt ) = f (0, x) + Ex

 t 0

∂f + As f ∂s

 (s, Xs ) ds.

(4.51)

2: In the homogeneous case let for f ∈ CK

Af (x) = lim t ↓0

Ex f (Xt ) − f (x) t

be the infinitesimal generator. Then it follows from (4.51) using the theorem of majorized convergence: 1 Af (x) = lim Ex t ↓0 t



t

Af (Xs ) ds = Af (x).

0

2 , let X be a solution of the stochastic differential Corollary 4.95 Let f ∈ CK equation

dXt = b(Xt ) dt + σ (Xt )dBt and let b, σi,j be bounded on supp f . Then it follows: Af (x) = Af (x) =

 i

bi (x)

∂f 1 ∂ 2f + aik (x) . ∂xi 2 ∂xi ∂xk i,j

The infinitesimal generator A of the diffusion process given by stochastic differential equation is identical with the corresponding diffusion operator A.

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187

Remark 4.96 (a) Corollary 4.95 also holds in the nonstationary case. 2 and u(t, x) := E f (X ) it holds: For f ∈ CK x t ∂ u = Au. ∂t

u(0, x) = f (x),

u is, therefore, the solution of the Cauchy problem. For 1 (Ex f (Xt +u ) − f (Xt )) u   f (Xt +u ) − f (z)  = Ex E  Xt = z −→ Ex Af (Xt ) = Au(t, x), u→0 u 2. if the exchange of expectation and limit is possible, such as for f ∈ CK As a special case we have the (one-dimensional) heat equation

1 ∂ 2u ∂u , = ∂t 2 ∂x 2

u(0, x) = f (x).

(4.52)

The transition density of the Brownian motion 1 p(t, x, y) = √ exp(−(x − y)2 /2t) 2ut is a fundamental solution of the above equation 1 ∂ 2p ∂p . = ∂t 2 ∂x 2 If



e−ax |f (x)| dx < ∞ then 2

 u(t, x) := Ex f (Bt ) =

f (y)p(t, x, y) dy

1 for 0 < t < 2a is a solution of the heat equation (4.52). Similarly, for d ≥ 1 one can construct the solution with the multivariate Brownian motion. (b) Corollary 4.94 also holds for stopping times using the optional sampling theorem 2 , τ be a stopping time with E τ < ∞, then it holds: Let f ∈ CK x



τ

Ex f (Xτ ) = f (x) + Ex 0

Af (Xs ) ds

(Dynkin f ormula).

(4.53)

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4 Elements of Stochastic Analysis

Examples of the Connection with PDE  ∂  2 bi ∂xi , Let A be the semi-elliptic differential operator A = aij (x) ∂x∂i ∂xj + where a = (aij ) ≥ 0 is positive semidefinite. (1) Dirichlet problem Let D ⊂ Rd be a bounded domain and consider the Dirichlet problem on D " (D)

Au = 0 in D u = f on ∂D

(4.54)

for a continuous function f ∈ C(∂D). To construct a solution to this problem, let X be an Itô diffusion i.e. solution of dXt = b(Xt ) dt + σ (Xt )dBt ,

1 T σ σ = a. 2

We define by analogy with the case of Brownian motion for the classical Dirichlet problem: u(x) := Ex f (XτD ), where τD = inf{t; Xt ∈ ∂D}. Assuming that Px (τD < ∞) = 1, x ∈ D, i.e. X leaves the domain D in finite time, it follows Ex τD < ∞, ∀ x ∈ D. Thus it follows as in the classical case: u is a unique solution of the Dirichlet problem (D). The uniqueness proof from Remark 4.16 b) using the Itô formula can be transferred analogously. Thus u is a unique solution of the Dirichlet problem. Sufficient for the existence of a solution is the condition ∃ i : inf aii (x) > 0 x∈D

This condition implies in particular, the ellipticity condition, A > 0. It follows from the uniform ellipticity : x 2 A(x)x ≥ δ|x|2. (2) Cauchy problem Let f = f (x), g = g(t, x) ≥ 0, and let v ∈ C 1,2 be polynomially bounded in x and be a solution of the parabolic differential equation "

d − ∂v ∂t + kv = At v + g in [0, T ] × R v(T , x) = f (x).

(4.55)

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189

The parabolic differential equation (4.55) has time-dependent coefficients, a potential kernel k and a Lagrangian term g. The solution v has the Feynman– Kac representation    v(t, x) = Et,x f (XT ) exp −

k(u, Xu ) du t



T

+



T

"



T

g(s, Xs ) exp −

t

#

 k(u, Xu ) du ds .

t

In the special case of the Kolmogorov backward equation with g = k = 0 holds v(t, x) = Et,x f (XT ) = E(f (XT ) | Xt = x). In particular, as a consequence of the Feynman–Kac representation, the uniqueness of the solution is obtained. The proof of this representation follows by applying Itô’s formula to 



s

−

v(s, Xs ) exp

 k(u, Xu ) du .

t

The solution of parabolic differential equations of the form (4.55) is thus in a unique relation to the determination of expected values for functions of diffusion processes. (3) Poisson equation Let D ⊂ Rd be a bounded domain. The Poisson equation is a generalization of the Dirichlet problem. It is given by : "1

= −g in D u = f on ∂D.

2 u

(P )

The Poisson equation (P ) has the solution   u(x) = Ex f (BτD ) +

τD

 g(Bt ) dt ,

x ∈ D.

(4.56)

0

In general, for a diffusion operator A solutions for equations of the type "

Au = −g in D u=f on ∂D

are obtained in the form (4.56), where the Brownian motion B is to be replaced by a diffusion process X and where Ex τD < ∞, x ∈ D is assumed.

5

Option Prices in Complete and Incomplete Markets

Already in the introductory Chap. 1 an introduction is given in a non-technical way to the basic principles of the theory of arbitrage-free prices, the hedging principle and the risk-neutral price measure, which leads from the binomial price formula by approximation to the Black–Scholes formula. The necessary link from processes in discrete time (binomial model) to those in continuous time (Black–Scholes model) is given by approximation theorems for stochastic processes in Chap. 2. Theorems of this type allow an interpretation of continuous financial market models using simple discrete models such as the Cox–Ross–Rubinstein model. This chapter introduces the basics of general arbitrage-free pricing theory. Fundamental are the first and second fundamental theorems of asset pricing and the associated risk-neutral valuation formula, which is based on valuation by means of equivalent martingale measures. For their construction, Girsanov’s theorem proves to be very useful. For the standard options, it can be used to determine the corresponding pricing formulas (Black–Scholes formulas) in a simple way. The determination of the corresponding hedging strategies leads in the Black–Scholes model (geometric Brownian motion) to a class of partial differential equations, the Black–Scholes differential equations, going back to the fundamental contributions of Black and Scholes as well as the connection with stochastic calculus by Merton in 1969. This led in the period 1979–1983 to the development of a general theory of arbitrage-free pricing for continuous-time price processes and the important role of equivalent martingale measures for this purpose by Harrison, Kreps and Pliska. Central topics of this general theory are the completeness and incompleteness of market models, the determination of associated arbitrage-free price intervals via the equivalent martingale measures and the corresponding sub- or super-hedging strategies (optional decomposition theorem).

© Springer-Verlag GmbH Germany, part of Springer Nature 2023 L. Rüschendorf, Stochastic Processes and Financial Mathematics, Mathematics Study Resources 1, https://doi.org/10.1007/978-3-662-64711-0_5

191

192

5.1

5 Option Prices in Complete and Incomplete Markets

The Black–Scholes Model and Risk-Neutral Valuation

We consider a continuous-time financial market model with time horizon T and two securities, a risk-free bond (bond) St0 and a risky security (stock) St (Fig. 5.1). The general task of financial mathematics is to describe the performance of the stock in comparison with the bond and, in particular, to determine a ‘correct’ price for functionals of the stock (options). Example 5.1 (Black–Scholes Model) In the Black–Scholes model, the bond S 0 is described by dSt0 = rS0t dt

d. h. St0 = ert with a deterministic interest rate r.

The stock S is given by a geometric Brownian motion, i.e., a solution of dSt = St (μ dt + σ dBt ). S has the explicit representation   1  St = S0 exp σ Bt + μ − σ 2 t , 2

t ∈ [0, T ].

The geometric Brownian motion model has two parameters, the volatility σ and the drift μ. The ‘log returns’ ln SS0t of S are given by a Brownian motion with drift

a = μ − 12 σ 2 , and diffusion coefficient σ . Thus, the log returns have independent increments, but not the process S itself. The reason for this modeling is that empirically, the relative increments St − Su St St = − 1 ≈ ln Su Su Su

turn out for u ∼ t to be independent. This leads to the fact that ln SS0t is modeled as a process with independent increments. Therefore, a natural modeling of security

Fig. 5.1 Financial market model

5.1 The Black–Scholes Model and Risk-Neutral Valuation

193

processes is given by the geometric Brownian motion or more generally by a model with exponential Lévy processes St = exp(Xt ). A discrete analogue of geometric Brownian motion is the multiplicative model = S0 S (n) k n

k -

Yi(n)

i=1

= S0 exp

k 

(n)

ln Yi

 .

i=1

If one defines for suitable constants un ≥ r ≥ dn    pn , un (n) P Yi = = dn 1 − pn , then S (n) approximates the model of geometric Brownian motion S if  r  n(pn log un + (1 − pn ) log dn ) = nμn ≈ a = μ − σ 2 2   and n pn log u2n + (1 − pn ) log dn2 − μ2n ≈ σ 2 .

 √σ These conditions are satisfied, for example, if un = e n , dn = u1n , pn = 12 1 +  a √ . σ n This discrete model with one up and one down jump un resp. dn is the classical Cox–Ross–Rubinstein model used as an approximation for geometric Brownian motion in practice. In the general case, the market model is described by a semimartingale. We now introduce some basic terms of option price  theory.  A trading strategy ,  = (t ) = ϕt0 , ϕt ∈ L(P) is a predictable adapted process consisting of two parts: ϕt0 gives the share of bonds at time t in the portfolio ϕt gives the share of stocks at time t in the portfolio T The first general assumption is that ϕ 0 , ϕ are integrable: 0 |ϕt0 | dt < ∞, ϕ ∈ L0 (S). The value of the portfolio at time t then is Vt () = ϕt0 St0 + ϕt St . Self-financing Trading Strategies In discrete time self-financing trading strategies n are defined by the property that at time n only a regrouping of the portfolio is undertaken, i.e., the value of the

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5 Option Prices in Complete and Incomplete Markets

regrouped portfolio is Tn+1 · Sn is identical to the value at time n: Tn · Sn = Tn+1 · Sn ⇐⇒ Vn+1 () − Vn () = Tn+1 (Sn+1 − Sn ). The analogous term in continuous time is described by the equation dVt () = ϕt0 dSt0 + ϕt dSt  t  t ⇐⇒ Vt () = V0 () + ϕs0 dSs0 + ϕu dSu 0

0

$ % Let  :=  = (ϕ 0 , ϕ) ∈ L0 (S);  is a self-financing trading strategy be the set of all self-financing trading strategies. Discounted Price Process  t () := e−rt Vt () the St := e−rt St denotes the discounted price process, V discounted value of the portfolio  at time t. The reference variable St0 acts thereby as ‘numéraire’. Lemma 5.2 Let  = (ϕ 0 , ϕ) ∈ L0 (S), then it holds: t () = V0 () +  ∈  ⇐⇒ V



t

ϕu d Su ,

∀ t > 0.

0

t () = e−rt Vt () it holds with Proof “⇒”: (for S continuous) For  ∈  and V partial integration t () = −r V t () dt + e−rt dVt () dV   = −re−rt ϕt0 ert + ϕt St dt + e−rt ϕt0 d(ert )ϕt dSt   = ϕt − re−rt St dt + e−rt dSt = ϕt d St . Thus it follows t () = V0 () + V



t

ϕu d Su

0

“⇐”: analogously

 

5.1 The Black–Scholes Model and Risk-Neutral Valuation

195

Remark 5.3 If  ∈  and ϕ 0 , ϕ are of finite variation, then, with partial integration, it holds dVt () = d(ϕt0 St0 ) + d(ϕt St ) = ϕt0 dSt0 + St0 dϕt0 + ϕt dSt + St dϕt , as [ϕ, S] = 0, = ϕt0 dSt0 + ϕt dSt . It follows St0 dϕt0 + St dϕt = 0, i.e. the change in value due to reallocation in the bond is identical to the negative of the change in value due to reallocation in the stock. A basic principle of price theory is the no-arbitrage principle, which states that no risk-free profit is possible in a ‘realistic market model’. A fundamental question is to describe price models that satisfy the no-arbitrage principle. Definition 5.4  ∈  is called an arbitrage strategy if V0 () = 0, VT () ≥ 0 [P ] and P (VT () > 0) > 0, i.e., there exists a risk-free profit from an initial value of zero. Remark 5.5 (Extended No-Arbitrage-Principle) For an extended portfolio consisting of securities (stocks, bonds) S i and derivatives Ci we assign to a strategy π = ((ϕ i ), (ai )) with ϕ i shares of S i , ai shares of Ci , ai = ai (t, ω) the value Vt (π) = ϕt0 St0 +



ϕti Sti +



ai (t, ·)Ci

to. The extended no-arbitrage principle states that no-arbitrage strategy exists in this extended market model. Definition 5.6 Let (S 0 , S, P ) be a market model. Q ∈ M 1 ( , A) is an equivalent (local) martingale measure to P , if: (a) Q ∼ P , the measures Q and P are equivalent. (b)  S ∈ Mloc (Q). Me = Me (P ) denotes the set of equivalent (local) martingale measures. Elements from Me are also called risk neutral measures.

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5 Option Prices in Complete and Incomplete Markets

For self-financing trading strategies and equivalent martingale measures Q ∈ Me it holds  t   ϕu d Su ∈ Mloc (Q); Vt () = V0 () + 0

the discounted value process is equal to the initial value plus the discounted profit () only a local martingale. process by trading from the stock. In general V The absence of arbitrage opportunities is a central economic requirement for a stochastic market model. However, already in discrete models arbitrage strategies typically exist for infinite time horizons, as can be seen from doubling strategies. In models in continuous time, such strategies already exist for finite time horizons. We, therefore, need to restrict the set of strategies appropriately. Definition 5.7 t () is a (a) For Q ∈ Me ,  ∈  is called Q-regular,  ∈ r (Q) if V supermartingale  concerning w.r.t. Q. Let r = Q∈Me r (Q) be the set of regular trading strategies. (b) Let Q ∈ Me (P ) then a trading strategy  ∈ b := { ∈  |∃ c ∈ t () ≥ c, ∀ t ∈ [0, T ]} is called Q-admissible, if V T () is a R, so that V martingale with respect to Q, Let a (Q) be the set of Q-admissible self-financing trading strategies and let a = Q∈Me (P ) a (Q) be the set of admissible strategies. Remark 5.8 t () ∈ Mloc (Q), and since (a) It holds that b ⊆ r , because for  ∈ b , V   Vt () ≥ c is bounded below, Vt () is a supermartingale. (b) For non-negative processes S i ≥ 0 it is sufficient to poseinstead of the boundedness from below the weaker condition V () ≥ − ni=1 ci S i with ci ≥ 0. The existence of equivalent martingale measures excludes the existence of regular arbitrage strategies. Theorem 5.9 Let Me (P ) = ∅ then no regular arbitrage strategy exists. t () is a Proof Let Q ∈ Me (P ),  ∈ r then the discounted value process V supermartingale with respect to Q. It follows that: T () ≤ EQ V 0 (). EQ V

(5.1)

5.1 The Black–Scholes Model and Risk-Neutral Valuation

197

0 () = 0, V T () ≥ 0, but this Suppose  would be an arbitrage strategy, then V implies according to (5.1) T () ≤ 0 also EQ V

T () = 0[Q]. V

T () = 0 [P ] and thus a contradiction to the definition of From this also follows V an arbitrage strategy.   In the discrete case, also the converse of Theorem 5.9 holds, i.e., no-arbitrage is equivalent to Me (P ) = ∅. In the continuous case, one needs a modification of the no-arbitrage notion. Definition 5.10 (a)  ∈  is δ-admissible, δ > 0 if V0 () = 0 and Vt () ≥ −δ, ∀ t ∈ [0, T ]. (b) S fulfills the NFLVR condition (No f ree lunch with vanishing risk) P

⇐⇒ ∀ δn ↓ 0 : ∀ n ∈  that are δn -admissible, it holds VT (n )−→0. Theorem 5.11 (First Fundamental Theorem of Asset Pricing) In the market model (S 0 , S, P ) let (S, P ) be a locally bounded semimartingale. Then it holds: S fulfills the NFLVR condition ⇐⇒ Me (P ) = ∅,

e.g. there is

an equivalent martingale measure. Proof “⇐” As in Theorem 5.9. “⇒” Proof idea (1) In the first step, reduction to bounded semimartingales by means of a ‘change of numeraire’ argument using Girsanov’s theorem. (2) Approximation by a discrete semimartingale model. (3) In the third step, an equivalent martingale measure is constructed using Hahn– Banach’s separation theorem (cf. Delbaen and Schachermayer (2006)).   Remark 5.12 In the case of general (non-locally bounded) semimartingales, the class Me (P ) of the equivalent martingale measures must be replaced by the larger class Mσ (P ) of equivalent σ martingale measures. A d-dimensional semimartingale S in Md is called σ -martingale with respect to Q if a ϕ in L(P) with values

198

5 Option Prices in Complete and Incomplete Markets

in R+ exists such that ϕ · S = (ϕ · S i ) ∈ M(Q).

(5.2)

Now the following characterization of the NFLVR condition holds by the existence of an equivalent σ -martingale measure (ESMM) (cf. Delbaen and Schachermayer (1998)): Theorem 5.13 ((General) First Fundamental Theorem of Asset Pricing:) Let S ∈ S d , then the following are equivalent: 1. (ESMM): ∃ Q ∼ P : S ∈ Mσ (Q) 2. NFLVR: S satisfies the NFLVR condition. In particular, it holds that Mloc ⊂ Mσ . We now show, as an application of Theorem 5.11 that the Black–Scholes model of geometric Brownian motion is arbitrage-free. 2 So let  St = e−rt St = S0 eXt with Xt = σ Bt + (μ − r − σ2 )t, volatility σ 2 and 2

drift μ − r − σ2 . To establish that S is arbitrage-free, we show that there exists an equivalent martingale measure. Proposition 5.14 Define Q ∈ M 1 ( , A) by the Radon–Nikodým density.   r −μ (r − μ)2 dQ := LT = exp BT − T . dP σ 2σ 2 Then it holds: (a)  S ∈ M(Q) i.e. Q ∈ Me (P ). 2 (b) (X,Q) is a Brownian motion with drift − σ2 and with volatility σ 2 . Proof B is a Brownian motion with respect to P and L is the exponential martingale. Therefore, by Girsanov’s theorem, it follows: r −μ t is a Brownian motion with respect to Q. σ   σ2 σ2 t = σ Bt − t Xt = σ Bt + μ − r − 2 2 B t := Bt −

5.1 The Black–Scholes Model and Risk-Neutral Valuation

199 2

is, therefore, a Brownian motion with volatility σ 2 and with drift − σ2 with respect to Q. It follows that  St = S0 eσ B t −

σ2 2 t

is an exponential martingale with respect to Q,

and thus  S ∈ M(Q).

 

Thus, there is no regular arbitrage strategy in the extended sense in the Black– Scholes model.

5.1.1

Risk-Neutral Valuation of Options

An option is a measurable functional of the process, i.e. an element of L(AT ). The basic options C = (ST −K)+ = European Call

and C = (K −ST )+ = European Put

are called vanilla options. For the European options, the (acquired) right to buy and sell can only be exercised at the end of the term (validity) of the option, in this case at time T . In contrast “American options”, such as an American Call C = (Sτ − K)+ respectively an American Put C = (K − Sτ )+ can be exercised during the entire term at any stopping time τ . Definition 5.15 (a) An option C ∈ L(AT ) is duplicable ⇐⇒ ∃  ∈ r with VT () = C. Each such  is called a hedge of C. (b) A hedge  of C is called admissible or martingale hedge of C, if  ∈ a , i.e. ∃ Q ∈ Me (P )

() ∈ M(Q). with V

(c) C is called strongly hedgeable, if there is a martingale hedge of C. Remark 5.16 If  is an admissible hedge strategy of C, then T () = e−rT C ∈ L1 (Q). V For what follows, it is of fundamental importance that two admissible hedge strategies for an option C have the same initial cost.

200

5 Option Prices in Complete and Incomplete Markets

Proposition 5.17 Let ,  ∈ be admissible hedge strategies of the option C, then it holds: V0 () = V0 (). Proof Let Qi ∈ Me (P ) and  ∈ a (Q1 ),  ∈ a (Q2 ) with x := V0 () and y := V0 () the initial costs of these strategies. Then it holds:   T () − V T () − EQ1 V T (). T () = EQ1 V 0 = EQ 1 V    =C

 =C

T () = EQ1 V 0 () = x. It is EQ1 V () is a supermartingale w.r.t. Q1 . From this Further  ∈ r and, therefore, V follows T () ≤ EQ1 V 0 () = y. EQ 1 V Altogether, as a result we obtain T () − EQ1 V T () ≥ x − y. 0 = EQ 1 V So x ≤ y. Conversely, one obtains analogously y ≤ x and thus x = y.

 

Remark 5.18 (a) More generally, it holds for admissible , : VT () ≥ VT () implies Vt () ≥ Vt (),

0 ≤ t ≤ T.

(b) If ,  ∈ a (Q) are martingale hedges of C with the same martingale measure Q, then even without the assumption of regularity V0 () = V0 (). According to these preparations, we can now introduce the basic no-arbitrage price as a consequence of the no-arbitrage principle. Definition 5.19 (No-arbitrage Price) Let Q ∈ Me (P ) and let C ∈ L1 (AT , Q) be a claim. If  ∈ a (Q) is an admissible hedge strategy of C, then p(C) := V0 () = ϕ00 S00 +ϕ0 S0  =1

5.1 The Black–Scholes Model and Risk-Neutral Valuation

201

is called no-arbitrage price (Black–Scholes price) of C (at the time t). p(C, t) := Vt () = ϕt0 er,t + ϕt St0 is called no-arbitrage price at time t. Remark 5.20 (a) According to Proposition 5.17, the no-arbitrage price of a hedgeable claim is uniquely defined. If Q1 , Q2 ∈ Me (P ) and C ∈ L1 (Q1 ) ∩ L1 (Q2 ) have admissible hedge strategies , , then V0 () = V0 (). Thus pQ1 (C) = pQ2 (C) is defined independently of Qi . Thus, the no-arbitrage price is defined for strongly hedgeable claims C ∈ L(AT ), i.e. there exists a Q ∈ Me (P ) and a  ∈ a (Q) such that C = VT (). (b) Economic justification of the no-arbitrage price. The generalized no-arbitrage principle, which also includes non-hedgeable claims with their prices, states that in a realistic market model no risk-free profit is possible. If the price p(C) of a hedgeable option C were not equal to the no-arbitrage price V0 (), p(C) = V0 (), then there would be a risk-free profit: If p(C) < V0 (), then a risk-free profit strategy consists of the following steps: (1) short selling from the stock portfolio, (2) buying the option C, (3) investing the difference without risk. This strategy leads to risk-free profit. If p(C) > V0 (), then the analogous dual strategy leads to risk-free profit: (1) short selling from the claim, (2) buying from the stock portfolio, (3) investing the difference without risk. This again results in a risk-free profit as opposed to the no-arbitrage principle. An equivalent formulation of the no-arbitrage principle is: Two financial instruments with the same payoff at time T are equally expensive at time t, ∀ t ≤ T . Of fundamental importance is the following risk-neutral valuation formula. The noarbitrage price can be determined as the expected value of the option with respect to an equivalent martingale measure without having to explicitly determine hedge strategies. Theorem 5.21 (Risk Neutral Valuation Formula) Let Q ∈ Me (P ) and let C ∈ L(AT , Q) be a duplicable option with admissible hedge  ∈ a (Q). Then it holds:  p(C) = EQ C,

 = e−rT C. C

 | At ). More generally: p(C, t) = e−r(T −t ) EQ (C | At ) = ert EQ (C

202

5 Option Prices in Complete and Incomplete Markets

Remark 5.22 The expected value is with respect to the martingale measure Q, but not with respect to the underlying statistical model measure P . It turns out, however, that real markets behave ‘approximately’ like risk-neutral markets. To calculate the price of an option one does not need to know the hedging strategy. Proof Let  ∈ a (Q) be a hedge strategy of C. Then  VT () = V0 () + 0

T

 ϕs dSs + 0

T

ϕs0 dSs0 = C.

Equivalent is T () = V0 () + V



T

 (cf. Lemma 5.2). ϕ d S=C

0

Now t () = V0 () + V



t

ϕu d Su ∈ M(Q), as  is admissible with respect to Q.

0

It follows  = EQ V T () = EQ V0 () = V0 () = p(C). EQ C Similarly it follows T () | At ) t = ert EQ (V p(C, t) = Vt () = ert V = ert EQ (C | At ).   Basic Questions (1) For which models is each claim strongly hedgeable? This question leads to the notion of a complete market model. (2) How is a meaningful price determination for non-hedgeable claims possible? With the no-arbitrage principle, one obtains only an interval of compatible prices in general (not complete) models. We will describe this so-called no-arbitrage interval in more detail below. Proposition 5.23 (Trivial Price Bounds) Let Me (P ) = ∅, S ≥ 0, K ≥ 0, and let C := (ST − K)+ be a European Call. Then, for any no-arbitrage price p(C) in the sense of the generalized no-arbitrage principle, it holds: (S0 − K)+ ≤ p(C) ≤ S0 .

5.1 The Black–Scholes Model and Risk-Neutral Valuation

203

Proof If p(C) > S0 then consider the following strategy: (1) At time t = 0 buy the stock, (2) shortselling of option C and (3) investing the difference risk-free in bonds. At time t = T the value of the stock is ST ≥ (ST − K)+ . The profit is then > 0 because of the risk-free investment of the difference in bonds. If 0 ≤ p(C) < (S0 − K)+ , then an analogous argument yields a risk-free profit again.   Remark 5.24 Forwards are OTC contracts (OTC—over the counter), i.e. contracts between parties that only meet when the contract is concluded. There is no more trading in the further course until maturity. There is no no-arbitrage argument with trading strategies for the valuation of forwards. However, the extended no-arbitrage principle can be used for valuation in an analogous manner. Example Two parties, one receiving at time T the underlying S 1 the other receives at time T the forward price K. Let K be such that the contract costs nothing at time 0. The payoff at maturity is C := ST1 − K at time T from the perspective of party 1. Financing occurs through the underlying security and a bond maturing in T , i.e., a security S 0 , which at time T pays out the value 1 (a freely traded zero coupon bond). The constant portfolio ϕ = (ϕ 0 , ϕ 1 ) = (−K, 1) has at maturity the value C. The initial value of this portfolio is V0 () = −KS00 + S01 . From the extended noarbitrage principle, it follows for the ‘fair price’: V0 () = 0 or equivalently: K =

S01 S00

is the unambiguous fair

forward price of the base paper S 1 . For the Black–Scholes model, all options can be uniquely priced by the no-arbitrage price. Theorem 5.25 In the Black–Scholes model with interest rate r and with equivalent martingale measure Q it holds: (a) Each claim C ∈ L1 (AT , Q) is strongly hedgeable. (b) ∀ C ∈ L2 (AT , Q) there exists exactly one martingale hedge of C.

204

5 Option Prices in Complete and Incomplete Markets

Proof The discounted price process  S has the representation  St = e−rt St = S0 eσ B t −

σ2 2 t

,

where B is a Brownian motion with respect to Q and AB = AB . For C ∈ t = EQ (C|A  t ), the martingale generated by C  = e−rT C, L1 (AT , Q) and with C 0 then it holds by the martingale representation theorem: ∃ fs ∈ L (B) (∃ !f ∈ L2 (B), if C ∈ L2 (Q)) such that t = EQ C + C



t

fs dB s . 0

It holds d  St = σ  St dB t since  S is exponential martingale with respect to Q. We define:  t ft + St , and  := (ϕ 0 , ϕ). ϕt := , ϕt0 := EQ C fs dB s − ϕt  σ St 0 Then  ∈ a (Q) is an admissible hedge of C, because:  according to the definition of ϕ 0 . 0 () = ϕ 0 + ϕ0 (1) V S0 = EQ C 0   (2) Vt () = Ct ∈ M(Q) because t = e−rt Vt () = e−rt (ϕt0 ert + ϕt St ) = ϕt0 + ϕt  St V  t  t + + = EQ C fs dB s − ϕt  fs dB s St + ϕt  St = EQ C 0

0

t ∈ Mc (Q), and V t ∈ =C

M2c (Q),

if C ∈ L (AT , Q). 2

() is, therefore, a continuous martingale. V t ∈ M(Q) for C ∈ L1 (AT , Q). t () = C (3)  is regular, because V  it (4)  is self-financing, because with the help of the above representation of V  follows from the Doléans–Dade equation for S t () = V0 () + V



t

fs dB s 0

 = V0 () +

t

0

 = V0 () +

fs  σ Ss dB s σ Ss  d Ss

t

ϕs d Ss .

0

So  is according to Lemma 5.2 self-financing.

5.1 The Black–Scholes Model and Risk-Neutral Valuation

205

T () = C T = C  and, therefore, it (5)  is a martingale hedge for C, because V follows VT = C. From (1) to (5) it follows:  ∈ a (Q). The uniqueness statement in (b) follows from the corresponding uniqueness statement for a Brownian motion.   As a consequence of the above theorem, we obtain: Every contingent claim C ∈ L1 (Q) in the Black–Scholes model is strongly hedgeable. This statement applies analogously to claims C ≥ c bounded from below. In particular, it can be applied to the example of the “European Call”. Let C = (ST − K)+ be a European Call with strike K and term T . As a consequence of the risk-neutral valuation formula in Proposition 5.21 we obtain the classical Black–Scholes formula for the valuation of a European Call option. Theorem 5.26 (Black–Scholes Formula) The no-arbitrage price of a European Call C = (ST − K)+ with strike K and running time T is given by p(C) = EQ e−rT C = S0 (d1 ) − Ke−rT (d2 ) =: p(S0 , T , K, σ ). Here  is the distribution function of the standard normal distribution N(0, 1), d2 :=

log SK0 + (r − √ Tσ T

σ2 2 )

√ and d1 := d2 + σ T .

Proof Using the risk-neutral valuation formula in Theorem 5.21 the following holds for the no-arbitrage price p (C) p(C) = EQ e−rT (ST − K)+ = EQ e−rT (S0 eσ B T +(r− = EQ e−rT (S0 eZ − K)+

σ2 2 )T

− K)+

 σ2 T. with Z := σ B T + r − 2 2

Z is normally distributed, QZ = N((r − σ2 )T , σ 2 T ). In general, for a normally distributed random variable Z ∼ N(a, γ 2 ) the following formula holds: E(beZ − c)+ = bea+

γ2 2

 

log bc + a + γ 2 γ

From this, by specialization, the assertion follows.



 − c

 log bc + a . γ  

206

5 Option Prices in Complete and Incomplete Markets

Remark 5.27 (a) Analogously it follows for the price process: p(C, t) = p(St , T − t, K, σ ). For the proof, substitute in the above formula S0 with St , T by the residual time T − t. (b) Approximation formula For the calculation of the Black–Scholes formula, in practice a (very good) approximation of the distribution function  is used: 1 (x) ≈ 1 − ϕ(x)(a1k + a2 k 2 + · · · + a5 k 5 ), where k := 1+γ x , γ = 0.23167, a1 = 0.31538, a2 = . . . This approximation yields six-digit precision for the price p. Using the following call-put parity yields the no-arbitrage price for the European Put option. Proposition 5.28 (Call-Put Parity) Let cT := p((ST − K)+ ) and pT := p((K − ST )+ ) be the prices of the European Call and Put. Then it holds: pT = cT − S0 + Ke−rT . Proof It holds: (K − ST )+ = (ST − K)+ − ST + K. Therefore, it follows from the no-arbitrage valuation formula with the equivalent martingale measure Q pT = p((K − ST )+ ) = EQ e−rT (K − ST )+ = EQ e−rT (ST − K)+ − EQ e−rT ST +Ke−rT  S0

= cT − S0 + Ke

−rT

.  

5.1.2

Discussion of the Black–Scholes Formula

(a) Dividend payments A rule of thumb for price changes in dividend payments is: Following a dividend payment of size d causes the share price to fall by 80% of the dividend paid, i.e., by 0.8 d.

5.1 The Black–Scholes Model and Risk-Neutral Valuation

207

For valuation purposes, the stock price can be decomposed into two components. Into a risk-free component, which is equal to the sum of the discounted dividend payments multiplied by 0.8; and into the remaining risk component, which is valued by the Black–Scholes formula. (b) Hedging portfolio—hedge strategy Let p(x, t, K, σ ) be the price of a call C = (ST − K)+ with market value x, remaining (residual) time t and strike K. p depends only on the volatility parameter σ but is independent of the drift parameter μ of the model. For t → 0, i.e., the remaining term approaches zero, the following applies p(x, t, K, σ ) → (x − K)+ . The Black–Scholes formula p(x, t, K, σ ) = x(d1 ) − e−rt K(d2 ) can also be interpreted as a hedging portfolio : The first component x(d1 ) corresponds to the value of a portfolio with (d1 ) shares in stocks at the time t, the second component corresponds to a short position of K(d2 ) shares in bonds, i.e. ((d1 ), −K(d2 )) is a hedging portfolio for the option C with residual maturity t. It remains to show that these strategies are self-financing, i.e. ϕt0 = (d1 ), ϕt = −K(d2 )e−r(T −t ) defines a hedging strategy for the call C. (c) The ‘Greeks’ In the following, we discuss the dependence of the Black–Scholes price p(x, t, K, σ ) on the parameters x = market value, t = residual maturity, K = strike and σ = volatility. These relationships have names with Greek letters and are called the ‘Greeks’. Useful for this is the following relation for the density ϕ of the standard normal distribution. It is valid with di = di (x, t, K, σ ) according to the Black– Scholes formula: xϕ(d1 ) − Ke−rt ϕ(d2 ) = 0. (c1) From (5.3) and

∂d1 ∂x

=

∂d2 ∂x

(5.3)

it follows

  ∂d1 ∂p = (d1 ) + xϕ(d1) − Ke−rt T ϕ(d2 ) = (d1 ) > 0. ∂x

 ∂x

(5.4)

=0

So p increasing in the spot price x.  := ∂p ∂x is the Delta of the option.  describes the share of equities in the hedging portfolio with residual maturity t (Delta-hedging).

208

5 Option Prices in Complete and Incomplete Markets 2

∂d1 (c2)  := ∂∂xp2 = ∂ ∂x = ϕ(d1 ) ∂x > 0 is the Gamma of the option and describes the sensitivity in the hedging portfolio as a function of the market value x. The Black–Scholes price p is strictly convex in x. (c3)

 σ ϕ(d2 )  ∂p = Ke−rt r(d2 ) + √ > 0. ∂t 2 t

(5.5)

The price p is monotonically increasing in the residual time t. √ ∂p (c4)  = ∂σ = xϕ(d1 ) t > 0 is the Lambda of the option. The price p is increasing in the volatility σ . (c5) ∂p = Kte−rt (d2 ) > 0. ∂r

(5.6)

p is monotonically increasing in the rate of interest r. (c6) ∂p = −e−rt (d2 ) < 0, ∂K

(5.7)

p is monotonically decreasing in the strike K. (d) The volatility σ is to be estimated from market data. To do this, one uses historical data, such as the data of the last 30–180 trading days. Now, let σ be the implied volatility, i.e., the solution of the equation p(x, t, K, σ ) = p where p denotes the current market price. Then, if the Black–Scholes model correctly describes the market, it should be that σ is independent of the strike K of the traded options. However, in empirical data one actually observes a dependence in the form of a ‘smile’ (cf. Fig. 5.2), an indication that the Black– Scholes model does not describe stock prices correctly, in particular that it underestimates tail probabilities: For x ≈ K (at the money) is σ < σ , for x 3 K and x ( K (out of the money) is σ > σ . That is, stocks with a strike K 3 x or K ( x are more expensive than indicated by the Black–Scholes model. The BS model underestimates the large jumps in prices, i.e., the tails of the distributions. (e) Examples of options The basic options (vanilla options) are the European Call and Put. With the help of these basic options, a wealth of other options (derivatives) can be constructed and valued by simple combinations.

5.1 The Black–Scholes Model and Risk-Neutral Valuation

209

Fig. 5.2 Smile effect

(e1) European Call: (ST − K)+ = f (ST ) Let c = cT be the no-arbitrage price, then the buyer’s profit is VT = f (ST ) − cT as in the adjacent form. Thus, a call provides a profit if the price of the underlying security rises.

(e2) European Put: f (ST ) = (K − ST )+ pT = cT − S0 + Ke−rT is the price of the European Put. Thus the profit of the buyer is VT = f (ST ) − pT as in the adjacent form. A put thus yields a profit if the price of the underlying security falls.

210

5 Option Prices in Complete and Incomplete Markets

(e3) A straddle is a combination of a call and a put with the same strike K. The cost of the straddle is c = cT + pT and with f (ST ) = |ST − K| the profit is VT is of the adjacent form. Thus, a straddle provides a profit when a larger change in price is expected in an unknown direction.

(e4) A strangle is a combination of a call and a put with different strikes. This leads to the profit VT = (ST − K2 )1(ST >K2 ) + (K1 − ST )1(ST K1 . This leads to a profit profile of the form VT = (K2 − K1 )1(ST ≥K2 ) + (ST − K1 )1(K1 z,  x 2z−x P (Bt ≤ x|Mt < z) = ( √t ) − 1 +  √t ⎪ ⎩ , x ≤ z. P (Mt < z) This conditional distribution has the density     2z − y  1 y h(y) = √ ϕ √ −ϕ √ , tP (Mt < z) t t

y ≤ z.

This gives  P (Xt ≤ x, Zt < z) =

f (y)h(y) dyP (Mt < z)

with a2

f (y) = 1(−∞,x] (y)eay− 2 t     z  2z − y  a2 1 y 1(−∞,x] (y)e−ay− 2 t dy = √ ϕ √ −ϕ √ t t t −∞  z a2 1  − y2 (2z − y)2  e 2t − e dy = eay− 2 t √ 2t 2πt −∞  x − at   x − 2z − at  = √ − e2az  . √ t t  

5.1 The Black–Scholes Model and Risk-Neutral Valuation

215

Theorem 5.31 (Price of a Barrier Option) The no-arbitrage price of the barrier option CB = (ST − K)+ 1{inf0≤t≤T St >B} in the Black–Scholes model is p(CB ) = EQ e−rT CB = p(S0 , T , K, σ ) −

e2αβ p(S0 , T , γ K, σ ). γ

r σ Here p is the Black–Scholes price of the European Call, with α := − , 2 σ  2   S0 S0 1 , and γ := β := log . B σ B 

2



σ B t + r− σ2 t

Proof It is St = S0 e with respect to Q, with a Brownian motion (B, Q). Thus it follows by Proposition 5.30 EQ e−rT (ST − K)+ 1{inft≤T St >B}

 =:f (ST )

 =

{ST >B}

 f (ST )dQ − e

2αβ

= EQ e−rT (ST − K)+ − = p(S0 , T , K, σ ) −



ST f γ {ST >B}

 dQ

e2αβ EQ e−rT (ST − γ K)+ γ

e2αβ p(S0 , T , γ K, σ ). γ

For the second equality it is used that Q(St ≥ x, infs≤t Ss > B) = Q(St ≥ x) − e2αβ Q( Sγt ≥ x).   Example 5.32 (American Option) Execution at stopping times 0 ≤ τ ≤ T . Let Z = (Zt ) be an adapted process and let Zt be the payoff at time t. The payoffs available are C(Z, τ ) := Zτ with the value p(C(Z, τ )). The optimal payoff is given by p(Z) :=

sup

τ ≤T , τ a stopping time

p(C(Z, τ ))

:= sup EQ e−rτ Zτ . τ

Thus, determining the American Call leads to an optimal stopping problem.

216

5 Option Prices in Complete and Incomplete Markets

Proposition 5.33 (American Call) The no-arbitrage price of the American Call option is identical to the price of the European Call. The following, therefore, holds sup EQ e−rτ (Sτ − K)+ = EQ e−rT (ST − K)+

τ ≤T

= p(St , T , K, σ ). Proof We show that e−rt (St −K)+ a submartingale with respect to Q. The function f (x) = (x − K)+ is convex. So according to the Jensen inequality it follows EQ (e−rt (St − K)+ | As ) ≥ e−rt (EQ (St | As ) − K)+ = (EQ ( St | As ) − e−rt K)+ = ( Ss − e−rt K)+ ≥ e−rs (Ss − K)+ , since (x − αK)+ ≥ (x − K)+ for α ≤ 1, K ≥ 0. It follows that e−rt (St − K)+ is a submartingale. For all stopping times τ ≤ T the following applies according to the optional sampling theorem EQ e−rτ (Sτ − K)+ ≤ EQ e−rT (ST − K)+ . The price of the American Call is, therefore, identical to the price of the European Call.   For the American Put, however, the price is different from that of the European Put. To get an idea of the price of an American (or European) Put, consider an C, example from Geske and Johnson (1984) with period T = 7 months, S0 = 40 B the strike K = 45, the interest rate r = 4.88% and the volatility σ = 0.3. In this case, the Black–Scholes formula yields the price of the European Put pE (C) = 5.97 B C, whereas the price of the American Put is pA (C) = 6.23 B C.

5.1.3

Hedging Strategies and Partial Differential Equations

For the European Call, a hedging portfolio for the call had resulted as a consequence of the Black–Scholes formula: p(x, t, K, σ ) = x (d1 (x, t, K, σ )) −e−rt K(d2 (x, t, K, σ )) .



 ∂p =:h(x,t )= ∂x =

=:g(x,t )

Here, t is the residual time (to be replaced by T − t for the price at time t). h(x, t) = ∂p ∂x =  the delta of the option indicates the proportion of shares in the hedging portfolio. In the following proposition, we give general conditions for a trading strategy (h(St , t), g(St , t)) to be self-financing.

5.1 The Black–Scholes Model and Risk-Neutral Valuation

217

Proposition 5.34 In the Black–Scholes model let h, g ∈ C 2,1 be solutions of the differential equation 

xhx + ert gx 1 2 2 2 σ x hx

=0

(5.8)

+ xht + ert gt = 0.

Let ϕ := h(St , t), ϕt0 := g(St , t). Then the trading strategy is  = (ϕ 0 , ϕ) is self-financing, i.e.  ∈ . Proof Define f (x, t) = g(x, t)ert + xh(x, t), then Vt = Vt () = f (St , t). Using the Itô formula, it follows 1 dVt = fx (St , t)dSt + ft (St , t) dt + fxx (St , t) d&S't 2   1 = fx (St , t) dSt + ft (St , t) + fxx (St , t)σ 2 St2 dt. 2 To be shown is the self-financing condition: dVt = h(St , t) dSt + rert g(St , t) dt. So to show is: fx = h, ft + 12 fxx σ 2 x 2 = rert g. (1) fx = h + xhx + ert gx = h using (5.8).

 =0

(2) ft + 12 fxx σ 2 x 2 = xht + gt ert + rgert + 12 σ 2 x 2 hx = rgert It follows that the trading strategy  is self-financing.

by (1) by assumption.  

Remark 5.35 The above conditions in (5.8) an g, h hold for the hedging portfolio in the Black–Scholes model for European options. It follows that: The delta hedge strategy is self-financing and hence an admissible hedge strategy. More generally, the following theorem for options of type C = f (ST , T ) has an analytical approach to determine self-financing trading strategies.

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5 Option Prices in Complete and Incomplete Markets

Theorem 5.36 (Black–Scholes Differential Equation) In the Black–Scholes model let f ∈ C 2,1 be a solution of the Black–Scholes differential equation 1 2 2 σ x fxx + rxfx + ft − rf = 0. 2

(5.9)

Let g := e−rt (f − xfx ), h := fx , then  = (ϕ 0 , ϕ)

with ϕt0 := g(St , t), ϕt := h(St , t),

self-financing with value process Vt = f (St , t). Proof The value process Vt () has the form Vt () = g(St , t)ert + h(St , t)St = (f (St , t) − St fx (St , t)) + fx (St , t)St = f (St , t). To prove  ∈  we verify the conditions from Proposition 5.34 to: xhx + ert gx = xfxx + ert (e−rt (fx − fx − xfxx )) = 0 and 1 2 2 σ x hx + xht + ert gt 2 1 = σ 2 x 2 fxx + xfxt + ert (−r)e−rt (f − xfx ) + ert e−rt (ft − xfxt ) 2 1 2 2 = σ x fxx + xfxt − rf + rxfx + ft − xfxt 2 1 2 = σ xfxx − rf + rxfx + ft = 0 2 by the Black–Scholes differential equation.   Consequence In order to evaluate and hedge an option C = f (ST , T ), the following steps are necessary: (1) Solve the Black–Scholes differential equation (5.9) with boundary condition f (x, T ).

5.2 Complete and Incomplete Markets

219

This is a Cauchy boundary value problem. Then the no-arbitrage price of C is: p(C) = f (S0 , 0) the no-arbitrage price (2) Determine the hedge strategy: ϕt0 = g(St , t), ϕt = h(St , t) with g = e−rt (f − xfx ), h = fx from Theorem 5.36. For the explicit solution of the Black–Scholes equation in the case of a European Call C = (x − K)+ = f (x, T ) the Black– Scholes equation is transformed into the heat equation. This can then be solved explicitly. Remark 5.37 (1) The above PDE approach for the Black–Scholes formula was used in the original Black–Scholes paper (1973). The elegant martingale argument goes back to the work of Harrison and Pliska (1981). The no-arbitrage argument for determining prices was introduced by Merton (1973). Shortly after the Black– Scholes work (since 1973), options were traded on the Chicago Board of Trade (CBO). (2) Let now be more general C = f (XT , T ) an option with value process VT = EQ (C | At ) and let (X, Q) be a diffusion process with infinitesimal producer A. Let h be a solution of the boundary value problem 

 ∂ + A h = 0, ∂t

h(·, T ) = f (·, T ),

(5.10)

then it follows with the Itô formula: 

t

h(Xt , t) = f (X0 , 0) +

hx (Xs , s) dXs . 0

From this, analogously to Theorem 5.36: ϕsC = hx (Xs , s) is a self-financing hedge strategy for C = f (XT , T ). The corresponding value process is Vs = h(Xs , s).

5.2

Complete and Incomplete Markets

We consider a market model S = (S 0 , S 1 , . . . , S k ) with a bond S 0 continuous, increasing, St0 > 0, S00 = 1 and with semimartingales Sti . Let (At ) = (ASt ) be the Si filtration (S 0 , S 1 , . . . , S k ) and let  Sti = t0 be the discounted stocks. St

Let ϑ = (x, π) be a trading strategy with initial capital x, π = (π 0 , . . . , π k ), π i the shares of security i in the portfolio.

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5 Option Prices in Complete and Incomplete Markets

t holds: ϑ is called self-financing, if for the discounted value process V t (ϑ) = x + V

k   i=1

t 0

πui d Sui .

We make the general assumption : Me (P ) = ∅. This assumption implies the arbitrage-free nature of the market model. Definition 5.38 The market model S is called complete, if for all Q ∈ Me (P ) and for all claims C ∈ L1 (AT , Q) a martingale hedge ϑ ∈ a (C) exists, i.e. t (ϑ) ∈ M(Q) and VT (ϑ) = C. V Proposition 5.39 Let S ∈ S k be a semimartingale model, then it holds: S is complete ⇐⇒ ∀ Q ∈ Me (P there exists a representation of the form  t ∈ M(Q) ), ∀ M Mt = EQ MT + ki=1 0 ϕui d Sui , ϕ ∈ L0 ( S). ⇐⇒ ∀ Q ∈ Me (P ), ∀ F ∈ L(AT , Q) there exists an integral representation    = EQ F + ki=1 T ϕui d Sui , ϕ ∈ L0 ( S). F 0 Proof The proof is similar to the proof of Theorem 4.27. For F ∈ L(Q, At ) let M be the corresponding generated martingale, Mt :=  | At ). If F is bounded, then the hedge property VT (ϑ) = F is equivalent E Q (F  . Because of the denseness of B(AT ) in L1 (AT , Q) it follows by to VT (ϑ) = F approximation the representability for F ∈ L1 (AT , Q) from the bounded case. The approximation argument uses stopping times and continuity.   Definition 5.40 C ∈ L(AT ) is a contingent claim, if C ≥ 0 and ∃ Q ∈ Me (P )  ∈ L1 (Q). such that C Corollary 5.41 If S is complete and C is a contingent claim, then C is strongly =V (π). hedgeable, i.e. ∃ Q ∈ Me (P ), ∃ π ∈ a (Q) so that C  Proof Let Mt be the martingale generated by C,  | At ), Mt := EQ (C

M ∈ M(Q).

According to Proposition 5.39 M has a stochastic integral representation 

t

Mt = Vt (π) = V0 (π) +

πu d Su .

0

π is, therefore, a martingale hedge and π is regular, since Mt ≥ 0.

 

5.2 Complete and Incomplete Markets

221

Remark 5.42 Thus, in complete markets, all contingent claims can be assigned a unique no-arbitrage  price.  = x + t πu d T (π) with C  ∈ L1 (Q) holds: For C Su = V 0  p(C) = x = EQ C. Example 5.43 (Diffusion Model) Let S be a multivariate diffusion model such that dSt = St 4 b(t, St ) dt + σ (t, St ) dBt , where 4 represents the component-wise product, B = (B 1 , . . . , B k ), is a k-dimensional Brownian motion and St0 = ert . Then S has the explicit representation   t    t 1 2  bs − St = S0 4 exp σij (s) ds + σs · dBs , 2 0 0

bs = b(s, ·).

j

Assumption: Let a(t, x) := σ (t, x) · σ (t, x)T be positive definite, let RT := T   2 exp 0 f (u,  Su ) · dBu − 12 Su ) du be the exponential martingale with fi (u,  fi (u,  Su ) :=

k  (σ −1 )ij (bj (u,  Su ) − r).

(5.11)

i=1

Let ERT = 1 and define dQ dP = RT . Then by Girsanov’s theorem follows: t  B t := Bt − 0 f (u, Su ) du is a Brownian motion with respect to Q. According to (5.11), the relation b − r · 1 = σf follows, and thus S 4 {(b − r · 1 − σf ) dt +  σ (t,  St ) · dB t d St =  = S 4 σ (t,  St ) · dB t

with respect to Q.

 So  S is an exponential martingale with respect to Q,  S ∈ M(Q) and AS = AB since σ is positive definite. S) =  | A If C is a contingent claim, C ∈ L1 (AT , Q) and Mt := EQ (C t B  EQ (C | At ) the generated martingale, then it follows by according to the martingale representation theorem (for k-dim martingales)

+ Mt = EQ C



t

gs dB s 0

with g ∈ L0 (B)

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5 Option Prices in Complete and Incomplete Markets

and according to (5.11) it holds: i

dB t =

k 

(σ −1 )ij

j =1

j d St . j  St

From this follows + Mt = EQ C

k   i=1

t 0

j iu d Su

k 

with iu =

gui (σ −1 )ij

j =1

1 j

.

Su

This results in the representation T (ϑ) = x +  = MT = V C

 j

T

u d Su j

j

0

 with ϑ = (x, 1 , . . . , k ), x = EQ (C).  is duplicable by a regular martingale hedge. Thus, the diffusion model is So C complete. For semimartingale models, completeness is closely coupled to the existence of a unique equivalent martingale measure. The following theorem deals with the case of continuous models. Theorem 5.44 (Second Fundamental Theorem of Asset Pricing, Completeness) Let S be a continuous martingale model, Me (P ) = ∅, and S = (S 0 , . . . , S k ) with a bond S 0 , and k stocks S 1 , . . . , S k . Then it holds: S is complete ⇐⇒ |Me (P )| = 1. Proof “⇒”: Let Q0 , Q1 ∈ Me (P ) be equivalent martingale measures, then Q := 1 0 1 0 2 (Q + Q ) ( Q and Q is an equivalent martingale measure, and Q ∈ Me (P ). dQ Let R := dQ0 , then the assertion is equivalent to: R ≡ 1. Let Rt := EQ0 (R | Ast ) ∈ M(Q) be the martingale generated by R. Because of completeness and according to Proposition 5.4 it holds: ∃ ϕ ∈ L0 ( S) so that Rt = 1 +



t

ϕu d Su

0

By stopping with τm := inf{t ≥ 0; | St | ≥ m} ∧ T we can w.l.g. assume that  Si 0 i 0  are martingales with respect to Q and Q , (S ) ∈ M(Q) ∩ M(Q ).

5.2 Complete and Incomplete Markets

223

From this follows ∀ B ∈ ASs

and

Sti = EQ 1B  Ssi . ∀ i holds EQ 1B 

However, it is: EQ 1B  Sti = EQ0 1B  Sti Rt

EQ 1B  Ssi = EQ0 1B  Ssi Rs .

and

Therefore, it follows: EQ0 1B  Sti Rt = EQ0 1B  Ssi Rs for B ∈ ASs . Sti Rt ) ∈ M(Q0 ). Therefore, ( Sti Rt ) is a martingale with respect to Q0 , ( i The martingales  S and R are, therefore, orthogonal with respect to Q0 or equivalently, the predictable quadratic covariation is zero, & S i , R't = 0, ∀ t [Q0 ]. It follows that the quadratic variation of R is zero &R'T =

k   i=1

T 0

ϕui d& S i , R'u = 0, i.e. Rt2 ∈ M(Q0 ).

It follows that: Rt = RT = 1 [Q0 ], so also Q a.s. and P almost surely. It follows Q = RQ0 = Q0 , i.e. there exists only one equivalent martingale measure. “⇐” Suppose there exists only one equivalent martingale measure Q, i.e. Me (P ) = {Q}.  is representable by a martingale To show: ∀ C ∈ B(AST ) with |C| ≤ K is C hedge.  | ASt ) be the martingale generated by C  in H2 . For this let Mt := EQ (C By Kunita–Watanabe’s theorem, M has a representation of the form + Mt = EQ C



t

ϕu d Su + Zt ,

0 2 with ϕ ∈ L ( S) and an L2 -martingale Z with respect to Q, with EQ Zt = 0 and i & S , Z' = 0, ∀ i. To show Z = 0 let σm be the stopping time defined by

  t " #      σm := inf t ≥ 0 :  ϕu dSu  ≥ m ∧ T . 0

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5 Option Prices in Complete and Incomplete Markets

Because of the continuity of  S i is Ztσm = Zt ∧σm is bounded by K 1 := 2K + m. i σ i Further & S , Z m 't = & S , Z'σt m = 0, ∀ m and, therefore,  S i Z σm is a martingale, i σ  S · Z m ∈ M(Q). 1 By Rt := 1 (K 1 + Ztσm ) a non-negative process is defined with EQ Rt = 1, K

 ≥0

Rt ≥ 0, ∀ t and  S i R is a Q-martingale,  S i R ∈ M(Q). With respect to the probability measure Q0 defined by the Radon–Nikodým density RT , Q0 := RT Q S i ∈ M(Q0 ). The argument to this is as in the first part of then  S i is a martingale,  the proof. However, this means that Q0 is also an equivalent martingale measure. By assumption, therefore, it follows Q0 = Q. So Rt = R = 1 [Q]. According to the definition of R then follows Z σm = 0 [Q], a.s. ∀ m. σm is a sequence of stopping times converging to infinity. From the above Kunita– Watanabe-representation follows then the representability-property of M, + Mt = EQ C



t

ϕu d Su ,

0

 

thus the completeness of S.

General Second Fundamental Theorem The second fundamental theorem can be generalized to general not necessarily continuous semimartingales. This requires the notion of predictable representation property (PRP). M ∈ M2loc has the PRP if the generated stable subspace F (M) := {ϕ · M; ϕ ∈ 2 L (M)} = M2 is, i.e. every element H ∈ M2 has a representation of the form  H = H0 +

ϕ dM,

ϕ ∈ L2 (M).

The property ϕ ∈ L2 (M) is important. Even for continuous local martingales, T representations can be found, e.g., of the form 1 = 0 ϕt dMt . But necessarily T then ϕ ∈ L2 (M), because otherwise E( 0 ϕt dMt ) = 0. For a semimartingale model S now it holds: Theorem 5.45 ((General) Second Fundamental Theorem) The following statements are equivalent: (i) S is complete (ii) |Me (P )| = 1 (iii) ∃ Q ∈ Me , such that S has the PRP with respect to Q.

5.2 Complete and Incomplete Markets

225

Proof (i) ⇒ (ii): Because of completeness, there exists to A ∈ A a martingale hedge T ϑ ∈ a (1A ), i.e. 1A = c + 0 ϑt dSt . Therefore, for all Q ∈ Me , Q(A) = EQ 1A = c, i.e. Q is unambiguously determined. (ii) ⇒ (iii): for this elaborate part, we refer to Protter (1990, Section IV.3). (iii) ⇒ (i): Let H ∈ B(AT ) be a bounded claim. Then for the associated Qmartingale Mt = EQ (H | AT ) it holds by the PRP a representation of the form  t Mt = M0 + ϑn dSn . 0

Since M is bounded, it follows that M is a martingale and hence ϑ ∈ a is a martingale hedging strategy.   It is remarkable that not only continuous semimartingales such as diffusion processes, e.g. Brownian motion or Brownian bridge, are complete. There are also (purely) discontinuous processes which are complete. In complete models, a contingent claim is assigned a unique price by the no-arbitrage argument. In non-complete models, prices are no longer uniquely determined by the no-arbitrage argument. Definition 5.46 Let S = (S 0 , . . . , S k ) be a market model with St0 = ert and let C ∈ L+ (AT ) be a contingent claim. A price p = pC for C is called arbitrage-free if a Q ∈ Me (P ) and an adapted process S k+1 ∈ M(Q) exist, such that S0k+1 = p,

Stk+1 ≥ 0,

∀ t ≤ T and STk+1 = C,

(5.12)

and the extended market model & S = (S 0 , . . . , S k , S k+1 ) is arbitrage-free in the sense of NFLVR. Let C be the set of arbitrage-free prices for C and let π∗ (C) := inf p and π ∗ (C) := sup p p∈C

p∈C

denote the lower and the upper arbitrage-free price The following theorem states that the set of arbitrage-free prices forms an interval. Each arbitrage-free price is described by the expected value with respect to an equivalent martingale measure.

226

5 Option Prices in Complete and Incomplete Markets

Theorem 5.47 (Arbitrage-Free Prices) Let Me (P ) = ∅ and C ∈ L+ (AT ) be a contingent claim. Then:  Q ∈ Me (P ), EQ C < ∞} (a) C = {EQ C;  π ∗ (C) = supQ∈M (P ) EQ C.  (b) π∗ (C) = infQ∈Me (P ) EQ C, e Proof (a) According to the First Fundamental Theorem 5.13, p ∈ C is an arbitrage-free price for C if and only if C can be hedged in the extended arbitrage-free model  S, and hence, equivalently, there exists for the discounted extended martingale & S (Q) with p = E C, & & & ∈ M S i ∈ M(Q), model  S a martingale measure Q &  e Q  T k+1 k+1 = St . 1 ≤ i ≤ k + 1 and C ST = p + 0 1 d Thus, C is hedgeable with a martingale hedge in the extended model & S. In & ∈ Me (P ) and p = EQ  by the risk-neutral valuation formula in particular, Q &C  Q ∈ Me (P ), EQ C < ∞}. Theorem 5.21, i.e. C ⊂ {EQ C;  < ∞ for a Q ∈ Me (P ). For the reverse inclusion, let pC = EQ C Then define a new process Stk+1 as the martingale with respect to Q generated by C Stk+1 = EQ (C | At ),

t ≤ T.

It then holds S0k+1 = pC and STk+1 = C. The extended market model & S is arbitrage-free and Q is a martingale measure for the discounted extended model  & S. So pC ∈ C is an arbitrage-free price. Hence the assertion follows  Q ∈ Me (P ), EQ C < ∞}. C = {EQ C; (b) The representation of the lower arbitrage-free price stated in (b), π∗ (C) =  follows directly from (a). infQ∈Me (P ) EQ C, The representation of π ∗ (C) needs an additional argument: We show: If EQ∗ C = ∞ for a Q∗ ∈ Mc (P ), then for all c > 0 there exists a p ∈ C with p > c. For this, let n be such that EQ∗ C ∧ n =: p  > c and define Stk+1 := EQ∗ (C ∧ n | At ),

t ≤ T.

Q∗ is a martingale measure for the extended model & S = (S 0 , . . . , S k+1 ); so this is arbitrage-free.

5.2 Complete and Incomplete Markets

227

Let w.l.g. C =  and let Q ∈ Me (P ) and EQ C < ∞. We consider Q as a new underlying measure instead of P .  ∼ Q such that According to the first fundamental theorem, there exists Q &  S ∈ M(Q). The proof of the fundamental theorem is based on a Hahn–Banach  Q can be separation argument (L1 − L∞ -duality) and implies that the density ddQ chosen boundedly. It follows: EQ C < ∞  ∈ Me (S) according to part (a) of the proof p = EQ Because of Q C is an arbitrage-free price for C. Further it holds: k+1 p = EQ = S0k+1 = p  > c.  C ≥ EQ  C ∧ n = EQ ST

 

From this follows the assertion.

Remark 5.48 (a) The set of all equivalent martingale measures Me (P ) is convex. Therefore, it follows, from Theorem 5.47 that the set of arbitrage-free prices of C is also convex, i.e., it is an interval (see Fig. 5.4). (b) Thus, the arbitrage-free prices determine an interval, the no-arbitrage interval [(π∗ (C), π ∗ (C))]. In general, it is not clear whether the lower or upper bound is attained. Prices below π∗ (C) favour the buyer, prices above π ∗ (C) favour the seller. It turns out that unique prices exist precisely for hedgeable claims. It holds: claim C ∈ L+ (AT ) is strongly hedgeable ⇐⇒ π∗ (C) = π ∗ (C). In incomplete models, claims C are generally not hedgeable. The following extension of the hedging idea leads to a fundamental connection with the arbitrage-free price principle and the no-arbitrage interval. Definition 5.49 Let S = (S 0 , . . . , S k ) be a market model, Me (P ) = ∅ and let C ∈ L+ (AT ) be a contingent claim. Fig. 5.4 Convex price interval

π∗ (c)

π ∗ (c)

228

5 Option Prices in Complete and Incomplete Markets

(a) ϑ = (x, π) is a super-hedging strategy of C, if T (ϑ) ≥ C  [P ], V0 (ϑ) = x. π ∈ r and V Let A+ C := {ϑ = (x, π); ϑ is super − hedging strategy for C} be the set of super-hedging strategies for C. (b) p (C, P ) := inf{x; ∃ (x, π) ∈ A+ C } is called upper hedging price from C. Theorem 5.50 Let S be a locally bounded semimartingale model with Me (P ) = ∅.  < ∞ be the upper arbitrageLet C ∈ L+ (AT ) and π (C) = supQ∈Me (P ) EQ C free price. Then: π (C) = p (C, P ) i.e., the upper hedging price of C is identical to the upper arbitrage-free price of C. Proof The proof is based on the optional decomposition theorem (cf. Kramkov (1996)). Optional Decomposition Theorem Let X ∈ S be a locally bounded semimartingale and V = (Vt ) ≥ 0 an adapted process to AX , then it holds: V is a supermartingale ∀ Q ∈ MX e (P ) ⇐⇒ ∃ decomposition Vt = V0 + (ϕ · X)t − Lt , t ≥ 0 with ϕ ∈ L0 (X), Lt is an optional increasing process, L0 = 0. L is an adapted optional process; it can in general not be chosen predictable. One direction of the proof of Theorem 5.50 is simple: is a super-hedging strategy for C and if Q ∈ Me (P ) “⇐”: If ϑ = (X, ϕ) ∈ A+ C  ≤ x + T ϕu d then it follows that C Su . 0 t   Because of C ≥ 0 0 ϕu dSu is a supermartingale. Therefore, it follows:  ≤ x + EQ EQ C



T

ϕu d Su ≤ x.

0

This gives the inequality π ∗ (C) ≤ p∗ (C, P ). “⇒”: For the proof of the inverse inequality define  | At ). Vt := ess sup EQ (C Q∈Me (P )



Here At = ASt and ess sup is the essential supremum.

5.2 Complete and Incomplete Markets

229

It follows that V = (Vt ) is a supermartingale with respect to Q, ∀ Q ∈ Me (P ) =  MSe (P ). This property is somewhat involved to show and remains here without proof. After the optional decomposition theorem follows the decomposition  Vt = x 0 +

t

ϕu d Su − Lt

0

with ϕ ∈ L0 ( S), 0 ≤ Lt , L0 = 0. Lt is monotonically increasing, adapted and optional. For t = 0 follows from this representation V0 = x 0 =

sup

Q∈Me (P )

 = π (C). EQ C

By definition Vt ≥ 0 and, therefore, ϑ0 = (x0 , ϕ) ∈ r . So ϑ0 = (x0 , ϕ) is a super-hedging strategy ϑ0 ∈ A+ C . This results in the inverse inequality x0 = π (C) ≥ p (C, P ).   Remark 5.51  ≤ K, then |Vt | ≤ K. Thus ∀ Q ∈ Me (P ): EQ LT ≤ x0 + K. From this (a) If |C| follows EQ sup |(ϕ ·  S)t | ≤ x0 + K + EQ LT < ∞, t ≤T

increasing in t. It follows that ϕ ·  S ∈ M(Q), i.e. because Lt is monotonically t   Vt (ϑ0 ) = x0 + 0 ϕu dSu is a martingale with respect to Q. Thus, for bounded options, one can dispensewith the regularity assumption for strategies and use as a strategy class a = Q∈Me a (Q). (b) Similarly, the lower hedge price s∗ (C, P ) can also be determined over the subhedging strategies and by analogy to Theorem 5.50 the equality of the lower hedge price and the lower arbitrage-free price hold: s∗ (C, P ) = π∗ (C). In the hedging problem for a claim H ∈ L+ (AT ) also a consumption process C can be introduced as an additional component.

230

5 Option Prices in Complete and Incomplete Markets

Definition 5.52 For the market model (S, (At ), P )  π = (x, ϕ, C) (is called extended) portfolio process, if ϕ ∈ L0 (S) and the consumption process (Ct ) is an adapted increasing process with C0 = 0.  t := x + t ϕs d V S s − Ct is called (discounted) value process (capital process 0 with consumption) to the portfolio  π. t ≥ 0, ∀ t.  π is called admissible portfolio process, if V Changes in the value process are driven by the portfolio strategy ϕ and the consumption C. Remark 5.53 (Characterization of Capital Processes) The optional decomposition theorem now has the following interpretation: Let Vt ≥ 0 be an adapted process. Then it holds:  is a capital process with consumption of a generalized portfolio process. V  is a Q-supermartingale, ∀ Q ∈ Me (P ). ⇐⇒ V  is a capital process (without consumption) of a portfolio Correspondingly V  process, exactly if V ∈ Mloc (Q), ∀ Q ∈ Me (P ). Definition 5.54 Let H ∈ L+ (AT ) be a claim. (a) A portfolio process  π = (x, ϕ, C) is a super-hedging portfolio for H , if T ≥ H . V & with capital process V & is a minimal super(b) A portfolio process & π = (& x, & ϕ , C) hedging portfolio, if t [P ], &t ≤ V V

). ∀ t ≤ T and for all super-hedging portfolios ( π, V

Theorem 5.55 Let H ∈ L+ (AT ) be a contingent claim with bounded price supQ∈Me (P ) EQ H < ∞ and let S be a locally bounded semimartingale. Then & and the corresponding there exists a minimal super-hedging portfolio & π = (& x, & ϕ , C) & has the representation capital process V   &t = & &t = ess sup EQ (H | At ). V x+ ϕ &·  S t −C Q∈Me (P )

 and Q ∈ Me (P ) the Proof For all admissible portfolio processes  π = (x,  ϕ , C)  is a supermartingale. Therefore: associated capital process V t ≥ ess sup EQ (V T |At ). V Q∈Me (P )

If  π is a super-hedging portfolio, then it follows that t ≥ ess sup EQ (H | At ). V Q∈Me (P )

5.2 Complete and Incomplete Markets

231

However, according to the optional decomposition theorem, the lower bound is & ≥ 0 adapted (even optionally), such attained, i.e., there exist & x, & ϕ ∈ L0 (S) and C & that for the associated value process V with consumption it holds: &t = & &t = ess sup EQ (H | At ). V x + (& ϕ · S)t − C Q∈Me (P )

  The boundedness of the semimartingale S can be replaced by the weaker condition of w-admissibility. Definition 5.56 (w-Admissibility) Let w ∈ L+ (A), w ≥ 1, Q0 ∈ Meσ where EQ0 w < ∞ and a > 0. ϕ ∈ L(P) is called (a, w)-admissible ⇐⇒ ∀ Q ∈ Meσ , ∀ t ≥ 0 : (ϕ · S)t ≥ −aEQ (w | At ) ϕ is called w-admissible ⇐⇒ ∃ a > 0 : ϕ is (a, w)-admissible. Remark 5.57 (Semimartingale Topology) On the set of semimartingales S d (P ) the Emery metric of two semimartingales X, Y is defined by d(X, Y ) = sup



|ϑ|≤1 n

2−n E min(1, |(ϑ · (X − Y ))n |),

(5.13)

t (ϑ · X)t = 0 ϑs dXs denotes the (vector) stochastic integral over ϑ ∈ L(P) with |ϑ| ≤ 1. Let L(S) be the space of S-integrable processes (L(S) the set of equivalence classes of such processes). Then it holds: (a) ϑ ∈ L(S) ⇐⇒ (ϑ1{|ϑ|≤n} · S) is a Cauchy sequence with respect to the semimartingale topology on S d (P ) generated by the Emery metric d. (b) {ϑ · S; ϑ ∈ L(S)} is closed in S d (P ) and (c) (L(S), d) is a complete topological vector space (cf. Memin (1980)). There exist S ∈ Sp and ϑ ∈ L(S) such that ϑ · S ∈ Mloc . More exactly it holds for S ∈ Sp , S = M + A: ϑ ∈ L(S) ⇐⇒ ϑ · M ∈ Mloc and ϑ · A exists as Lebesgue–Stieltjes integral The following theorems by Jacka (1992) and Ansel and Stricker (1994) generalize the super-hedging result in Theorem 5.50.

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5 Option Prices in Complete and Incomplete Markets

Theorem 5.58 Let S ∈ Sp and let C ≥ −w be a claim, w ≥ 1 a weight function. Then it holds: (a) sup EQ C = inf{α; ∃ ϑ ∈ L(P) w-admissible C ≤ α + (ϑ · S)∞ }

Q∈Meσ EQ w −∞}. This implies that dom(u) = (x, ∞) or dom(u) = [x, ∞).

© Springer-Verlag GmbH Germany, part of Springer Nature 2023 L. Rüschendorf, Stochastic Processes and Financial Mathematics, Mathematics Study Resources 1, https://doi.org/10.1007/978-3-662-64711-0_6

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6

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6 Utility Optimization, Indifference Pricing

Classical examples of utility functions are • u(x) = 1 − e−px exponential utility function, p > 0 xp • u(x) = , x > 0, p ∈ (−∞, 1) \ {0} power utility function p • u(x) = log x, x > 0 logarithmic utility function The trader’s objective is, given the initial capital x, to maximize his utility by investing in the financial market. With a class E of admissible trading strategies and market price model S, this leads to the utility-based portfolio optimization problem   V (x) = sup Eu x + ϑ∈E



T

ϑt dSt .

(6.2)

0

The optimal hedging of a claim B leads utility-based to the hedging optimization problem   V (x − B) = sup Eu x + ϑ∈E

T

 ϑt dSt − B

(6.3)

0

The solution to these problems is rendered possible by a duality theorem from convex calculus. Definition 6.1 The convex conjugate function u∗ : R+ → R of u is defined by u∗ (y) = sup (u(x) − xy). x∈R

(6.4)

It holds: u∗ (y) = u(I (y)) − yI (y)

with I := (u )−1 .

(6.5)

With (6.1) it follows: I takes values in (x, ∞) an and I (0) = ∞. p

Remark 6.2 For the power utility function xp the logarithmic utility function log x ∗ and the exponential utility function 1 − e−px the corresponding   conjugates u (y) p p−1 p−1 y y y are given by − p y , − log y − 1 and 1 − p + p log p . From the definition ∗ of u it follows a form of Fenchel’s inequality −xy ≤ u∗ (y) − u(x).

(6.6)

The utility indifference method is based on the comparison between the optimal behavior under the alternative scenarios of selling a claim and receiving a premium for it resp. not selling the claim.

6.2 Minimum Distance Martingale Measures

235

Definition 6.3 (Utility Indifference Price) The utility indifference price π (of the seller) of a claim B is defined in a market without trading opportunities as a solution π = π(B) of the equation u(x) = Eu(x + π − B).

(6.7)

In a market with trading opportunities E it is defined as a solution π = π(B) of the equation V (x) = V (x + π − B).

(6.8)

The utility indifference price  π of a buyer is defined analogously as the solution of the equation V (x) = V (x −  π + B).

6.2

Minimum Distance Martingale Measures

A minimum distance martingale measure minimizes the distance of the underlying statistical measure to the class of martingale measures (or a suitable subset of them). For this purpose f -divergences are considered as distances. Definition 6.4 Let Q, P ∈ P with Q ( P and let f : (0, ∞) → R be a convex function. The f -divergence between Q and P is defined by ⎧   ⎨ f dQ dP , if the integral exists, f (Q  P ) := dP ⎩ ∞, otherwise, where f (0) = limx↓0 f (x). A detailed treatment of f -divergences can be found in Liese and Vajda (1987). For f (x) = x log x the f -divergence is given by the relative entropy ⎧    ⎨ E dQ log dQ , if Q ( P , H (Q  P ) = dP dP ⎩ ∞, else. For f (x) = − log x the reverse relative entropy is obtained, for f (x) = |x√ − 1| the total variation distance, and for f (x) = − x the Hellinger distance. Let K be a convex dominated subset of probability measures on ( , A).

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6 Utility Optimization, Indifference Pricing

Definition 6.5 A probability measure Q∗ ∈ K is called f -projection (or minimum distance measure) of P on (in) K if f (Q∗  P ) = inf f (Q  P ) =: f (K  P ).

(6.9)

Q∈K

f -projections are uniquely determined. Proposition 6.6 (Uniqueness of f -Projections) Let the function f be strictly convex and f (K  P ) < ∞. Then there exists at most one f -projection of P on K. & := Proof Let Q1 and Q2 be two f -projections of P on K with Q1 = Q2 . Then Q 1 2 (Q1 + Q2 ) is a probability measure. Because of the strict convexity it holds & P) = f (K  P ) = inf f (Q  P ) ≤ f (Q Q∈K


0. According to Theorem 6.8 (i) it holds

 f





dQ∗ dP





∗

dQ −

f





dQ∗ dP

 dQ ≤ 0.

 ∗  But since Q dQ = 0 > 0 and according to the assumption f  (0) = −∞ we get dP a contradiction.   As an application, this yields a characterization of the minimum distance martingale measure. The set of martingale measures can be described as a generalized moment family. Let   G := (ϑ · S)T : ϑ i = H i 1(si ,ti ] , si < ti , H i bounded, Fsi -measurable ∪ {1B : P (B) = 0} and  Gloc := (ϑ · S)T : ϑ i = H i 1(si ,ti ] 1[0,T&i ] , si < ti ,

 H i bounded, Fsi -measurable, T&i ∈ γ i ∪ {1B : P (B) = 0} ,

$ i % & stopping time; (S i )T&i is bounded . where γ i := T The following theorem gives a necessary condition for an f -projection on the set M of martingale measures. Theorem 6.10 Let Q∗ ∈ Mf ∩ M and f  projection of P on M, then f





dQ∗ dP





T

=c+

ϑt dSt



dQ∗ dP



∈ L1 (Q∗ ). If Q∗ is the f -

Q∗ -almost surely

(6.12)

0

for a process ϑ ∈ L(S, Q∗ ) such that

· 0

ϑt dSt is a martingale with respect to Q∗ .

Proof The class M of martingale measures can be characterized as a momentfamily generated by G, M = {Q ∈ P : G ⊂ L1 (Q) and EQ g = 0

∀g ∈ G}

240

6 Utility Optimization, Indifference Pricing

Let F be the vector space generated by 1 and G. The conditions of Theorem 6.8 (ii) are satisfied and it, therefore, follows that   dQ∗ = ξ Q∗ -almost surely. f dP for a ξ ∈ L1 (F, Q∗ ), the L1 (Q∗ )-closure of f . Using a closure theorem for stochastic integrals of Yor (1978, Corollary 2.5.2), (for a multidimensional version, see Delbaen and Schachermayer (1999, Theorem 1.6)) we can identify the closure L1 (G, Q∗ ) with a class of stochastic integrals which are Q∗ -martingales; it holds that L1 (G, Q∗ ) ⊂ {(ϑ · S)T : ϑ ∈ L(S, Q∗ ), such that ϑ · S is a Q∗ martingale}. This result holds even without the assumption of a complete filtration, see Jacod (1979, Proposition 1.1). According to Schaefer (1971, Proposition I.3.3) we obtain the necessary condition since f is generated by 1 and G.   In Theorem 6.10 it is enough to assume that S is a general Rd+1 -valued semimartingale. In the next theorem, we show that the necessary condition in Theorem 6.10 also holds for the set of local martingale measures, under the additional condition that the price process S is locally bounded. Theorem 6.11 (Density of f -Projection)  ∗ Let S be a locally bounded semimartingale. Let Q∗ ∈ Mf ∩ Mloc and f  dQ ∈ L1 (Q∗ ). If Q∗ is the f -projection of dP P on Mloc , then f





dQ∗ dP





T

=c+

Q∗ -almost surely

ϑt dSt

(6.13)

0

for a process ϑ ∈ L1loc (S, Q∗ ) such that

· 0

ϑt dSt is a Q∗ -martingale.

Proof Consider the following characterization of Mloc as a moment-family Mloc = {Q ∈ P : Gloc ⊂ L1 (Q) and EQ g = 0 ∀g ∈ Gloc }. The assertion now follows analogously to the proof of Theorem 6.10.

 

Remark (1) If Q∗ ∼ P and if additionally −f 



dQ∗ dP



is bounded below, then according to  ∗ Goll and Rüschendorf (2001) the representability of f  dQ as a stochastic dP integral holds using Theorem 6.8 (i), the martingale representation result Theorem 3.4 from Jacka (1992) and Theorem 3.2 from Ansel and Stricker (1994). (2) If S is (locally) bounded, then Mloc is closed with respect to the variation distance. This can be seen by the characterization of Mloc with the help

6.2 Minimum Distance Martingale Measures

241

of Gloc (see Lemma 7.8 in Rheinländer and Sexton (2011)). If in addition limx→∞ f (x) x = ∞ then by Theorem 6.7 there exists an f -projection from P on Mloc . In particular, this condition is satisfied for the relative entropy. Next we give some sufficient conditions for an f -projection of P on Mloc . Proposition 6.12 Let Q∗ ∈ Mf ∩ Mloc and let f for

T 0



dQ∗ dP





T

=c+

ϑt dSt

P −almost surely

(6.14)

0

ϑt dSt ∈ Gloc . Then Q∗ is the f -projection of P on Mloc .

Proof Let F be the vector  space that generated by 1 and Gloc . Because of the ∗ T as an integral and because of 0 ϑt dSt ∈ Gloc one representation of f  dQ dP can apply Theorem 6.8 (iii).   The condition from Proposition 6.12 can be generalized as follows. Proposition 6.13 Let Q∗ ∈ Mf ∩ Mloc such that for a predictable process ϑ ∈ L(S) f

  T dQ∗ =c+ ϑt dSt P −almost surely, dP 0  . − ϑt dSt is P −almost surely bounded below,



0



T

EQ ∗

 ϑt dSt = 0

0

holds. Then Q∗ is the f -projection of P on Mloc . Proof By assumption ϑ ∈ L(S) is a predictable S-integrable process with respect to P and we obtain by Proposition 7.26 (b) from Jacod (1979) that ϑ in dimension d = 1 is S-integrable with respect to any Q ∈ Mloc . Using Proposition 3 from Jacod (1980) one can extend this result to dimensions d ≥ 1. According to the definition of Mloc := {Q ( P : Q is a (local) martingale measure} the pricing process S is a (local) Q-martingale. By Corollary 3.5 from Ansel and Stricker (1994) −ϑ · S is a local Q-martingale and since it is bounded below it is also a Q-supermartingale for every Q ∈ Mloc .

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6 Utility Optimization, Indifference Pricing

So EQ (ϑ · S)T ≥ EQ (ϑ · S)0 = 0 and it follows EQ f 



dQ∗ dP

 = c + EQ (ϑ · S)T ≥ c + EQ (ϑ · S)0  = c + EQ∗ (ϑ · S)T = EQ∗ f







=0

 dQ∗ . dP

=0

Since Q ∈ Mloc was chosen arbitrarily, the assertion follows with Theorem 6.8 (i). 

6.3

Duality Results

The utility maximization problem is related by a duality theorem of convex optimization to the problem of determining the minimum of the f -divergence distance. More precisely, the minimum distance martingale measures with respect to the convex conjugate of the utility function turn out to be equivalent to minimax measures. A characterization of minimum distance martingale measures then leads to a characterization of optimal portfolio strategies.

6.3.1

Minimum Distance Martingale Measures and Minimax Measures

Let K ⊂ M(P ) be a convex set of P -continuous martingale measures on ( , A) and let u be an utility function as in Sect. 6.1. For Q ∈ K and x > x we define UQ (x) := sup{Eu(Y ) : Y ∈ L1 (Q), EQ Y ≤ x, Eu(Y )− < ∞}.

(6.15)

One can interpret UQ (x) as the maximum expected utility that can be achieved with a capital stock X if the market prices are computed with Q. In doing so, one would like to avoid that the expected utility is −∞ that is to say, that ruin occurs.   Lemma 6.14 Let Q ∈ K and let EQ I λ dQ dP < ∞, ∀λ > 0. Then it holds   (i) UQ (x) = infλ>0 {Eu∗ λ dQ dP + λx}.

6.3 Duality Results

243

  (ii) For the equation EQ I λ dQ dP = x there is a unique solution for λ denoted by λQ (x) ∈ (0, ∞) and it holds    dQ . UQ (x) = Eu I λQ (x) dP Proof Let Y ∈ L1 (Q) with EQ Y ≤ x and Eu(Y )− < ∞. Then it holds for λ > 0: Eu(Y ) ≤ Eu(Y ) + λ(x − EQ Y )   dQ Y + λx = Eu(Y ) − E λ dP   dQ + λx ≤ Eu∗ λ dP       dQ dQ dQ (6.6) = Eu I λ −E λ I λ + λx dP dP dP       dQ dQ = Eu I λ + λ x − EQ I λ . dP dP   Equality holds exactly if Y = I λQ (x) dQ dP with λQ (x) a solution of the equation   dQ EQ I λ dP = x. According to (6.1) follows I (0) = ∞ and I (∞) = x. Together with the strict concavity of u it follows we obtain with the  that I is strictly decreasing.   Overall, dQ dQ condition EQ I λ dP < ∞ for all λ > 0 that EQ I λ dP is a function continuous in λ and monotonically decreasing with values in (x, ∞). Since x > x the unique existence of λQ (x) is guaranteed. −    It remains to be verified that E u I λQ (x) dQ < ∞. With a reformuladP tion of the Fenchel inequality in (6.6) u(x) − xy ≤ u(I (y)) − yI (y) one obtains       dQ − dQ dQ I λ < ∞. −λ E u I λ dP dP dP With the inequality    − dQ u I λ dP          dQ − dQ − dQ dQ dQ I λ I λ ≤ u I λ + λ −λ dP dP dP dP dP

(6.16)

244

6 Utility Optimization, Indifference Pricing

−    we obtain that the condition E u I λQ (x) dQ < ∞ is fulfilled. dP

 

Remark 6.15   (1) One can interpret I λQ (x) dQ dP as an optimal claim, which is financeable under the pricing measure Q with the capital x.   (2) If for Q ∈ K there exists a λ > 0 with Eu∗ λ dQ < ∞, then we can use dP Lemma 6.14 (i) to state that UQ (x)  for all x > x. If for Q ∈ K with  < ∞ UQ (x) < ∞ the assumption EQ I λ dQ dP < ∞   satisfied, then Eu∗ λQ (x) dQ dP < ∞.

∀λ > 0 from Lemma 6.14 is

(3) The convex conjugate of the utility function u(x) = 1 − e−x is u∗ (y) = 1 − y + y log y. The u∗ -divergence distance of the exponential utility function is thus Eu∗



dQ dP



   dQ dQ dQ =E 1− + log dP dP dP    dQ dQ =E log = H (Q  P ) dP dP

the relative entropy distance. Similarly, we see that the divergence distance associated with the logarithmic utility function i.e. the u∗ -divergence distance is the reverse relative entropy. p p p−1 (4) For log x, xp , 1−e−x the convex conjugate functions are − log x−1, − p−1 p x and 1 − x + x log x. Therefore, for u(x) = 1 − e−x the u∗ -divergence distance is the relative entropy, for u(x) = log x the reverse relative entropy and for u(x) = −x −1 the Hellinger distance. Definition 6.16 A measure Q∗ = Q∗ (x) ∈ K is called minimax measure for x and K if it minimizes the mapping Q → UQ (x) over all Q ∈ K, i.e. if holds UQ∗ (x) = U (x) := inf UQ (x). Q∈K

In general, the minimax measure Q∗ depends on x. For standard utility functions p such as u(x) = xp (p ∈ (−∞, 1)\{0}), u(x) = log x and u(x) = 1 − e−px (p ∈ (0, ∞)) the minimax measure is independent of x. Throughout this chapter we assume that ∃x > x with U (x) < ∞,   dQ < ∞, EQ I λ dP

∀λ > 0,

(6.17)

∀Q ∈ K.

(6.18)

6.3 Duality Results

245

Remark 6.17 (1) The assumption (6.18) is satisfied for u(x) = log x : Here I (y) = obtain     dQ 1 dP 1 EQ I λ = EQ = < ∞. dP λ dQ λ

1 y

and we

If for any Q ∈ K, u∗ (Q  P ) < ∞, then the assumption (6.18) is satisfied p likewise for u(x) = 1 − e−px for p ∈ (0, ∞) and u(x) = xp (p ∈   (−∞, 1)\{0}). For u(x) = 1 − e−px is I (y) = − p1 log py and we get      λ dQ 1 dQ = EQ − log EQ I λ dP p p dP      1 λ dQ 1 = − log < ∞. − EQ log p p p dP

 0 : u∗λ (Q  P ) < ∞}. We denote by ∂U (x) the subdifferential of the function U at x. For f (x) = u∗ (λ0 x) we denote the corresponding f -divergence by u∗λ0 (·  ·). The following proposition establishes an important connection between minimax measures and minimum distance martingale measures. Proposition 6.18 Let x > x, λ0 ∈ ∂U (x), λ0 > 0. Then it holds (i) U (x) = u∗λ0 (K  P ) + λ0 x. (ii) If Q∗ ∈ K is a u∗λ0 -projection of P on K, then Q∗ is a minimax measure and λ0 = λQ∗ (x). (iii) If Q∗ ∈ K is a minimax measure, then Q∗ is a u∗λ ∗ (x) -projection of P on Q K,λQ∗ (x) ∈ ∂U (x) and it holds UQ∗ (x) = inf UQ (x) = sup{Eu(Y ); sup EQ Y ≤ x} Q∈K

where K(x) := {Q ∈ K; UQ (x) < ∞}.

Q∈K(x)

246

6 Utility Optimization, Indifference Pricing

The proposition shows that minimax measures can be described by minimum distance measures. The converse is also true that u∗λ0 -projections can be interpreted in terms of utility maximization. Proof (i) By Lemma 6.14 it holds  # "  dQ U (x) = inf UQ (x) = inf inf Eu∗ λ + λx dP Q∈K Q∈K λ>0 $ ∗ % = inf uλ (K  P ) + λx .

(6.19)

λ>0

Define H : (0, ∞) → R ∪ {∞}by H (λ) := u∗λ (K  P ). By Remark 6.15, the assumptions (6.17) and (6.18) guarantee that there is a λ > 0 with H (λ) < ∞. Therefore, from the representation of U (x ) in (6.19), it follows that U (x) < ∞ for all x ∈ dom(u) = dom(U ). Thus we obtain H (λ) < ∞ for all λ > 0. Now to show is: H is a convex function. Let ε > 0 and let Q1 , Q2 ∈ K such that   dQ1 H (λ1 ) + ε ≥ Eu∗ λ1 and dP   dQ2 . H (λ2 ) + ε ≥ Eu∗ λ2 dP Herewith and with the convexity of u∗ it holds: γ H (λ1 ) + (1 − γ )H (λ2 ) + 2ε     dQ1 dQ2 ∗ ∗ ≥ γ Eu λ1 + (1 − γ )Eu λ2 dP dP   dQ1 dQ2 + (1 − γ )λ2 ≥ Eu∗ γ λ1 dP dP = Eu∗ ((γ λ1 + (1 − γ )λ2 )   (1 − γ )λ2 γ λ1 dQ1 dQ2 + · γ λ1 + (1 − γ )λ2 dP γ λ1 + (1 − γ )λ2 dP   (∗) dQ ≥ inf Eu∗ (γ λ1 + (1 − γ )λ2 ) = H (γ λ1 + (1 − γ )λ2 ) . dP Q∈K To (∗): Since K is a convex set and since for Q1 , Q2 ∈ K it holds (1 − γ )λ2 γ λ1 dQ1 dQ2 + ∈ K. γ λ1 + (1 − γ )λ2 dP γ λ1 + (1 − γ )λ2 dP

6.3 Duality Results

247

Since H is a convex function, we can use the equivalence (a) ⇐⇒ (b) from Theorem 23.5 from Rockafellar (1970) and so infλ>0 {H (λ) + λx} takes its infimum in λ at λ = λ0 if and only if −x ∈ ∂H (λ0 ) or equivalently exactly if λ0 ∈ ∂U (x). We thus obtain by Eq. (6.19) that U (x) = u∗λ0 (K  P ) + λ0 x. (ii) This statement follows from Lemma 6.14:  # "  dQ∗ UQ∗ (x) = inf Eu∗ λ + λx λ>0 dP $ ∗ % = inf uλ (K  P ) + λx λ>0

= U (x). (iii) If Q∗ ∈ K is a minimax measure, it follows by Lemma 6.14 (i) that Q∗ a u∗λ ∗ (x)-projection of P on K, since one obtains similarly to Eq. (6.19): Q

 # "  dQ ∗ + λx Eu λ U (x) = inf UQ (x) = inf inf dP Q∈K Q∈K λ>0 $ ∗ % = inf uλ (K  P ) + λx λ>0

= UQ∗ (x). For the second part of the statement it holds:  ∗  uλ (x) = −λI (λx),

(6.20)

since    ∗  uλ (x) = u∗ (λx) = (u(I (λx)) − λxI (λx)) = u (I (λx))I  (λx)λ − λI (λx) − λxI  (λx)λ = u ((u )−1 (λx))I  (λx)λ − λI (λx) − λxI  (λx)λ = λxI  (λx)λ − λI (λx) − λxI  (λx)λ = −λI (λx). By Remark 6.17 it holds K(x) = {Q ∈ K : UQ (x) < ∞} = {Q ∈ K : u∗λQ∗ (x) (Q  P ) < ∞}. That is, the conditions of Theorem 6.8 are satisfied. Because of the first part of (iii) Q∗ is a u∗λ ∗ (x) -projection and with Theorem 6.8 (i) and Eq. (6.20) it Q

248

6 Utility Optimization, Indifference Pricing

follows. E

Q∗

    dQ∗ dQ∗ ∗ ∗ ≥ EQ I λQ (x) . I λQ (x) dP dP

With Lemma 6.14 (ii) it follows UQ∗ (x) = sup{Eu(Y ) : sup EQ Y ≤ x}. Q∈K(x)

  As a corollary of Proposition 6.18 we obtain Corollary 6.19 Let x > x, λ0 ∈ ∂U (x) and λ0 > 0. Further, let U be differentiable in x. Then Q∗ is a minimax measure if and only if Q∗ is the u∗λ0 -projection of P on K, where λ0 = ∇U (x). Proof Because of the assumption of differentiability, the subdifferential consists of only one point, which is the derivative at the point x. Now the statement follows directly from Proposition 6.18.   In general one knows U (x) often not explicitly and, therefore, the verification of the condition λ0 ∈ ∂U (x) is difficult. In the case of the standard utility functions p u(x) = 1 − e−px (p ∈ (0, ∞)), u(x) = xp (p ∈ (−∞, 1)\{0}) and u(x) = log x the minimax measure does not depend on X and accordingly the u∗λ -projection does not depend on λ and so it does not pose the problem to determine λ. Proposition 6.20 Let x = 0 and let u be bounded from above. Then U is differentiable in every x > 0.  Proof Sketch According to Theorem 26.3 from Rockafellar (1970), the main argument is to show that H (λ) = u∗λ (K  P ) is strictly convex. The strict convexity of H (λ) is shown by means of the convexity of K, Alaoglu’s theorem, the theorem of dominated convergence and the strict convexity of u∗ .   The differentiability condition from Corollary 6.19 is satisfied if for all λ > 0 there exists a u∗λ -projection. Proposition 6.21 Let K be closed with respect to the topology of the variational distance, and let dom(u) = (−∞, ∞). Then for all λ > 0 there is a u∗λ -projection of P on K. ∗

Proof According to Theorem 6.7 it suffices to show that limx→∞ u (λx) = ∞.  x   Since u∗ (λx) = sup{u(y) − λxy} ≥ u − nλ + nx it follows with u − nλ > −∞, that

y u∗ (λx) limx→∞ x

≥ n for each n ∈ N.

 

6.3 Duality Results

249

The sufficient conditions for projections from Sect. 6.2 provide a way to determine the parameter λ0 ∈ ∂U (x) and hence also the f -divergence distance associated with a minimax measure. Proposition 6.22 Let Q∗ ∈ M(Mloc), λ > 0 where u∗λ (Q∗  P ) < ∞ such that for an S-integrable process ϑ it holds:    T dQ∗ =x+ ϑt dSt I λ dP 0

P -almost surely.

If, in addition, the sufficient conditions for a u∗λ -projection from Proposition 6.12 hold (Theorem 6.13), then Q∗ is a minimax measure for x and λ ∈ ∂U (x). Remark 6.23 Note that the conditions in Proposition 6.12 and in Proposition 6.13 are formulated for (u∗λ ) (x) = −λI (λx). Proof In the following K stands either for M or for Mloc  . If the conditions T from Theorem 6.13 hold, then it is already true that EQ∗ 0 ϑt dSt = 0. If the conditions from Proposition 6.12 6.11  · hold then Theorem 6.10 or Theorem ∗ . So it holds ϑ dS is a martingale under Q can beapplied and it follows that t t 0  ∗

= x and with Lemma 6.14 (ii) we obtain that λ = λQ∗ (x). Because EQ∗ I λ dQ dP of Proposition 6.12 (Theorem 6.13) is Q∗ the u∗λ -projection of P on K, i.e. the assumptions of Theorem 6.8 are satisfied. From Theorem 6.8 and from Eq. (6.20) it follows for all measures Q ∈ K with u∗λ (Q  P ) < ∞, that      dQ∗ dQ∗ ≤ EQ u∗λ dP dP     ∗ ∗ dQ dQ EQ I λ ≤ EQ ∗ I λ = x. dP dP

  EQ∗ u∗λ ⇐⇒



  ∗ is financeable with respect to all measures Q ∈ K. The That is, the claim I λ dQ dP maximum expected utility of claims which are Q ∈ K financeable is at least  w.r.t. dQ∗ as large as the expected utility of the claim I λ dP .    ∗ It holds UQ (x) ≥ Eu I λ dQ = UQ∗ (x) for all Q ∈ K. dP According to Remark 6.15 it holds because of assumption (6.18) that {Q ∈ K : u∗λ (Q  P ) < ∞} = {Q ∈ K : UQ (x) < ∞}. Thus UQ∗ (x) < ∞ and it follows that Q∗ is the minimax measure for X and K. By Proposition 6.18 it thus follows λ = λQ∗ (x) ∈ ∂U (x).  

250

6 Utility Optimization, Indifference Pricing

6.3.2

Relationship to Portfolio Optimization

The topic of this subsection is to show a connection of the results in Sect. 6.3.1 on minimum distance martingale measures and minimax measures to the portfolio optimization problem in (6.2). In general, we assume the conditions (6.17) and (6.18) for K = Mloc as well as for x > −∞. For a solution of the portfolio optimization problem, admissibility conditions must be imposed on the class of strategies to exclude doubling strategies. Definition 6.24 (Admissible Trading Strategies) The set of admissible trading strategies is given by # "   !0 = ϑ ∈ L(S)  ϑ dS is uniformly bounded from below , " #   !1 = ϑ ∈ L(S)  ϑ dS is a Q∗ −supermartingale , " #   f !2 = ϑ ∈ L(S)  ϑ dS is a Q−supermartingale for allQ ∈ M . Furthermore, let the sets !1 and !2 be defined by replacing in !1 respectively !2 the condition ‘supermartingale’ by ‘martingale’. !0 represents the class of admissible trading strategies for a trader with finite credit This definition is used by Goll and Rüschendorf (2001) and here it holds  d limit. i S i = x + t ϑ dS for each t ∈ [0, T ]. The sets ! and ! are used for the ϑ i=0 t t 1 2 0 exponential utility in Rheinländer and Sexton (2011). A discussion of admissible strategies can be found in Schachermayer (2003). For x = −∞ the optimal portfolio is in general not bounded below and thus the optimal portfolio strategy is not included in !0 . The following theorem relates minimum distance martingale measures to minimax measures and to optimal portfolio strategies.  Theorem 6.25 Let Q∗ ∈ Meloc such that u∗λ0 (Q∗  P ) < ∞ and I λ0 L1 (Q∗ ),

dQ∗ dP



let S be (locally) bounded, and let λ0 ∈ ∂U (x). Then

(i) The following statements are equivalent: (a) Q∗ isa (local) minimumdistance martingale measure. ∗ ∗ (b) EQ I λ0 dQ ≤ EQ∗ I λ0 dQ , ∀Q ∈ Mloc with u∗λ0 (Q  P ) < ∞. dP dP    ∗ T =x+ 0 & (c) I λ0 dQ ϑ dS and & ϑ ∈ !1 . dP

∈

6.3 Duality Results

251

(ii) If (c) holds, then for i = 0, 1, 2 it holds   sup Eu x + ϑ∈!i



T

ϑ dS

  = Eu x +

0

T

& ϑ dS

 = UQ∗ (x) = U (x),

0

and & ϑ ∈ !i for i = 0, 1, 2. Furthermore & ϑ is the optimal strategy in !1 . ∗ (iii) If (a) holds then Q is a (local) minimax measure. Proof According to Eq. (6.20) it holds (u∗λ ) (x) = −λI (λx). (i) The equivalence (a) ⇔ (b) follows by Theorem 6.8 (i): Q∗ ∈ Meloc with u∗λ0 (Q∗  P ) < ∞ is a (local) minimum distance martingale measure if and only if   EQ∗ u∗λ0



dQ∗ dP



  ≤ EQ u∗λ0



dQ∗ dP

 ,

∀Q ∈ Mloc

with u∗λ0 (Q  P ) < ∞. By Eq. (6.20) we get  dQ∗ dP     ∗ dQ dQ∗ ⇐⇒ EQ∗ I λ0 ≥ EQ I λ0 . dP dP   EQ∗ u∗λ0



dQ∗ dP



  ≤ EQ u∗λ0



(a) ⇒ (c): If Q∗ is a (local) minimum distance martingale measure, then from 6.10 or Proposition 6.12 and Eq. (6.20) the representation of   Theorem · ∗ follows with a Q∗ -martingale 0 & ϑ dS for a S-integrable, preI λ0 dQ dP  dictable process & ϑ , i.e. & ϑ ∈ !1 . So it remains to show the direction (c) ⇒ (a). From I : R → (x, ∞) and the assumption that x > −∞ we obtain that   ϑ · S T ≥ x > −∞. Moreover, since & x+ & ϑ · S is a Q∗ -martingale and Q∗ ∼ P , it follows that −& ϑ · S is P -almost surely bounded from below. Thus, we obtain by Theorem 6.13 that Q∗ is the u∗λ0 -projection of P on Mloc . (ii) First, we show that the strategy & ϑ is in the respective set of admissible ϑ also in !1 . strategies. According to (c) & ϑ ∈ !1 and since !1 ⊂ !1 is & As in the proof of (i), for a process & ϑ satisfying condition (c), we can conclude that & ϑ · S is bounded from below and thus & ϑ ∈ !0 . Further, using 3.5  Corollary  from Ansel and Stricker (1994) it can be concluded that & ϑ · S T is a Qsupermartingale for all Q ∈ Mf and thus it is in !2 . It remains to show that & ϑ is the optimal strategy. According to Corollary 3.5 from Ansel and Stricker (1994) ϑ · S is a local Q∗ -martingale and for each ϑ ∈ !0 it is also bounded from below, and thus is a Q∗ -supermartingale. For ϑ ∈ !1 , !2 and ϑ ∈ !1 the supermartingale property follows directly.

252

6 Utility Optimization, Indifference Pricing

Because of the supermartingale property it holds EQ∗ [x + (ϑ · S)T ] ≤ EQ∗ [x + (ϑ · S)0 ] = x. By analogy with Lemma 6.14 (ii), for i = 0, 1, 2 we can conclude that   sup Eu x +



T

ϑ dS

  = Eu x +

0

ϑ∈!i

T

& ϑ dS



0

and thus & ϑ is an optimal portfolio strategy. In the same way, it follows that & ϑ is the optimal strategy in !1 . Furthermore, according to (c).   Eu x +

T 0

& ϑ dS



   dQ∗ = Eu I λ0 = UQ∗ (x). dP

Therefore, according to Proposition 6.18, it follows that UQ∗ (x) = U (x). (iii) Since Q∗ is a (local) minimum distance martingale measure and the preconditions of Proposition 6.18 are satisfied, we obtain by applying Proposition 6.18 (ii) that Q∗ is a (local) minimax measure.   Remark 6.26   ϑ · S T ≥ −∞ holds, from which it (1) If x = −∞, then in (i) and (ii) only x + & does not follow in part (i) that −& ϑ · S is P -almost surely bounded from below. Furthermore, in part (ii) we cannot conclude that & ϑ ∈ !0 . Consequently, it is not sufficient in case x = −∞ to use !0 as the set of admissible strategies. In this case, one uses the larger class !2 (cf. Schachermayer 2003). (2) For the exponential utility, the optimal strategy & ϑ is also in !2 . For general  utility functions, however, it is not in !2 . Corollary 6.27 Let Q∗ be a minimax measure for X and Mloc .   &  P < ∞ for a measure Q & ∈ Me then Q∗ ∼ P . (i) If u∗ Q loc ∗ (ii) If Q ∼ P and S is (locally) bounded, then    T dQ∗ & I λQ∗ (x) ϑ dS, =x+ dP 0 where & ϑ is an optimal portfolio strategy and it holds U (x) = UQ∗ (x) = sup{Eu(Y ) : EQ Y ≤ x f or all Q ∈ Mloc }.

6.3 Duality Results

253

Proof (i) Proposition 6.18 (iii) shows that Q∗ is a u∗λ ∗ (x) -projection. Since the u∗  & Q   &  P = Eu∗ d Q distance is finite, u∗ Q dP < ∞ and the assumption (6.18) is satisfied, we obtain, by Remark 6.15, UQ & (x) < ∞ and thus according to   &  P < ∞. Because of Eq. (6.20) and I (0) = ∞ Remark 6.17, that u∗λ ∗ (x) Q  Q    it holds u∗λ ∗ (x) (0) = −λQ∗ (x)I λQ∗ (x) · 0 = −∞. Now the conditions Q of Corollary 6.9 are satisfied and it follows Q∗ ∼ P . (ii) By Proposition 6.18 (iii) it follows that Q∗ is a u∗λ ∗ (x) -projection, λQ∗ (x) ∈ Q ∂U (x) with UQ∗ (x) < ∞ and thus u∗λ ∗ (x) (Q∗  P ) < ∞. With Theorem 6.25 Q (i), therefore, it follows that     dQ∗ I λQ∗ (x) =x+ & ϑ ·S T , dP

(6.21)

where & ϑ ∈ !1 . With Theorem 6.25 (ii) it then follows that   sup Eu x + ϑ∈!i



T

ϑ dS

  = Eu x +

0

T

& ϑ dS

 = UQ∗ (x) = U (x)

0

for i = 0, 1, 2 and & ϑ ∈ !i for i = 0, 1, 2. Likewise, & ϑ is the optimal strategy in !1 . So it remains to show that UQ∗ (x) = sup{Eu(Y ) : EQ Y ≤ x for all Q ∈ Mloc }. According to Corollary 3.5 from Ansel and Stricker (1994) & ϑ ·S is a local Qmartingale and because of the representation (6.21) and since I : R → (x, ∞) it is a Q-supermartingale for each Q ∈ Mloc . Thus, for all Q ∈ Mloc we obtain '   ( '   ( EQ x + & ϑ · S T ≤ EQ x + & ϑ · S 0 = x. Therefore, the optimal claim from (6.21) is financeable with respect to any price measure Q ∈ Mloc .    dQ∗ ∗ (x) = Eu I λQ∗ (x) . Suppose By Lemma 6.14 it holds that U Q dP   dQ∗  ) =  with EQ Y ≤ x for all Q ∈ Mloc and Eu(Y there was a Y = I λQ∗ (x) dP

sup{Eu(Y ) : EQ Y ≤ x for all Q ∈ Mloc }. Since Q∗ ∈ Mloc it also ) which is a  ≤ x and thus as a consequence UQ∗ (x) = Eu(Y holds EQ∗ Y   ∗

contradiction to the fact that I λQ∗ (x) dQ dP

is an optimal claim. So it holds

% $ UQ∗ (x) = sup Eu(Y ) : EQ Y ≤ x for all Q ∈ Mloc .  

254

6 Utility Optimization, Indifference Pricing

Remark 6.28 If one computes derivative prices using a minimax or a minimumdistance martingale measure Q∗ , then Theorem 6.25 shows, that the optimal claim, i.e., the solution to problem (6.15) for Q = Q∗ can be hedged by a portfolio strategy & ϑ . Thus, no trading in derivatives increases the maximum expected utility relative to an optimal portfolio if prices are calculated by Q∗ . It then holds     ϑ · S T ≥ Eu(Y ) Eu x + & for all claims Y such that EQ∗ Y ≤ x, i.e. for all claims that can be financed under the price measure Q∗ . Davis proposes another method for pricing a claim. If the price of a contingent claim is defined by Davis’ fair price, then one cannot increase the maximum expected utility of the final asset by taking an infinitesimal long or short position of the derivative in comparison to an optimal portfolio. Under regularity assumptions the fair price of Davis of a contingent claim B is obtained as a function of the initial financial x by p(B) =

'     ( E u x + & ϑ ·S T B const

(6.22)

,

where & ϑ is the optimal portfolio strategy. If one puts in formula (6.22) the representation of the density of the minimum distance martingale measure from Theorem 6.25 (i), then one obtains Davis’ fair price as the expected value of the claim B with respect to the minimum distance martingale measure. Thus, neither infinitesimal nor general trading increases the maximum expected utility. Corollary 6.29 Let the assumptions from Theorem 6.25 hold. If Q∗ is the minimum distance martingale measure, then Davis’ fair derivative price is given by p(B) = EQ∗ B.

(6.23)

If we consider the exponential utility, then the duality results also hold for !2 . For this we consider for p > 0 the portfolio optimization problem (6.2) for i = 0, 1, 2 and likewise for !1 and !2 , i.e.     sup E 1 − exp −p x +



T

ϑt dSt

(6.24)

0

ϑ∈!i

and the associated minimum distance problems, respectively. For these, we obtain: Theorem 6.30 It holds   inf  E exp −p ϑ∈!i

0



T

ϑt dSt

  = exp −H (Q∗  P ) ,

(6.25)

6.4 Utility-Based Hedging

255

where Q∗ is the minimum entropy martingale measure and the infimum is attained ϑ ∈ !i for i = 1, 2. by − p1 & Proof Sketch First, the result for !1 is shown. Here the representation    T dQ∗ & = exp x + ϑt dSt , dP 0

& ϑ ∈ !1 ,

of the density of the minimal entropy martingale measure (see Rheinländer and Sexton 2011, Theorem 7.14) is fundamental. As an additional tool, the Jensen inequality is used. Then the result is extended to !2 . The same procedure as for the first case yields Eq. (6.25). Then it must be shown that − p1 & ϑ ∈ !2 . To do this, one shows using Barron’s inequality, see $ Proposition % 7.21 from Rheinländer and Sexton (2011), the τ uniform integrability of 0 & ϑt dSt τ , where τ is from the set of stopping times with  values in [T , T ]. It then follows with a result of Chou et al. (1980) that & ϑ dS is a Q-martingale for all Q ∈ Mf .  Thus, for the exponential utility, the   entropic duality is valid for !2 , thef set of admissible strategies such that & ϑ · S T is a Q-martingale for all Q ∈ M . The duality results for !2 , however, cannot be extended to general utility functions u : R → R because the class !2 is too small to contain the optimal strategy (cf. Schachermayer 2003, Proposition 4).  

6.4

Utility-Based Hedging

For the utility-based hedging problem (6.3)   sup Eu x +

ϑ∈E

T

 ϑt dSt − B

0

with E = !i , i = 0, 1, 2 of a contingent claim B ≥ 0 there is a similar duality theory as for the corresponding portfolio optimization problem in Sect. 6.3.2. We are looking for an admissible hedging strategy which, given the initial capital x hedges the claim B with respect to the cost function u as well as possible. u describes the risk aversion of an investor. If x := inf{x ∈ R; u(x) > −∞} ≥ 0, then the criterion (6.3) only admits superT hedging strategies, since in this case x + 0 ϑ dS − B ≥ 0. To allow for more general strategies, suppose that −∞ < x < 0

and

x−x >

sup EQ B.

Q∈Meloc

256

Here

6 Utility Optimization, Indifference Pricing

sup EQ B is the minimum cost of a super-hedging strategy. We choose x

Q∈Meloc

small enough to not exclusively allow super-hedging strategies, and large enough to not exclude super-hedging strategies. Analogously to Sect. 6.3.2 we define for Q ∈ K and x > x B UQ (x) := sup{Eu(Y − B); EQ Y ≤ x, Eu(Y − B)− < ∞}. B (x) corresponds to the maximum utility to be expected from utility-based UQ hedging of the claim B, which can be achieved with a capital x that can be achieved when market prices are computed by Q. The following lemma generalizes Lemma 6.14, and the proof is similar.

    + B < ∞ ∀λ > 0. Then it Lemma 6.31 Let Q ∈ Mloc and EQ I λ dQ dP holds      B (x) = inf ∗ λ dQ + λ x − E B . (i) UQ λ>0 Eu Q   dP   (ii) For the equation EQ I λ dQ dP + B = x there exists a unique solution for λ denoted by λQ (x) ∈ (0, ∞) and it holds B UQ (x)

   dQ = Eu I λQ (x) . dP

  1 − Proof Let I λ dQ dP + B ∈ L (Q) with EQ Y ≤ x and Eu(Y − B) < ∞. Then λ>0 Eu(Y − B) ≤ Eu(Y − B) + λ(x − EQ Y )   = Eu(Y − B) + λ x − EQ [Y − B + B]     dQ (Y − B) + λ x − EQ B = Eu(Y − B) − E λ dP     dQ ≤ Eu∗ λ + λ x − EQ B dP         dQ dQ dQ −E λ I λ + λ x − EQ B =E I λ dP dP dP         dQ dQ + λ x − EQ I λ +B . = Eu I λ dP dP (6.26)

6.4 Utility-Based Hedging

257

  Equality holds if and only if Y − B = I λQ (x) dQ dP with λQ (x) is a solution of the     equation EQ I λ dQ dP + B = x. The unambiguous existence of λQ (x) and the fact that    − dQ E u I λQ (x) 0 dP  # "    dQ∗ dQ∗ ∗ = inf E u λ B + λx −λ λ>0 dP dP   # "   dQ dQ = inf inf −λ B + λx E u∗ λ λ>0 Q∈Mloc dP dP  # "    dQ dQ B + λx = inf inf E u∗ λ −λ dP dP Q∈Mloc λ>0 =

inf

Q∈Mloc

B UQ (x) = U B (x).

Altogether it holds $ % U B (x) = sup Eu(Y − B) : EQ∗ Y ≤ x, Eu(Y − B)− < ∞    dQ∗ , = Eu I λ0 dP   and EQ∗ I λ0

dQ∗ dP





= x. Since & ϑ · S is a Q∗ -martingale it follows     ∗ +B =x+ & ϑ ·S T. c = x and it holds I λ0 dQ dP (c) ⇒ (b) With I : R → (x, ∞) and since B is nonnegative it holds that +B

    x+ & ϑ ·S T ≥ x + & ϑ · S T − B ≥ x. As in the proof of Theorem 6.25 & ϑ · S is a Q∗ -martingale and with Q∗ ∼ P , it follows that & ϑ · S is P almost surely bounded from below. According to Ansel and Stricker (1994, Corollary 3.5) it follows that & ϑ · S is a local Qmartingale and hence also a Q-supermartingale for every Q ∈ Mloc . Thus,

262

6 Utility Optimization, Indifference Pricing

using assumption (c) and the supermartingale property, we obtain       dQ∗ ϑ ·S T + B = x + EQ & EQ I λ0 dP   ϑ ·S 0 =x ≤ x + EQ &     dQ∗ = EQ∗ I λ0 +B . dP (ii) First, we show that the strategy & ϑ is in the respective set of admissible ϑ satisfying (c), strategies. According to (c) & ϑ ∈ !1 ⊂ !1 . For a process & we can conclude, as in (i), that & ϑ · S is P -almost surely bounded below, and  thus & ϑ ∈ !0 . As above, it follows that & ϑ · S T is a Q-supermartingale for all ϑ ∈ !2 . Q ∈ Mf is and thus & If ϑ ∈ !i for i = 0, 1, 2 respectively ϑ ∈ !1 is an admissible strategy, then Eu (x + (ϑ · S)T − B)  '    ϑ ·S T −B ≤E u x+ &       ( ϑ · S T − B (ϑ · S)T − & ϑ ·S T +u x + &   dQ∗           ϑ ·S T =E u x+ & ϑ · S T − B + u I λ0 (ϑ · S)T − & dP    ∗        dQ ϑ ·S T − & ϑ ·S T =E u x+ & ϑ · S T − B + λ0 dP     & ≤ Eu x + ϑ · S T − B . Since u is concave −u convex; hence the The first  first∗ inequality holds. T & + B = x + inequality holds with the representation I λ0 dQ ϑ dS from 0 dP   −1 . (c). The second equality follows with I := u As before ϑ · S − & ϑ · S is a local Q∗ -martingale. For the last inequality, we differentiate in the argument according to whether we consider the supremum over ϑ ∈ !0 or over ϑ ∈ !i for i = 1, 2 resp. ϑ ∈ !1 respectively: If ϑ ∈ !0 , then ϑ · S is bounded below. As in part (i), we can conclude that −& ϑ · S is almost surely bounded from below. Thus also ϑ · S − & ϑ · S is almost ∗ surely bounded below and, therefore, since it is also a local Q  -martingale,  ( ' ∗ it is a Q -supermartingale. So it holds EQ∗ (ϑ · S)T − & ϑ ·S T ≤ '   ( EQ∗ (ϑ · S)0 − & ϑ · S 0 = 0. The last inequality follows, since λ0 > 0. If ϑ ∈ !i for i = 1, 2 respectively ϑ ∈ !1 then ϑ · S a Q∗ -supermartingale and it holds EQ∗ (ϑ · S)T ≤ EQ∗ (ϑ · S)0 = 0.

6.5 Examples in Exponential Lévy Models

263

Analogously to part (i) one sees that −& ϑ · S is almost surely bounded below, so it is also a Q∗ -supermartingale. Thus it holds '   ( '   ( EQ ∗ − & ϑ · S T ≤ EQ ∗ − & ϑ ·S 0 =0 

and the last inequality follows, since λ0 > 0.

Finally, we conclude this section by giving the duality result for the exponential utility with claim B (cf. Rheinländer and Sexton 2011). Corollary 6.34 For any x ∈ R holds     sup E 1 − exp −p x + ϑ∈!i

T

 ϑt dSt − B

0

   1 H (Q  P ) + x − EQ B = 1 − exp −p inf . Q∈Mf p

and the supremum is attained for & ϑ ∈ !2 . The proof of this result is based on an application of Theorem 6.30, on measure change and on the representation of the density of the minimal entropy martingale measure.

6.5

Examples in Exponential Lévy Models

Exponential Lévy processes are important models for pricing processes. In this chapter, using the results from Sects. 6.2 and 6.3 optimal portfolios and minimum distance martingales are determined for some standard utility functions. Exponential Utility Function The distance to be minimized for the exponential utility function u(x) = 1−e−px is the relative entropy given by the function f (x) = x log x. Using the necessary and sufficient conditions from the Propositions 6.10–6.13 we obtain for the minimum distance martingale measure a representation of the density of the form dQ∗ 1 = exp (−p (x + (ϑ · S)T )) . dP λ0

(6.29)

Theorem 6.13 provides a sufficient condition: If the stochastic integral from (6.29) is P -a.s. bounded from below and is a Q∗ -martingale, then Q∗ is the minimum entropy martingale measure, and ϑ is an optimal portfolio strategy. In general, the question is whether (ϑ · S)T is bounded from below, i.e., the admissibility of ϑ, is an interesting point.

264

6 Utility Optimization, Indifference Pricing

Let X = (X1 , . . . , Xd ) be a Rd -valued Lévy process, E be the stochastic exponential and let the positive price process S = (S 1 , . . . , S d ) be given by S i = S0i E(Xi ).

(6.30)

 i  for a These processes also admit a representation of the form S i = S0i exp X d  R -valued Lévy process X (see Goll and Kallsen 2000). Let (B, C, f ) be the characteristic triplet (the differential characteristic). of X with respect to a truncation function h : Rd → Rd . Furthermore, assume there exists a γ ∈ Rd with the following properties:  1.

b − cγ +

2.

|xe−γ

2x

− h(x)|F ( dx) < ∞,

   2 xe−γ x − h(x) F ( dx) = 0.

(6.31)

t  for i = 1, . . . , d, ϑt0 := x + 0 ϑs dSs − di=1 ϑti Sti for t ∈ (0, T ].      2 Define Zt = E −γ 2 Xsc + e−γ x − 1 ∗ μX − ν s . t The minimum distance martingale measure with respect to the relative entropy is described in the following corollary. Let ϑti :=

γi i St−

∗

Corollary 6.35 The measure Q∗ defined by dQ dP = ZT is an equivalent local martingale measure. If γ · X is bounded below, then Q∗ minimizes the relative entropy between P and Mloc , i.e. H (Q∗ | P ) = H (Mloc | P ). Proof In Kallsen (2000) it is shown that Z as defined above is a martingale and S i Z ∗ is a local martingale with respect to P for i ∈ {1, . . . , d}. The density ZT = dQ dP of Q∗ with respect to P is a stochastic exponential and has a representation as in (6.29) i with ϑti := γi . Furthermore EQ∗ (ϑ ·S)T = 0. This implies that the relative entropy St−  ∗ < ∞. between Q∗ and P is finite, i.e. EQ∗ log dQ dP If the process is ϑ · S = γ · X is bounded from below, then according to Theorem 6.13 it follows that Q∗ minimizes the relative entropy between P and Mloc .   The condition that γ · X = ϑ · S is bounded from below is not satisfied in general. In the context of portfolio optimization for the unconstrained case, this issue is discussed in Kallsen (2000) and Schachermayer (2001).

6.5 Examples in Exponential Lévy Models

265

Logarithmic Utility Function To the logarithmic utility function u(x) = log x the associated f -divergence distance with f (x) = − log x is the reverse relative entropy. The characteristic (B, C, ν) of the Rd -valued semimartingale (S 1 , . . . , S d ) with respect to a fixed truncation function h : Rd → Rd is given by 



·

B=

bt dAt , 0

·

C=

ct dAt ,

ν = A ⊗ F,

(6.32)

0

d where A ∈ A+ loc is a predictable process, B a predictable R -valued process, C a d×d -valued process whose values are nonnegative symmetric matrices, predictable R and f a transition kernel of ( × R+ , P) according to (Rd , Bd ). T hereby A+ loc describes the class of locally integrable, adapted, increasing processes, and P the predictable σ -algebra. Furthermore, let there exist a Rd -valued, S-integrable process H with the following properties:

1. 1 + Ht2 x > 0 for (A ⊗ F )-almost all (t, x) ∈ [0, T ] × Rd ,    x   2. − h(x)  Ft ( dx) < ∞ (P ⊗ A)-almost surely on × [0, T ], 2 1 + Ht x   3. bt − ct Ht +

 x − h(x) Ft ( dx) = 0 1 + Ht2 x

(6.33)

(P ⊗ A)-almost surely on × [0, T ]. Let for t ∈ (0, T ]  ϑti := xHti E



·

Hs dSs 0

|t−

 for i = 1, . . . d, ϑt0 := x +

t

ϑs dSs −

0

d 

ϑti Sti .

i=1

In Theorem 3.1 from Goll and Kallsen (2000) it is shown that ϑ, defined as above, is an optimal portfolio strategy for the logarithmic utility maximization problem. Based on this, Theorem 6.13 gives a characterization of the local martingale measure that minimizes the reverse relative entropy.      Corollary 6.36 If Zt := E −H · Ssc + 1+H1 2 x − 1 ∗ μS − ν s is a mart

tingale, then the associated measure Q∗ with density ZT is an equivalent local martingale measure and it minimizes the reverse relative entropy.

Proof According to Goll and Kallsen (2000, proof of Theorem 3.1) (Zt ) is a positive local martingale such that S i Z is a local martingale for i ∈ {1, . . . , d}. Further ZxT = x + (ϑ · S)T and ϑ · S is bounded from below. If as assumed Z is

266

6 Utility Optimization, Indifference Pricing

a martingale then ZT is the density of an equivalent local martingale measure Q∗ . For Q∗ it holds      dQ∗ 1 E − log = E log = E log (x + (ϑ · S)T ) − log x. dP ZT As in the proof of Lemma 6.14, it follows that 



E − log

dQ∗ dP



   dQ + log x = E log (x + (ϑ · S)T ) ≤ E − log +x dP

for all Q ∈ Mloc . If the reverse relative entropy of Q∗ is infinite, it follows from this inequality that all other measures Q ∈ Mloc also have infinite reverse relative entropy. Because of EQ∗ [x + (ϑ · S)T ] = E[ZT (x + (ϑ · S)T )] = x it follows by Theorem 6.13 that Q∗ minimizes the reverse relative entropy.

 

Derivative Pricing by the Esscher Transform In this subsection, we show that the solution to the utility maximization problem with respect to a special power utility function can be determined using the Esscher transform. The Esscher transform is a classical means of finding an equivalent martingale measure. Let the price process S = (St )t ≤T be given by a Lévy process X = (Xt )t ≤T with X0 = 0 such that St = eXt . Let M be the moment generating function of X with M(u, t) = M(u)t = EeuXt . Let M exist for |u|$ < C for a constant % C > 0. The Esscher transformation defines a set of measures Qϑ : |ϑ| < C by eϑXT dQϑ . = dP M(ϑ)T If & ϑ is a solution of 

 M(ϑ + 1) 0 = log , M(ϑ) &

(6.34)

then Qϑ is an equivalent martingale measure, the Esscher pricing measure (the Esscher transform). p Let u be the power utility function, u(x) = xp , p ∈ (−∞, 1)\{0}. Then the condition (c) from Theorem 6.25 (i) for a minimal distance martingale measure and

6.5 Examples in Exponential Lévy Models

267

for a strategy & ϕ is equivalent to dQ∗ ϕ · S)T )p−1 (x + (& = dP λ0

and & ϕ · S is a Q∗ martingale.

&

The Esscher martingale measure Qϑ , defined by &

&

dQϑ e ϑ XT = dP M(& ϑ )T

with p &= & ϑ + 1,

& with the constant strategy & ϕ = x and λ0 = x ϑ M(& ϑ) satisfies condition (c) of Theorem 6.25 (i) respectively, the assumption from Proposition 6.12. Therefore, as a consequence, we have the following corollary.

Corollary 6.37 In the model of an exponential Lévy process, the Esscher transform & Qϑ is a minimum distance martingale measure for the power divergence f (x) = p & & − p&−1 x p&−1 if & ϑ solves (6.34) and p &= & ϑ + 1 < 1. Furthermore, Qϑ is a minimax p &

measure for the power utility function

x p& p &. &

Corollary 6.37 shows that the Esscher measure Qϑ is the minimum distance p & martingale measure to the (special) power utility function u(x) = xp& where the parameter p & is determined such that K · eXT is equal to the value of the optimal portfolio at time T . The optimal portfolio strategy constantly invests the entire wealth in the risky security. In the next subsection, we consider the solution to this problem for general power utility functions. Power Utility Function Next, we determine the local martingale measure, which minimizes the f p p−1 divergence distance for the general power utility function f (x) = − p−1 p x (p ∈ (−∞, 1) \ {0}) if the discounted price process is S = (S 1 , . . . , S d ) is of the form S i = S0i E(Xi )

(6.35)

for a Rd -valued Lévy process X = (X1 , . . . , Xd ). According to Theorem 6.25 this problem is equivalent to the portfolio optimization problem for the utility function p u(x) = xp . Let (B, C, f ) be the differential triplet of X with respect to a truncation function h : Rd → Rd . Let there exist a γ ∈ Rd with the following properties: 1. F ({x ∈ Rd : 1 + γ 2 x ≤ 0}) = 0,

268

2.

6 Utility Optimization, Indifference Pricing

   

3.

x (1 + γ 2 x)1−p

  − h(x)F ( dx) < ∞,  

b + (p − 1)cγ +



x (1 + γ 2 x)1−p

− h(x) F ( dx) = 0.

(6.36)

 t d  γi  0 Let := i Vt − for i = 1, . . . , d, ϑt := x + ϑs dSs − ϑti Sti for t ∈ St − 0 i=1  is the value process with respect to ϑ i.e. for f = u∗ . (0, T ], where V ϑti

    Define Zt := E (p − 1)γ 2 Xsc + (1 + γ 2 x)p−1 − 1 ∗ (μX − ν)s t , then it holds Corollary 6.38 The measure Q∗ defined by

dQ∗ dP

= ZT is an equivalent local marp

p−1 . tingale measure and it minimizes the f -divergence distance for f (x) = − p−1 p x

Proof In the proof of Theorem 3.2 in Kallsen (2000), it is shown that Z is a positive martingale such that S i Z is a local martingale with respect to P for i ∈ {1, . . . , d}. ∗ ∗ The density process ZT = dQ dP of Q with respect to P has the representation ZT = ϑti :=

(x + (ϑ · S)T )p−1 E (x + (ϑ · S)T )p−1

γi  i Vt − . Furthermore EQ∗ (ϑ ·S)T St−

= 0 and, therefore, it follows f (Q∗ | P )
0 Q∈Mf

0

ϑ∈!2

% $ ∗ uλ (Q  P ) + λx , (6.37)

  V (x + π(B) − B) = sup Eu x + π(B) +

 ϑt dSt − B

0

ϑ∈!2

=

T

$ % inf u∗λ (Q  P ) + λ(x + π(B) − EQ B) .

λ>0 Q∈Mf

(6.38)

The following results are based on the work of Owen and Žitkovi´c (2009), Biagini et al. (2011), and Rheinländer and Sexton (2011). Proposition 6.40 (Properties of the Utility Indifference Price) The utility indifference price is well-defined and satisfies the following properties: (1) (Translation invariance) π(B + c) = π(B) + c for c ∈ R; (2) (Monotonicity) π(B1 ) ≤ π(B2 ) if B1 ≤ B2 ; (3) (Convexity) Given two contingent claims B1 and B2 then for each γ ∈ [0, 1] π(γ B1 + (1 − γ )B2 ) ≤ γ π(B1 ) + (1 − γ )π(B2 ); (4) (Pricing of hedgeable claims) For a hedgeable claim B = b + ϑ ∈ !2 it holds:   π b+ 0

T

 ϑt dSt = b;

T 0

ϑt dSt with

270

6 Utility Optimization, Indifference Pricing

(5) (Pricing with entropic penalty term) π(B) allows the dual representation π(B) = sup {EQ B − α(Q)} Q∈Mf

with a penalty term α : Mf → [0, ∞), α(Q) = inf λ1 (u∗λ (Q  P )+λx −V (x)) λ>0

(6) (Price bounds) Let Q∗ be the minimum distance martingale measure, then it holds EQ∗ B ≤ π(B) ≤ sup EQ B, Q∈Mf

where EQ∗ B is Davis’s fair price; (7) (Strong continuity) If (Bn )n≥0 is a sequence of contingent claims such that sup EQ [Bn − B] → 0 and

Q∈Mf

inf EQ [Bn − B] → 0,

Q∈Mf

then it holds π(Bn ) → π(B); (8) (Fatou property) For a sequence of contingent claims (Bn )n≥0 it holds   π lim inf Bn ≤ lim inf π(Bn ); n→∞

n→∞

(9) (Continuity from below) If (Bn )n≥0 is a sequence of contingent claims, then it holds Bn ↑ B P -a.s. ⇒ π(Bn ) ↑ π(B). Remark (a) For the utility indifference price  π of the buyer according to Definition 6.3 one obtains in part (3) the concavity of the buyer’s utility indifference price instead of the convexity. In part (5), we obtain for the buyer’s utility indifference price the representation  π (B) =

inf {EQ B + α(Q)},

Q∈Mf

in part (6) one obtains as price bounds π (B) ≤ EQ∗ B inf EQ B ≤ 

Q∈Mf

and in part (9) one obtains continuity from above.

6.6 Properties of the Utility Indifference Price

271

(b) As a consequence of strong continuity, in Biagini et al. (2011), norm continuity is shown. The main argument is to apply an extension of the Namioka–Klee theorem (see Biagini and Frittelli 2009, Theorem 1). It also follows from this extension of the Namioka–Klee theorem that π is subdifferentiable. Proof For the existence and uniqueness of a solution of (6.1) consider V (x + p − B). Analogously to the proof of parts (2) and (3), one can show that this function in P is monotonically increasing and concave. Using the theorem on monotone convergence and the Fenchel inequality, we see that the function V (x + p − B) in P assumes values in (−∞, u(+∞)] and well-definedness follows. (1) Follows directly from the definition and the well-definedness of π. (2) Using the duality result (6.37) it follows that V (X) is monotonically increasing. Let x1 ≤ x2 and Q∗2 be optimal in the dual problem for the initial value x2 . Then it holds $ % V (x1 ) = inf u∗λ (Q  P ) + λx1 λ>0 Q∈Mf

$ % ≤ inf u∗λ (Q∗2  P ) + λx1 λ>0

$ % ≤ inf u∗λ (Q∗2  P ) + λx2 λ>0

=

inf

λ>0 Q∈Mf

$ ∗ % uλ (Q  P ) + λx2 = V (x2 ).

Thus the assertion follows from the monotonicity of V (X) and the definition of the utility indifference price. (3) From the duality result (6.37) it further follows that V (X) is concave. Let γ ∈ [0, 1] then it holds V (γ x1 + (1 − γ )x2 ) = = ≥

inf {u∗λ (Q  P ) + λ(γ x1 + (1 − γ )x2 )}

λ>0 Q∈Mf

inf {γ (u∗λ (Q  P ) + λx1 ) + (1 − γ )(u∗λ (Q  P ) + λx2 )}

λ>0 Q∈Mf

inf {γ (u∗λ (Q  P ) + λx1 )} + inf {(1 − γ )(u∗λ (Q  P ) + λx2 )}

λ>0 Q∈Mf

= γ V (x1 ) + (1 − γ )V (x2 ).

λ>0 Q∈Mf

272

6 Utility Optimization, Indifference Pricing

Similarly V (x, B) := V (x − B) = sup Eu(x + Because for γ ∈ (0, 1) it holds

v∈!2

T

ϑt dSt ) is concave in B.

0

V (x, γ B1 + (1 − γ )B2 ) = sup Eu(x + (ϑ · S)T − (γ B1 + (1 − γ )B2 )) ϑ∈!2

= ≥

sup

ϑ1 ,ϑ2 ∈!2

Eu(γ (x + (ϑ1 · S)T − B1 ) + (1 − γ )(x + (ϑ2 · S)T − B2 ))

sup [γ E(x + (ϑ1 · S)T − B1 ) + (1 − γ )E(x + (ϑ2 · S) − B2 ]

ϑ1 ,ϑ2 ∈!2

= γ V (x, B1 ) + (1 − γ )V (x, B2 ). From this follows V (γ x + (1 − γ )x + γ π(B1 ) + (1 − γ )π(B2 ) − γ B1 − (1 − γ )B2 ) = V (γ (x + π(B1 ) − B1 ) + (1 − γ )(x + π(B2 ) − B2 )) ≥ γ V (x + π(B1 ) − B1 ) + (1 − γ )V (x + π(B2 ) − B2 ) = γ V (x) + (1 − γ )V (x) = V (x). Altogether we get V (x) = V (x + π(γ B1 + (1 − γ )B2 ) − (γ B1 + (1 − γ )B2 )) ≤ V (x + γ π(B1 ) + (1 − γ )π(B2 ) − (γ B1 + (1 − γ )B2 )). Because of the monotonicity of V (X), therefore, it holds π(γ B1 + (1 − γ )B2 ) ≤ γ π(B1 ) + (1 − γ )π(B2 ), i.e. π is convex.  (4) Since ϑ ∈ !2 it follows that ϑ dS is a Q martingale for all Q ∈ Mf . Together with the duality result (6.38) we obtain    V x+π b+

T 0

=

 inf λ>0 Q ∈ Mf

  ϑt dSt − b −



T

ϑt dSt 0

   u∗λ (Q  P ) + λ x + π b +

T 0

  ϑt dSt − b .

6.6 Properties of the Utility Indifference Price

273

In order that    V x+π b+

T

  ϑt dSt − b −

0

T

 ϑt dSt = V (x)

0

  T holds, it follows because of (6.37) that already π b + 0 ϑt dSt = b must hold. (5) By Definition 6.3 of π(B) the dual representation (6.38) implies that V (x) = V (x + π(B) − B) = ≤ Therefore, π(B) ≥ ≥

inf {u∗λ (Q  P ) + λ(x + π(B) − EQ B)}

λ>0 Q∈Mf

inf {u∗λ (Q  P ) + λ(x + π(B) − EQ B)},

Q∈Mf

V (x) − inf λ Q∈Mf

"

# u∗λ (Q  P ) + x − EQ B , λ

V (x) u∗λ (Q  P ) − − x + EQ B, λ λ

∀λ > 0. ∀λ > 0

∀λ > 0, ∀Q ∈ Mf .

So it follows that for all Q ∈ Mf it holds " π(B) ≥ EQ B − inf

λ>0

# u∗λ (Q  P ) V (x) +x− , λ λ

with equality for the optimal measure Q∗ . Thus it follows " π(B) = sup

Q∈Mf

" EQ B − inf λ>0

## u∗λ (Q  P ) V (x) +x− . λ λ

 =:α(Q)

(6) The first inequality follows from the representation in (5) and from α(Q∗ ) = 0: π(B) = sup {EQ B − α(Q)} ≥ EQ∗ B − α(Q∗ ) ≥ EQ∗ B. Q∈Mf

The second inequality follows since the penalty term α(Q) ≥ 0 and we obtain as an upper bound the weak superreplication price supQ∈Mf EQ B.

274

6 Utility Optimization, Indifference Pricing

(7) This property follows from the representation of the utility indifference price in part (5), since. inf EQ [Bn − B]

Q∈Mf

= − sup EQ [B − Bn ] Q∈Mf

= − sup {(EQ B − α(Q)) − (EQ Bn − α(Q))} Q∈Mf

≤ − sup {EQ B − α(Q)} + sup {EQ Bn − α(Q)} Q∈Mf

Q∈Mf

= π(Bn ) − π(B) ≤ sup {(EQ Bn − α(Q)) − (EQ B − α(Q))} Q∈Mf

= sup EQ [Bn − B]. Q∈Mf

Thus it follows π(Bn ) → π(B). (8) This property follows from the representation of the utility indifference price in part (5) and Fatou’s lemma:       π lim inf Bn = sup EQ lim inf Bn − α(Q) n→∞

n→∞

Q∈Mf

% $ ≤ sup lim inf EQ [Bn ] − α(Q) Q∈Mf

n→∞

≤ lim inf sup n→∞

Q∈Mf

$ % EQ [Bn ] − α(Q)

= lim inf π(Bn ). n→∞

(9) This property follows directly from the Fatou property. It holds   π(B) = π lim inf Bn ≤ lim inf π(Bn ). n→∞

n→∞

  Remark (a) Because of the representation from part (5), the utility indifference price can be viewed as pricing with entropic penalty term.

6.6 Properties of the Utility Indifference Price

275

(b) For the case of the exponential utility function, the utility indifference price does not depend on the initial capital and the utility indifference price π(B; p) to the risk aversion parameter P according to (5) can also be represented as π(B; p) = sup

Q∈Mf



EQ B −

 1 H (Q  P ) − H (Q∗  P ) . p

(cf. Rheinländer and Sexton 2011, Eq. 7.15). In this case, the representation of the utility indifference price is also obtained from the dual representation of the utility maximization problem: 1  1 H (Q  P ) = inf H (Q  P ) + π(B; p) − EQ B . Q∈Mf p Q∈Mf p inf

(c) By Definition 6.3 it holds π(B) = − π (−B) with  π the buyer’s utility indifference price. Because of the convexity of the utility indifference price and π(0) = 0 it follows that π(B) ≥  π (B). (d) The utility indifference price is increasing with respect to risk aversion. If the investor is risk averse, he expects a risk premium for a risk is taken. That is, the more risk averse the investor, the higher the risk premium and hence the higher the utility indifference price. The next topic is the study of volume asymptotics for the mean of the seller’s utility indifference price for B units b > 0 for b → ∞ and b → 0 of the contingent claim B, i.e. for π(bB) b . Proposition 6.41 (Volume Asymptotics) Let B be a contingent claim and b > 0. Then π(bB) is a continuous increasing function in b. Furthermore, it holds b (i) EQ∗ B ≤

π(bB) b

(ii)

lim π(bB) b↑∞ b

(iii)

lim π(bB) b b↓0

≤ sup EQ B; Q∈Mf

= sup EQ B; Q∈Mf

= EQ∗ B.



Proof The continuity in b follows from the strong continuity from Proposition 6.40. Let 0 < b1 ≤ b2 . Set γ = bb12 , B1 = b2 B and B2 = 0. Because of the convexity of

276

6 Utility Optimization, Indifference Pricing

π(B) it holds π(b1 B) 1 = π b1 b1



  b1 b1  b2 B + 1 − B2 b2  b    2 =0 =B1 =γ

=(1−γ )

b1  1 b1 1 π(b2 B) 1− π(0) = ≤ π(b2 B) + . b1 b2 b1 b2  b2 =0

(i) Follows from Proposition 6.40, part (6).  ∈ Mf (ii) We prove this statement by contradiction. Suppose there exists a Q π(bB) such that EQ B > lim b . Then for each b > 0, b↑∞

V (x) = V (x + π(bB) − bB) inf {u∗λ (Q  P ) + λ(x + π(bB) − bEQ B)} λ>0 Q ∈ Mf       P + λx + λb π(bB) − EQ ≤ inf u∗λ Q B . λ>0 b

=

The right hand side goes for b ↑ ∞ to −∞ and, therefore, leads to a contradiction. (iii) Let π  (C, B) denote the directional derivative of π at C in direction B, i.e. π  (C, B) = lim b↓0

π(C + bB) − π(C) . b

For convex functions f on a Banach space X with values in R ∪ {∞} it holds for all continuity points x with f (x) < ∞ f  (x, d) = max{&x ∗ , d'; x ∗ ∈ ∂f (x)} (cf. Borwein and Zhu 2005). Thus, for the convex price function π and a continuity point C the representation π  (C, B) =

sup

EQ B follows.

Q∈∂π(C)

With C = 0 and π(0) = 0 we get lim b↓0

π(bB) = π  (0, B) = sup EQ B. b Q∈∂π(0)

6.6 Properties of the Utility Indifference Price

277

As a consequence of the dual representation of Proposition 6.40 part (5) (see Biagini et al. 2011, Proposition 4.2, eq. (4.4)), the subdifferential of π at zero i.e. ∂π(0) is equal to the minimizing measure of the dual problem (6.37), i.e., equal to Q∗ and the statement follows.   Remark 6.42 (a) For the limit in part (ii), we obtain the asymptotic behavior of the weak superreplication price for B. (b) Using Proposition 6.41, it follows for the exponential utility function with Corollary 7.28 (iii) from Rheinländer and Sexton (2011): lim π(B; p) = sup EQ B,

p↑∞

Q∈Me

lim π(B; p) = EQ∗ B. p↓0

If the risk aversion parameter P tends to infinity, then the utility indifference price converges to the superreplication price. If the risk aversion parameter P goes towards 0, then the linear pricing rule under the minimum entropy martingale measure Q∗ is obtained. The price with respect to the minimum entropy measure is identical to Davis’ fair price (cf. Corollary 6.29). As a corollary of Proposition 6.40 one obtains Corollary 6.43 The mapping  : B → π(−B) is a convex risk measure on the set of bounded claims. Proof Because of the translation invariance, monotonicity and convexity from Propositions 6.40 (1), (2), and (3) and π(0) = 0 the required properties of a convex risk measure are satisfied.  

7

Variance-Minimal Hedging

This chapter is devoted to the determination of optimal hedging strategies by the criterion of variance-minimal strategies. In incomplete market models, not every claim H is hedgeable. A natural question is, therefore: How good is H hedgebable? With respect to quadratic deviation, this question was studied in Föllmer and Sondermann (1986), in Föllmer and Schweizer (1991), and in Schweizer (1991). The answer to this question is based on the Föllmer–Schweizer decomposition, a generalization of the Kunita–Watanabe decomposition, and on the associated minimal martingale measure. Two types of hedging problems can be distinguished. In the first type, the initial capital x0 of the hedging strategy is fixed. In the second the hedging error of the hedging strategy is minimized over all initial values x. The criteria for this are given by the following hedging problems: Definition 7.1 (Mean-Variance Hedging Problem) Let H ∈ L2+ (AT , P ) be a contingent claim. (a) A strategy ϕ is called admissible if ϕ ∈ L2 (S). (b) A pair  = (x0 , ϕ ∗ ), ϕ ∗ admissible, is called variance-minimal hedging strategy for H , if EP (H − xo − VT (ϕ ∗ ))2 = inf . x,ϕ

(7.1)

(c) An admissible strategy ϕ ∗ ∈ L2 (S) is called variance-minimal for C to the initial capital x when EP (H − x − VT (ϕ ∗ ))2 = inf . ϕ

© Springer-Verlag GmbH Germany, part of Springer Nature 2023 L. Rüschendorf, Stochastic Processes and Financial Mathematics, Mathematics Study Resources 1, https://doi.org/10.1007/978-3-662-64711-0_7

(7.2)

279

280

7 Variance-Minimal Hedging

If (x, ϕ) is a solution of the hedging problem (7.1), then x + VT (ϕ) is the hedgeable part of the claim H . x is the associated ‘fair’ price of H with respect to the variance criterion.

7.1

Hedging in the Martingale Case

In the case of hedging with respect to a martingale measure Q, the Kunita–Watanabe decomposition for martingales provides an optimal hedging strategy. Let P = Q be a martingale measure for S, S ∈ Mloc (Q) and H ∈ L2 (AT , Q). Then H induces the martingale Vt = EQ (H | At ), 0 ≤ t ≤ T . Theorem 7.2 (Mean-Variance Hedging, Martingale Case) Let S ∈ Mloc (Q) and H ∈ L2 (AT , Q) and let 

T

H = x0 + 0

ϕu∗ dSu + LT

(7.3)

be the Kunita–Watanabe decomposition of V with ϕ ∗ ∈ L2 (S), L0 = 0, L ∈ M2 such that L is strictly orthogonal to V , i.e. &V , L' = 0. Then ∗ = (x0 , ϕ ∗ ) is a variance-minimal hedging strategy for H . Proof By the projection theorem for L2 (Q), ∗ = (x0 , ϕ ∗ ) is an optimal hedging strategy ⇐⇒ E(H − x0 − VT (ϕ ∗ ))(x + VT (ϕ)) = 0,

∀ x, ∀ ϕ ∈ L2 (S), E = EQ .

Equivalent to this is that x0 = EQ H and using the Kunita–Watanabe decomposition that ∗

E(H − x0 − VT (ϕ ))VT (ϕ) = ELT



T 0



T

=E 0

ϕu∗ dSu



ϕn∗ d&L, S'u = 0,

since L is strictly orthogonal to S, i.e. &L, S' = 0. ∗ = (x0 , ϕ ∗ ) is thus a varianceminimal hedging strategy for H .   An alternative approach to optimal hedging is made possible by considering generalized strategies, not necessarily self-financing, that hedge the claim H . Definition 7.3 (Generalized Hedging Strategy) A pair  = (ϕ 0 , ϕ), ϕ ∈ L2 (S), ϕ 0 an adapted càdlàg process, is called (generalized) hedging strategy for H ∈

7.1 Hedging in the Martingale Case

281

L2 (Q) if the value process satisfies Vt () = ϕt0 + ϕt St ∈ L2 (Q)

and

VT (Q) = H.

(7.4)

t () = ϕt0 + ϕt  Implicitly, the discussed value process V St is considered, so that here the interest rate rt ≡ 1 is chosen. It holds 

t

Vt () = Vo () +

ϕu dSu

(7.5)

0

if and only if  is self-financing, i.e. the value  t process Vt () is given by the initial value V0 () plus the gain process Gt (ϕ) := 0 ϕu dSu . If one defines the cost process as the difference between the value process and the gain process Ct := Ct () := Vt () − Gt (),

(7.6)

then  is self-financing, exactly when Ct = const. = V0 (). The cost process thus describes how much capital, in addition to trading with  has to be raised in order to realize the value process. The objective is then: minimize (quadratic) functionals of the cost process C among all generalized strategies that hedge the claim H , i.e. VT () = H . Under the assumption H ∈ L2 (P ), St ∈ L2 (P ), ∀ t, this leads to the following definition. Definition 7.4 (Risk Minimizing Strategy) (a)  = (ϕ 0 , ϕ) is called L2 -admissible (hedging) strategy for H , if VT () = H,

ϕ ∈ L2 (S)

and Vt ∈ L2 (P ), ∀ t.

(b) (residual) risk process Rt () := E((CT − Ct )2 | At )

(7.7)

is called (residual) risk process of . & for H is risk-minimizing, if (c) An L2 -admissible strategy  &) ≤ Rt (), Rt (

∀ t ≤ T , ∀  L2 −admissible.

& minimizes Among all L2 -admissible strategies for H a risk-minimizing strategy  the risk process at all times ≤ T . For hedgeable claims H and, in particular, in the case of a complete market model, risk-minimizing is equivalent to ϕ being self-

282

7 Variance-Minimal Hedging

financing and Rt () = 0, ∀ t ≤ T . Risk-minimizing strategies have the attenuated property of being ‘self-financing in mean’. Definition 7.5 (Self-Financing in Mean) An L2 -admissible strategy  for H is self-financing in mean if C() ∈ M(P ), i.e. for s ≤ t holds E(Ct () | As ) = Cs (). Proposition 7.6 If  is risk-minimizing for H , then  is self-financing in mean. Proof To s ∈ [0, T ] and  = (ϕ 0 , ϕ) let  ϕ ∈ L2 (S) and  η be such that  = Vt () for 0 ≤ t < s Vt ()  T     = E VT () − ϕu dSu  At . and Vt () t

 is an admissible strategy for H , because VT ( ) = Vt () = H and Then   CT () = CT ().  it holds: By construction of   − Cs ()  + E(CT ()  | As ) − Cs (). CT () − Cs () = CT () Thus it follows    + E(CT () | As ) − Cs () 2 . Rs () = Rs () Since  is risk-minimizing it holds ) ≤ 0. Rs () − Rs ( Thus it follows: E(CT () | As ) = Cs () a.s., i.e.  is self-financing in mean.

 

The existence and uniqueness of a risk-minimizing strategy is shown in Föllmer and Sondermann (1986). A description of this strategy follows from the Kunita– Watanabe decomposition (KW decomposition) of H 

T

H = EH + 0

ϕuH dSu + LH T ,

(7.8)

where LH ∈ M2 is an L2 -martingale orthogonal to S, i.e. &H, S' = 0. With the value process  VtH

t

= H0 + 0

ϕuH dSu + LH t

(7.9)

7.1 Hedging in the Martingale Case

283

it follows VtH = E(H | At ) and ϕH =

d&V H , S' . d&S'

(7.10)

To prove that ϕ H is risk-minimizing, we need one more definition. Definition 7.7 The process H 2 &t = E((LH & R T − Lt ) | At ) = Rt (H ),

0≤t ≤T

is called intrinsic risk process of H . & := Theorem 7.8 There exists a unique risk-minimizing strategy for H , viz.  H H H (V − ϕ · S, ϕ ) and it holds &t . & =R Rt () & is admissible for H , i.e. VT () & = H a.s. Let  be an Proof By definition  & admissible continuation of  at time t, i.e. &s for s < t s = 

&) = H. and VT () = VT (

Then it holds 

T

CT () − Ct () = VT () − Vt () − &0 + =V



T 0



T

= t

ϕu dSu t

ϕuH dSu + LH T − Vt () −



T

ϕu dSu t

H & (ϕuH − ϕu ) dSu + (LH T − Lt ) + (Vt () − Vt ())

Since S and LH are orthogonal it follows Rt () = E((CT () − Ct ())2 | At )   T & + (Vt () & − Vt ())2 (ϕuH − ϕu )2 d&Su ' | At + Rt () =E t

& ≥ Rt (); (7.11) & a risk-minimal strategy for H . therefore, 

284

7 Variance-Minimal Hedging

 = ( To prove uniqueness, let  η,  ϕ ) be another risk-minimizing strategy for H . Then it follows from (7.11) that ϕtH = & ϕt a.s. for 0 ≤ t ≤ T .  ) = V ( ) − G() is a martingale it follows that Since by Proposition 7.6 C( ) is a martingale. Because of VT ( & ) = V T ( ) = H it, therefore, follows V ( & ) = V t ( &), 0 ≤ t ≤ T and, therefore,  &= . V t (   In particular, the following corollary is obtained for hedgable claims. Corollary 7.9 (Hedgable Claims) The following statements are equivalent: The risk-minimizing strategy is self-financing &t (H ) = 0, for all t ≤ T ⇐⇒ The intrinsic risk of H is 0, R ⇐⇒ H is hedgeable, i.e. H has a representation of the form 

T

H = EH + 0

ϕuH dSu a.s.

However, it is shown below that in the incomplete case the risk-minimizing strategy ϕ H = ϕQ depends on the chosen martingale measure Q. Hence, the question of choosing an ‘appropriate’ martingale measure arises, as well as the question of hedging under the (statistical) measure P .

7.2

Hedging in the Semimartingale Case

Let now S = (St )0≤t ≤T a semimartingale with Doob–Meyer decomposition S = S0 + M + A

(7.12)

with M ∈ M2 and A ∈ V predictable with bounded variation |A|. Then E(S02 + &X'T + |A|2T ) < ∞ and &X' = &M'. The cost process of a strategy  = (ϕ 0 , ϕ) is given by 

t

Ct () = Vt () − 0



t

ϕs dMs −

ϕs dAs .

(7.13)

0

In contrast to the martingale case, there is no direct generalization of the KW decomposition of H into a stochastic integral and an orthogonal component. Schweizer (1991) introduced the concept of a local risk-minimizing strategy, i.e. a strategy that minimizes risk under small perturbations. It is shown that local risk-minimizing strategies are self-financing (cf. Sect. 7.2.1). To determine such optimal (= risk-minimizing) strategies, there is a characterization via an extended decomposition, the Föllmer–Schweizer decomposition, which leads to the solution

7.2 Hedging in the Semimartingale Case

285

of an optimality equation for the optimal strategy (cf. Sect. 7.2.1). A second approach by Föllmer and Schweizer (1991) reduces the determination of the optimal strategy in the semimartingale case with respect to the statistical measure P to the determination of an optimal strategy in the martingale case, with respect to a suitable equivalent martingale measure, the ‘minimum martingale measure’ (cf. Sect. 7.2.2).

7.2.1

Föllmer–Schweizer Decomposition and Optimality Equation

To introduce locally risk-minimizing strategies, the notion of a ‘small perturbation of a strategy ϕ is needed. Definition 7.10 A strategy  = (ε, δ) is called (small) perturbation (strategy) if (1)  δ is bounded, T (2) δs d |A|s is bounded, and 0

(3) δT = εT = 0. In particular, for an admissible strategy  and a (small) perturbation  it holds that  +  is admissible and the restriction of  to a subinterval is again a small perturbation. For a finite decomposition τ = (ti )1≤i≤N of [0, T ] with 0 = t0 < t1 < · · · < tN = T let |τ | = max |ti − ti−1 |. A sequence (τn ) of decompositions is called increasing if τn ⊂ τn+1 , ∀ n; denomination : (τn ) ↑.    For a perturbation  = (ε, δ) let the restriction (s,t ] = (ε(s,t ], δ (s,t ]) be   defined by δu (s,t ](u, w) = δu (w)1(s,t ](u) and ε(s,t ](u, w) = ε − u(w)1(s,t ](u). Definition 7.11 (a) For a strategy , a perturbation  and a partition τ let r τ (, )(w, t) :=

 Rti ( +  |(ti ,ti+1 ] −Rti () (w)1(ti ,ti+1 ] (t) E(&M'ti+1 − &M'ti | Ati ) t ∈τ i

be the risk ratio. (b) An admissible strategy  for H is called locally risk-minimizing if for all perturbations  lim r τn (, ) ≥ 0.

n→∞ (τn )↑, |τn |→0

(7.14)

286

7 Variance-Minimal Hedging

The risk ratio describes the relative change in risk when  is affected by a perturbation  along a partition τ . There is an analogue of Proposition 7.6 for locally risk-minimizing strategies. Proposition 7.12 Let for P almost all ω the carrier of the measure d&M'(ω) be equal to [0, T ]. Then, if  is locally risk-minimizing,  is self-financing in mean.  = ( ϕ ) be a Proof (Idea) The proof is similar to that of Proposition 7.6. Let  ϕ0 ,  t self-financing strategy with  ϕ = ϕ and  ϕ0,t = E(CT () | At ) + 0 ϕu dSu − ϕt St .  −  ia a perturbation. Using a similar reasoning as in the proof of Then  =  Proposition 7.6 it then follows that for a suitable partition τ the risk ratio r τ (, ) is not P a.s. greater than or equal to 0 in contradiction to the assumption that  is risk-minimizing.   In particular, to find a locally risk-minimal (LRM)-strategy only the determination of ϕ is to be done, since ϕ 0 then follows unambiguously from the martingale property of C(). We now pose the following regularity assumption: Assumption (A) (1) A is continuous (2) A ( &M' with density α ∈ L log+ L P × &M' a.s. (3) S is continuous in T P a.s. The following characterization theorem of Schweizer (1991) we state without proof: Theorem 7.13 (Characterization Theorem of (LRM)) Under the assumption A) holds for an H -admissible strategy :  is (LRM) ⇐⇒  is self-financing in mean and the martingale C() is orthogonal to M. We refer to (LRM) strategies as optimal using Theorem 7.13. Definition 7.14 An admissible in mean self-financing strategy  for H is called ‘optimal’ if the martingale C() orthogonal to M. The following basic theorem gives a characterization of optimal strategies and is an extension of the Kunita–Watanabe decomposition. Theorem 7.15 (Föllmer–Schweizer Decomposition, (FS Decomposition) & for H is equivalent to the existence of Theexistence of an optimal strategy  the FS-decomposition 

T

H = H0 + 0

ϕuH dSu + LH

(7.15)

7.2 Hedging in the Semimartingale Case

287

H 2 H with H0 = E(H | A0 ), ϕ H ∈ L2 (S) and LH t = E(LT | At ) ∈ M such that L H is orthogonal to M is, i.e. &L , M' = 0.

Proof & be “⇐”: If H has a representation of the form (7.15) and let the strategy  t H H H H & ϕ defined by  = (V − ϕ · S, ϕ ) with Vt = H0 + 0 ϕu dSu + Lt . Then & is a martingale with is admissible for H and the cost process Ct = H0 + LH t & is optimal for H (cf. Definition 7.14). respect to P orthogonal to M, i.e.  & = (ϕ0 , ϕ H ) is optimal for H , then Ct = E(CT | At ) a martingale “⇒”: If  orthogonal to M. Thus the FS decomposition holds & = CT + H = VT ()

 

T 0 T

= H0 + 0

ϕuH dSu ϕuH dSu + LH T

H with LH T = CT − C0 and L is orthogonal to M under P .

  By Theorem 7.15, determining optimal and hence locally risk-minimizing hedging strategies for H is equivalent to determining the FS decomposition of H . A direct approach to this involves the following three steps: (1) Application of the KW-decomposition of H with respect to the martingale M ∈ M2 provides 

T

H = N0 +

μs dMs + N H a.s. with respect to P

(7.16)

0

with the martingale NtH = E(NTH | At ). It is EN H = 0 and N H is orthogonal to M with respect to P .  (2) Application of the KW decomposition to ϕ H dA provides 

T 0

ϕ

ϕsH dAs = N0 +

ϕH



T 0

ϕH

H

μϕs dMs + N ϕ

H

(7.17)

ϕH

H

5 with Nt = E(NT | At ), a martingale with ENt = 0 and N ϕ orthogonal to M. (3) From the FS decomposition (7.15) and the KW decomposition (7.16) it follows: ϕH

H = (H0 + N0 ) +



T 0

H

ϕH

(ϕsH + μϕs ) dMs + (LH T + NT ).

(7.18)

288

7 Variance-Minimal Hedging

Since the KW decomposition of H is unique, it follows for an optimal strategy &:  & is a solution of the optimality equation:  H

ϕsH + μϕs = μs ,

(7.19)

The integrand ϕ H of the FS-decomposition thus differs in general from the integrand μ of the KW-decomposition by the compensation term μϕH . For some classes of examples this program is carried out in Schweizer (1990, 1991).

7.2.2

Minimal Martingale Measures and Optimal Strategies

The determination of optimal strategies can alternatively be reduced to the martingale case by selecting a suitable martingale measure, the minimal martingale measure. Definition 7.16 (Minimal Martingale Measure) An equivalent martingale measure P& ∈ Me (P ) is called minimal, if (1) P& = P on A0 and (2) L ∈ M2 and &L, M' = 0 P a.s. implies L ∈ M(P&). An equivalent martingale measure P& is uniquely determined by the right-continuous & ∈ M2 (P ) defined by martingale G  &  &t = E dP | At . G dP

(7.20)

With respect to P the Doob–Meyer decomposition of S equals S = S0 + M + A. & is given by Therefore, the Doob–Meyer decomposition of M with respect to P M = −S0 + S + (−A).

(7.21)

Using the Girsanov transformation, it follows that the predictable process −A ∈ V & in the form can be determined with the help of G 

t

−At = 0

1 & s, d&M, G' &s − G

0 ≤ t ≤ T.

(7.22)

& ( &M' = &S', A is absolutely continuous with respect to &S' and, Since &M, G' therefore, has a representation of the form 

t

At =

αu d&S'u , 0

0 ≤ t ≤ T.

(7.23)

7.2 Hedging in the Semimartingale Case

289

This gives an existence and uniqueness statement for the minimal martingale measure. We assume in the following theorem that S is a continuous semimartingale. Theorem 7.17 (Existence and Uniqueness) Let S be a continuous semimartingale. (1) The minimal martingale measure P& is uniquely determined if it exists. (2) P& exists if and only if    t  1 t 2 &t = exp − αs dMs − αs d&S's G 2 0 0

(7.24)

& ∈ M2 (P ). In this case: defines an L2 -martingale under P , i.e. G dP& &T . =G dP

(7.25)

(3) The minimal martingale measure P& preserves the orthogonality of L2 martingales, i.e.: If for an L ∈ M2 it holds: &L, M' = 0 with respect to P , then &L, S' = 0 also with respect to P&. Proof & ∈ M2 (P ) then according to (1) Let P& be a minimal martingale measure and let G KW  t & & βs dMs + Lt , 0 ≤ t ≤ T , Gt = G0 + 0

& holds with L ∈ M2 (P ), &L, M' = 0, β ∈ L2 (M). W.r.t P 

t

−At = 0

1 &s= d&M, G' & Gs −



t 0

1 βu d&S'u . & Gs −

Therefore, it follows: α=−

β . &− G

(7.26)

& > 0[P ] because of P& ≈ P and &M' = &S' and, therefore, it holds that It is G T 2 & & α o u d&S'u < ∞ P a.s. Since by assumption P is minimal, it follows G0 = & dP E dP | A0 = 1 and L ∈ M(P&) according to Definition 7.16. Therefore, it & = 0 and, therefore, &L' = &L, G' & = 0, thus L = 0. follows: &L, G'

290

7 Variance-Minimal Hedging

& therefore, solves the equation G, &t = 1 + G



t

&s − (−αs ) dMs . G

(7.27)

0

The solution for a continuous martingale M of (7.27) is given by the stochastic & in (7.24). From this follows the uniqueness of the minimal exponential G martingale measure & defined in (7.24) be a L2 -martingale, G ∈ M2 (P ). To prove that P& is (2) Let G minimal,it is to show : L ∈ M2 (P )

and &L, M' = 0 with respect to P ⇒ L ∈ M(P&).

& ∈ M2 (P ), it follows by the maximal inequality Since L, G EP& sup |Lt | = E



0≤t ≤T

≤E

 &T sup |Lt | G

0≤t ≤T



sup L2t

1 2

0≤t ≤T

&2T ) 2 ≤ 4(EL2T ) 2 (E G &2T ) 2 < ∞. (E G 1

1

1

From this follows that L ∈ M(P&). (3) To prove that P& preserves the orthogonality, let L ∈ M2 and &L, M' = 0 with respect to P . For semimartingales Y , Z the quadratic covariation is [Y, Z] = &Y c , Z c ' +



Ys Zs .

s

Since S and A are continuous, it holds: &L, S' = &Lc , S' + &Ld , S' = &Lc , S' = &L, S' = [L, M] + [L, A] = [L, M] with respect to P&. Because of the continuity of M it holds [L, M] = &Lc , M' = &L, M' = 0 with respect to P and, therefore, [L, M] = 0 with respect to P&.   The minimal martingale measure is described by the exponential density in (7.24). According to Corollary 6.35 is P& thus is, under the conditions there, a projection of P on the set of martingale measures with respect to the relative entropy H (Q  P ) =

⎧ ⎨ ⎩

log

∞,

dQ dQ, for Q ( P , dP otherwise.

7.2 Hedging in the Semimartingale Case

291

The following theorem gives an independent proof of this important result. Theorem 7.18 (Minimal Martingale Measure and Entropy) (a) The minimal martingale measure P& minimizes the functional 1 H (Q  P ) − EQ 2



T 0

αu2 d &S'u

(7.28)

w.r.t. Q on the set Me , α the &S'-density of A from (7.23). (b) For Q ∈ Me it holds: H (Q  P ) ≥

1 EQ 2



T 0

αu2 d &S'u .

(7.29)

For Q = P& holds equality in (7.29). (c) P& minimizes the relative entropy H (·  P ) on the set of martingale measures Q ∈ Me with 

T

EQ 0

 αu2 d &S'u

≥ EP&

T 0

αu2 d &S'u .

Proof Let Q ∈ Me ; then M has with respect to Q the Doob–Meyer decomposition (cf. (7.21))   t  αu d &S'u . Mt = St − S0 + −

(7.30)

0

 &T dP < ∞. &T = dP& ∈ L2 (P ) and, therefore, H (P&  P ) = G &T log G By (7.24) G dP e & T = Now let Q ∈ M be the minimal martingale measure, then it follows with G & dQ dP from (7.30): & + H (P&  P ) = H (P&  Q) & + = H (P&  Q)



T dP& log G

 



T

−

& + 1 EQ = H (P&  Q) & 2



< ∞ by assumption.

0 T 0

1 αs dMs − 2  αu2 d &Su '



T 0

 & αu2 d &S'u dQ

292

7 Variance-Minimal Hedging

This implies  1 H (P&  P ) − EQ & 2



T 0

 & ≥ 0. αu2 d &S'u = H (P&  Q)

(7.31)

From this follows (7.28) and (7.29). & = 0 exactly then, when P& = Q. & From this follows In Eq. (7.31) is H (P&  Q) c).   As a result, we now obtain that the FS decomposition can be determined via the KW decomposition with respect to the minimum martingale measure. This also gives a representation of the optimal strategy. Theorem 7.19 (Optimal Strategy) & and thus the FS decomposition in (7.15) is uniquely (a) The optimal strategy  determined. It can be determined with the help of the KW decomposition of H in (7.3) with respect to the minimal martingale measure P&. & is a right continuous version of the martingale EP&(H | At ) then the (b) If V & is given by optimal strategy  & − ϕ H · S) with ϕ H = & = (ϕ H , V 

&, S' d&V & w.r.t. P d&S'

(cf. Definition 7.11). & P Proof Since H ∈ L2 (P ) and ddP ∈ L2 (P ), it follows H ∈ L2 (P&), and the &t is well defined. Now let martingale EP&(H | At ) = V



t

Vt = H 0 + 0

ϕuH dSu + LH t

(7.32)

t be an FS-decomposition of H under P , then LH ∈ M(P&) and 0 ϕuH dSu ∈ M(P&) &t = E &(H | At ) = Vt . With P = P × d&X' defined on the σ and so it holds V P algebra P of the predictable sets in = × [0, T ], P (dω, dt) = P (dω)d&S't (ω) and analogously P&, it holds: &LH , S' = 0 [P ] and, therefore, also according to Theorem 7.17 (3): &LH , S' = 0 [P&]. Therefore, the process LH from the FS-decomposition in (7.32) is also a P&-martingale orthogonal to S. The FS decomposition in (7.32) with respect to P is, therefore, identical to the KW-decomposition of H under the minimal martingale measure P&. & = (ϕ H , V − ϕ H · S) is an optimal strategy.  By Theorem 7.8, it follows that  

7.2 Hedging in the Semimartingale Case

293

Remark In Föllmer and Schweizer (1991), the present approach to optimal strategies is extended to the case with incomplete information, i.e., there is a filtration t , 0 ≤ t ≤ T such that H is representable with respect to this larger At ⊂ A  filtration A 0 + H =H



T 0

uH dSu , ϕ

 is predictable with respect to (A t ). The optimal strategy is then obtained H ∈ L(P) ϕ from the projection of the optimal strategy  ϕ H onto P the predictable σ -algebra with respect to (At ), with respect to the minimal martingale measure ϕuH = EP&( ϕuH | P) and ϕ0H = V − ϕuH · S, Vt = EP&(H | At ).

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Index

A Absolutely continuous, 133 Adapted, 28 Admissible δ-, 197 portfolio process, 230 strategy, 196 trading strategy, 250 American Call, 2, 199, 216 option, 215 Put, 199 Approximation continuous, 39 formula, 206 Riemann-, 51 Arbitrage strategy, 5, 195 Arc-sine distribution, 24 law, 24 Arzela–Ascoli criterion, stochastic, 171 Asset pricing first fundamental theorem, 197 second fundamental theorem, 222 Attainment theorem, 232

B Barrier option, 213 price, 215 Basic terms option pricing theory, 193 Bear spread, 212 Binomial pricing formula, 9 Black–Scholes differential equation, 218 formula, 10, 205 model, 10, 192 price, 3, 201

Bond stock and bond, 192 Borel’s σ -algebra, 19 Bracket process, angular, 80 Branching diffusion, 164 Branching process, Galton–Watson, 165 Brownian bridge, 158 Brownian motion, 146 geometric, 104, 152, 153 properties, 41 recurrence, 109 simulation of the, 12 Bull spread, 211

C Càdlàg, 28 process, 28, 79 Call, 2 American, 2, 199, 216 European, 2, 199, 205, 209 Call-put parity, 206 Cameron–Martin space, 146 theorem, 145, 147 Capital process, 230 with consumption, 230 Cauchy net, 54 Cauchy problem, 188 Chaos representation, Wiener, 126 Chapman–Kolmogorov equation, 173 Characteristic differential, 264 semimartingale, 93 triplet, 94 Clark formula, 149 Class monotone, theorem, 38 Closure theorem, 68

© Springer-Verlag GmbH Germany, part of Springer Nature 2023 L. Rüschendorf, Stochastic Processes and Financial Mathematics, Mathematics Study Resources 1, https://doi.org/10.1007/978-3-662-64711-0

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300 Compensator, predictable, 63 Completeness theorem, 119 Contingent claim, 220 Continuous absolutely, 133 local, 133 martingale part, 83, 93 process, 28 semimartingale, 79 Convergence in distribution, 19 dominated, 78 positivity range, 24 set, 70 stochastic, 51 Cost process, 281 Covariation, 88 predictable, 90 predictable quadratic, 64, 84 quadratic, 82 Cox–Ingersoll–Ross model, 164 Cox–Ross–Rubinstein model, 10 Criterion, stochastic Arzela–Ascoli, 171

D Davis’ fair (derivative) price, 254 Decomposition canonical, 83 Föllmer–Schweizer (FS), 286 Kunita–Watanabe (KW), 127, 282 semimartingale, 96 Decomposition theorem, 84 optional, 228 special, 85 δ-admissible, 197  of the option, 207 Derivative price, Davis’ fair, 254 Development, general discrete for securities of, 4 Differential equation Black–Scholes, 218 stochastic, 152 Diffusion equation non-Markovian, 153 matrix, 174 model, 221 process, 108, 173 generalized, 153 Dirichlet class, 60 local, 60 Dirichlet problem, 188 uniqueness, 108

Index Distance measure, minimal, 236 Distribution, arc-sine, 24 Dividend payment, 206 Doléans, 73 measure, 43, 73 restricted, 43, 73 Doléans–Dade formula, 102 Donsker’s invariance principle, 19, 20 theorem, 10, 11 Doob–Meyer decomposition, 62, 63 Doob’s maximal inequality, 34 Doob’s theorem, 31 Drift coefficient, 174 Dual predictable projection, 63 Dynkin formula, 187

E Embedding theorem, Skorohod, 15 Emery metric, 231 Entropy reverse relative, 235 Equation of Chapman–Kolmogorov, 173 Equivalent, 133 locally, 133 (local) martingale measure, 195 Erd˝os and Kac theorem, 24 ESMM, 198 Esscher martingale measure, 267 Esscher price measure, 266 Esscher transform, 266 European Call, 2, 199, 205, 209 Put, 199, 209 Exponential generalized stochastic, 102 stochastic, 99, 101, 152

F Fair price, 3, 8 Feller property, 176 Feller’s branching diffusion, 164 Fenchel inequality, 234 Feynman–Kac representation, 189 Financial market model, 192 Flow property, 164 Föllmer–Schweizer (FS) decomposition, 286 Forwards, 1, 203 f -projection, 236 existence of, 236 uniqueness, 236

Index Function, convex conjugate, 234 Fundamental theorem asset pricing, 197, 222 second general, 224 Futures, 1

G Galton–Watson branching process, 165  of the option, 208 Generalized moment family, 236 Generator, infinitesimal, 108, 176 Girsanov pair, 145 theorem, 132, 140 transform, 140 Greeks, 207 Gronwall’s lemma, 157

H Harmonic, 107 Hartmann–Wintner’s law, 17, 18 Heat equation, 187 Hedge, 199 admissible, 199 Hedgeable, strongly, 199 Hedging optimization problem, 234 Hedging price upper, 228 Hedging, quadratic, 130 Hedging strategy, 207 generalized, 280 variance-minimal, 279 Hellinger distance, 235 Hermitian polynomials, 114 Hilbert space, subset, 117 Hille–Yosida theory, 175

I Integral elementary processes, 42 infinite intervals, 53 Lebesgue–Stieltjes, 231 multiple stochastic, 124 pathwise L2 -processes, 53 stochastic, 45, 46, 71, 74, 86 vector, 231 stopping times, 47 Integrands L2 , 43

301 Integration formula, partial, 82, 89 semimartingale, 82 stochastic, 27 Interpolation, linear, 20 Interval infinite, integral, 52 no-arbitrage-, 227 stochastic, 38 Invariance principle, Donsker’s, 19, 20 Isometry property, 119 Itô formula, 95 general, 98 Itô integral, 40 Itô theory, 175

J Jump process, 90 Jumps of semimartingales, 90

K Kazamaki condition, 105 Kolmogorov equations, 175 forward equation, 181 Kunita–Watanabe (KW) decomposition, 127, 282 Kunita–Watanabe inequality, 67

L L2 -bounded martingale, 67 L2 -integrands, 43 L2 -martingale, 29  of the option, 208 Law arc-sine, 24 iterated logarithm, 12 of large numbers, strong, 12 Lebesgue–Stieltjes integral, 231 Lemma, Gronwall’s, 157 Limit, stochastic, 54 Limit theorem central, 12, 16 functional, 19 Localization, 59 Locally equivalent, 133 Logarithm iterated, law, 12 stochastic, 102, 141

302 M Markov time, 30 Martingale, 27, 28 bounded, 46 continuous local, 77 exponential, 100 L2 -bounded, 46, 67 local, 34 local L2 -, 34 locally square-integrable, 129 problem, 108 local, 169 σ -, 197 Martingale case, 280 Martingale hedge, 199 Martingale measure equivalent (local), 195 Esscher, 267 minimum distance, 233, 235 Martingale part, continuous, 83, 93 Martingale problem, local, 169 Martingale property, 60 local, 60 Martingale representation theorem, 117, 121 Mean-variance hedging, 280 Measure minimal, 288 random, 36 risk neutral, 195 Metric, Emery-, 231 Minimal martingale measure, 288 super-hedging portfolio, 230 Minimum distance martingale measure, 233, 235 Modification, 28 Moment family, generalized, 236

N No-arbitrage interval, 227 No-arbitrage price, 200, 201 No-arbitrage principle, 4, 195 extended, 195 generalized, 201 No free lunch with vanishing risk (NFLVR), 198 No free lunch with vanishing risk (NFLVR) condition, 197 Non-Markovian diffusion equations, 153 Norm for class H2 , 68 Normal distribution density, 114 Novikov condition, 105 Numéraire, 194

Index O Optimality equation, 288 Optimization problem hedging-, 234 portfolio, 234 Option American, 199, 215  of, 207 duplicable, 199  of the, 208  of the, 208 Optional sampling theorem, 31 for local martingales, 54 Option pricing theory, basic terms, 193 Ornstein–Uhlenbeck model, 161 Ornstein–Uhlenbeck process, 161, 173 OTC contract, 203

P Perturbation strategy, 285 Poisson equation, 189 Polynomial Hermitian, 114 standardized Hermitian, 115 Portfolio, 207 duplicating, 3 hedging, 207 optimization problem, 234 value, 193 Portfolio process, 230 admissible, 230 Positivity range, 24 convergence, 24 Power utility function, 234 Predictable compensator, 63 covariation, 90 process, 27, 38 projection, 63, 150 dual, 63 theorem, 47 quadratic covariation, 64, 84 quadratic variation, 56, 84 set, 38 stopping time, 38 Predictable representation property (PRP), 224 Price arbitrage-free, 225, 226 barrier option, 215 Black–Scholes, 3 Davis’ fair (derivative), 254 fair, 3 lower arbitrage-free, 225

Index upper arbitrage-free, 225 Price bound, trivial, 202 Price measure, Esscher, 266 Price model, 3 exponential, 9 simple, 3 Price process, discounted, 194 Process predictable, 38 càdlàg, 28 continuous, 28 elementary previsible, 38 finite quadratic variation, 55 of finite variation, 35 integrable increasing, 36 integrable variation, 36 locally bounded, 36 locally integrable, 36 Ornstein–Uhlenbeck, 173 orthogonal, 107 pathwise L2 -, integral, 53 predictable, 27 Projection, 236 dual predictable, 63 predictable, 63, 150 Put, 2 American, 199 European, 199, 209

Q Q-admissible, 196 Quadratic covariation, 82 predictable, 64 predictable, 84 hedging, 130 variation, 40, 55, 61, 80, 88 predictable, 56, 84 process, 80

R Random walk, 23 Ratio, risk, 285 Recurrence, 109 of Brownian motion, 109 Regularity theorem, 28 Representation, canonical, 64 Return, 104 Riemann approximation, 51 Riemann integral, 51 Risk neutral measure, 195

303 valuation formula, 201 Risk process intrinsic, 283 residual, 281 Risk ratio, 285

S Sampling, optional, 31 Scaling, temporal, 20 Self-financing, 5, 220 strategy, 282 Semimartingal topology, 231 Semimartingale, 36, 64, 82 characteristic, 93 continuous, 79 decomposition, 96 jumps of, 90 special, 64, 83 Sequence, localizing, 34 Set, predictable, 38 Short selling, 201 σ -algebra Borel’s, 19 optional, 40 progressive, 40 of τ -past, 30 σ -martingale, 197 Simulation of the Brownian motion, 12 Skorohod embedding theorem, 11, 15 Skorohod representation, 12 Solution strong, 153, 165 weak, 153, 165, 169 Special decomposition theorem, 85 semimartingale, 64, 83 Stationarity condition, 184 Stochastic Arzela–Ascoli criterion, 171 convergence, 51 differential equation, 152 exponential, 99, 101, 102, 141, 152 integral, 45, 46, 71, 74, 75, 86 characterization, 74 as isometry, 75 multiple, 124 for semimartingales, 87 vector, 231 integration, 27 interval, 38 limit, 54 logarithm, 102, 141

304 Stock, 192 and bond, 192 Stopping time, 30 finite, 30 integral, 47 predictable, 38 reachable, 38 Straddle, 210 Strangle, 210 Strap, 211 Strategy admissible, 196 locally risk-minimizing, 285 LRM-, 286 optimal, 292 perturbation, 285 regular trading, 196 risk-minimizing, 281 self-financing in mean, 282 self-financing trading, 193 variance-minimal, 279 Strike, 3 Submartingale, 28 Subset dense, 118 Hilbert space, 117 total, 117 Subspace, stable, 224 Super-hedging portfolio, 230 minimal, 230 Super-hedging strategy, 228 Supermartingale, 28, 33

T Telescope sum, 37 Temporal scaling, 20 Theorem attainment, 232 Cameron–Martin, 145, 147 Donsker’s, 10, 11 Doob, 30, 31 Erd˝os and Kac, 24 Girsanov, 132, 140 Hille–Yosida, 179 monotone class, 38 optional sampling, 31 predictable projection, 47 Riesz, 75

Index Topology, semimartingale, 231 Total variation distance, 235 Trading strategy, 5, 130, 193, 196 admissible, 250 regular, 196 self-financing, 193 Transformation, Girsanov, 140 Transform, Esscher, 266 Triplet, characteristic, 94

U Uniqueness, pathwise, 168 Unique, weak, 168 Utility function exponential, 234 logarithmic, 234 power, 234 Utility indifference price, 233, 235

V Value, 5 Value process, discounted, 230 Vanilla options, 199, 208 Variance-minimal hedging strategy, 279 strategy, 279 Variation finite quadratic, 55 from A, 35 integrable, 36 predictable quadratic, 56, 84 quadratic, 40, 55, 80, 88 Variation process, quadratic, 80 Version, 28 Volatility, 104

W Walk, random, 23 Weakly unique, 168 Wiener chaos, 125 representation, 126

Z Zero coupon bond, 203