Semimartingales and stochastic integration

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Semimartingales and stochastic integration Spring 2011 Sergio Pulido∗

Chris Almost†

Contents

Contents

1

0 Motivation

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1 Preliminaries 1.1 Review of stochastic processes . . . . . . . . . . . . . . . . 1.2 Review of martingales . . . . . . . . . . . . . . . . . . . . . 1.3 Poisson process and Brownian motion . . . . . . . . . . . 1.4 Lévy processes . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Lévy measures . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Integration with respect to processes of finite variation . 1.8 Naïve stochastic integration is impossible . . . . . . . . . ∗ †

[email protected] [email protected]

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Contents 2 Semimartingales and stochastic integration 2.1 Introduction . . . . . . . . . . . . . . . . . . . . 2.2 Stability properties of semimartingales . . . . 2.3 Elementary examples of semimartingales . . 2.4 The stochastic integral as a process . . . . . . 2.5 Properties of the stochastic integral . . . . . . 2.6 The quadratic variation of a semimartingale 2.7 Itô’s formula . . . . . . . . . . . . . . . . . . . . 2.8 Applications of Itô’s formula . . . . . . . . . .

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24 24 25 26 27 29 31 35 40

3 The Bichteler-Dellacherie Theorem and its connexions to arbitrage 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Proofs of Theorems 3.1.7 and 3.1.8 . . . . . . . . . . . . . . . . . . . . 3.3 A short proof of the Doob-Meyer theorem . . . . . . . . . . . . . . . . 3.4 Fundamental theorem of local martingales . . . . . . . . . . . . . . . 3.5 Quasimartingales, compensators, and the fundamental theorem of local martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Special semimartingales and another decomposition theorem for local martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Girsanov’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43 43 45 52 54

4 General stochastic integration 4.1 Stochastic integrals with respect to predictable processes . . . . . .

65 65

Index

72

56 58 61

Chapter 0

Motivation Why stochastic integration with respect to semimartingales with jumps? ◦ To model “unpredictable” events (e.g. default times in credit risk theory) one needs to consider models with jumps. ◦ A lot of interesting stochastic processes jump, e.g. Poisson process, Lévy processes. This course will closely follow the textbook, Stochastic integration and differential equations by Philip E. Protter, second edition. We will not cover every chapter, and some proofs given in the course will differ from those in the text. The following numbers correspond to sections in the textbook. I Preliminaries. 1. Filtrations, stochastic processes, stopping times, path regularity, “functional” monotone class theorem, optional σ-algebra. 2. Martingales. 3. Poisson processes, Brownian motion. 4. Lévy processes. 6. Localization procedure for stochastic processes. 7. Stieltjes integration 8. Impossibility of naïve stochastic integration (via the Banach-Steinhaus theorem). II Semimartingales and stochastic integrals. 1–3. Definition of the stochastic integral with respect to processes in L. 5. Properties of the stochastic integral. 6. Quadratic variation. 7. Itô’s formula. 8. Stochastic exponential and Lévy’s characterization theorem. III Bichteler-Dellacherie theorem. (NFLVR) implies S is a semimartingale (NFLVR) and little investment if and only if S is a semimartingale IV Stochastic integration with respect to predictable processes and martingale representation theorems (i.e. market completeness). For more information on the history of the development of stochastic integration, see the paper by Protter and Jarrow on that topic. 3

Chapter 1

Preliminaries 1.1

Review of stochastic processes

The standard setup we will use is that of a complete probability space (Ω, F , P) and a filtration F = (F t )0≤t≤∞ of sub-σ-algebras of F . The filtration can be thought of as the flow of information. Expectation E will always be with respect to P unless stated otherwise. Notation. We will use the convection that t, s, and u will always be real variables, not including ∞ unless it is explicitly mentioned, e.g. {t|t ≥ 0} = [0, ∞). On the other hand, n and k will always be integers, e.g. {n : n ≥ 0} = {0, 1, 2, 3, . . . } =: N. 1.1.1 Definition. (Ω, F , F, P) satisfies the usual conditions if (i) F0 contains all the P-null sets. T (ii) F is right continuous, i.e. F t = s 0. PROOF: Assume T is a stopping time. Then for any t > 0, [ _ {T < t} = {T ≤ t − 1n } ∈ Fs ⊆ F t . st

n≥1

so T is a stopping time.

ƒ 4

1.1. Review of stochastic processes

5

1.1.4 Theorem. W V (i) If (Tn )n≥1 is a sequence of stopping times then n Tn and n Tn are stopping times. (ii) If T and S are stopping times then T + S is a stopping time. 1.1.5 Exercises. (i) If T ≥ S then is T − S a stopping time? (ii) For which constants α is αT a stopping time? SOLUTION: Clearly αT need not be a stopping time if α < 1. If α ≥ 1 then, for any t ≥ 0, t/α ≤ t so {αT ≤ t} = {T ≤ t/α} ∈ F t/α ⊆ F t and αT is a stopping time. Let H be a stopping time for which H/2 is not a stopping time (e.g. the first hitting time of Brownian motion at the level 1). Take T = 3H/2 and S = H, both stopping times, and note that T − S = H/2 is not a stopping time. ú 1.1.6 Definition. Let T be a stopping time. The σ-algebras of events before time T and events strictly before time T are (i) F T = {A ∈ F : A ∩ {T ≤ t} ∈ F t for all t}. (ii) F T − = F0 ∨ σ{A ∩ {t < T } : t ∈ [0, ∞), A ∈ F t }. 1.1.7 Definition. (i) A stochastic process X is a collection of Rd -valued r.v.’s, (X t )0≤t 0 : X t ∈ Λ} is a stopping time. (iii) If Λ ⊆ Rd is closed and X is càd and adapted then T := inf{t > 0 : X t ∈ Λ or X t− ∈ Λ} is a stopping time. (iv) If X is càdlàg and adapted and ∆X T 1 T 0 : X t ∈ B} is a stopping time. 1.1.12 Theorem. If X is an optional process then (i) X is (F ⊗ B([0, ∞)))-measurable and (ii) X T 1 T 1 then   p sup kX t k L p . k sup X t k L p ≤ p − 1 t≥0 t≥0 In particular if p = 2 then E[sup t≥0 X t2 ] ≤ 4 sup t≥0 E[X t2 ]. (iv) (Jensen’s inequality) If ϕ is a convex function, Z is an integrable r.v., and G is a sub-σ-algebra of F then ϕ(E[Z|G ]) ≤ E[ϕ(Z)|G ].

1.3. Poisson process and Brownian motion 1.2.10 Definition. Let X be a process and T be a random time. The stopped process is X tT := X t 1 t≤T + X T 1 t>T,T 0 a.s. The process Nt = n≥1 1 Tn ≤t is the counting process associated with (Tn )n≥1 . The random time T := supn≥1 Tn is the explosion time. If T = ∞ a.s. then N is a counting process without explosion. 1.3.2 Theorem. A counting process is an adapted process if and only if Tn is a stopping time for all n. PROOF: If N is adapted then {t < Tn } = {Nt < n} ∈ F t for all t and all n. Therefore, for all n, {Tn ≤ t} ∈ F t for all t, so Tn is a stopping time. Conversely, if all the Tn are stopping times then, for all t, {Nt ≤ n} = {t ≤ Tn } ∈ F t for all n. Since N takes only integer values this implies that Nt ∈ F t . ƒ 1.3.3 Definition. An adapted process N is called a Poisson process if () N is a counting process. (i) Nt −Ns is independent of Fs for all 0 ≤ s < t < ∞ (independent increments).

9

10

Preliminaries (d)

(ii) Nt − Ns = Nt−s for all 0 ≤ s < t < ∞ (stationary increments). Remark. Implicit in this definition is that a Poisson process does not explode. The definition can be modified slightly to allow this possibility, and then it can be proved as a theorem that a Poisson process does not explode, but the details are very technical and relatively unenlightening. 1.3.4 Theorem. Suppose that N is a Poisson process. (i) N is continuous in probability. (d) (ii) Nt = Poisson(λt) for some λ ≥ 0, called the intensity or arrival rate of N . In particular, Nt has finite moments of all orders for all t. (iii) (Nt − λt) t≥0 and ((Nt − λt)2 − λt) t≥0 are martingales. (iv) If F tN := σ(Ns : s ≤ t) and FN = (F tN ) t≥0 then FN is right continuous. PROOF: Let α(t) := P[Nt = 0] for t ≥ 0. For all s < t, α(t + s) = P[Nt+s = 0] = P[{Ns = 0} ∩ {Nt+s − Ns = 0}]

non-decreasing and non-negative

= P[Ns = 0] P[Nt+s − Ns = 0]

independent increments

= P[Ns = 0] P[Nt = 0]

stationary increments

= α(t)α(s) If t n ↓ t then {Nt n = 0} % {Nt = 0}, so α is right continuous and decreasing. It follows that either α ≡ 0 or α(t) = e−λt for some λ ≥ 0. By the definition of counting process N0 = 0, so α is cannot be the zero function. (i) Observe that, given " > 0 small, for all s < t, P[|Nt − Ns | > "] = P[|Nt−s | > "]

stationary increments

= P[Nt−s > "]

N is non-decreasing

= 1 − P[Nt−s = 0]

N is integer valued

=1−e

−λ(t−s)

→ 0 as s → t. Therefore N is left continuous in probability. The proof of continuity from the right is similar. (ii) First we need to prove that lim t→0 1t P[Nt = 1] = λ. Towards this, let β(t) := P[Nt ≥ 2] for t ≥ 0. If we can show that lim t→0 β(t)/t = 0 then we would have P[Nt = 1] 1 − α(t) − β(t) lim = lim = λ. t→0 t→0 t t It is enough to prove that limn→∞ nβ(1/n) = 0. Divide [0, 1] into n equal subintervals and let Sn be the number of subintervals with at least two ar(d) rivals. It can be see that Sn = Binomial(n, β(1/n)) because of the stationary and independent increments of N . In the definition of counting process the

1.3. Poisson process and Brownian motion

11

sequence of jump times is strictly increasing, so limn→∞ Sn = 0 a.s. Clearly Sn < N1 , so limn→∞ nβ(1/n) = limn→∞ E[Sn ] = 0 provided that E[N1 ] < ∞. We are going to gloss over this point. Let ϕ(t) := E[γNt ] for 0 < γ < 1. ϕ(t + s) = E[γNt+s ] = E[γNs γNt+s −Ns ] = E[γNs ] E[γNt ] = ϕ(t)ϕ(s) and ϕ is right continuous because N has right continuous paths. Therefore ϕ(t) = e tψ(γ) for some function ψ of γ. By definition, ϕ(t) =

X

γn P[Nt = n]

n≥0

e

tψ(γ)

= α(t) + γ P[Nt = 1] +

X

γn P[Nt = n].

n≥2

Differentiating, ψ(γ) = lim t→0 (ϕ(t) − 1)/t = −λ + γλ by the computations above. Comparing coefficients of γn in the power series shows that Nt has a Poisson distribution with rate λt, i.e. P[Nt = n] = e−λt (λt)n /n!. (iii) Exercise. (iv) See the textbook.

ƒ

1.3.5 Definition. An adapted process B is called a Brownian motion if (i) B t −Bs is independent of Fs for all 0 ≤ s < t < ∞ (independent increments). (d)

(ii) B t − Bs = Normal(0, t − s) for all 0 ≤ s < t < ∞ (stationary, normally distributed, increments). If B0 ≡ 0 then B is a standard Brownian motion. 1.3.6 Theorem. Let B be a Brownian motion. (i) If E[|B0 |] < ∞ then B is a martingale. (ii) There is a modification of B with continuous paths. (iii) (B 2t − t) t≥0 is a martingale when B is standard BM. (iv) Let Πn be a refining sequence of partitions of the interval [a, a + t] with P limn→∞ mesh(Πn ) = 0. Then Πn B := t n ∈Πn (B t i+1 − B t i )2 → t in L 2 and a.s. when B is standard BM. (v) For almost all ω, the function t 7→ B t (ω) is of unbounded variation. P Remark. To prove (ii) with the additional assumption that n≥0 mesh(Πn ) < ∞, but dropping the assumption that the partitions are refining, you can use the BorelCantelli lemma. The proof of (iv) uses the backwards martingale convergence theorem.

12

Preliminaries

1.4

Lévy processes

1.4.1 Definition. An adapted, real-valued process X is called a Lévy process if () X 0 = 0 (i) X t −X s is independent of Fs for all 0 ≤ s < t < ∞ (independent increments). (d) (ii) X t − X s = X t−s for all 0 ≤ s < t < ∞ (stationary increments). (iii) (X t ) t≥0 is continuous in probability. If only (i) and (iii) hold then X is an additive process. If (X t ) t≥0 is non-decreasing then X is called a subordinator. Remark. If the filtration is not specified then we assume that F = FX is the natural (not necessarily complete) filtration of X . In this case X might then be called an intrinsic Lévy process. 1.4.2 Example. The Poisson process and Brownian motion are both Lévy processes. Let W be a standard BM and define Tb := inf{t > 0 : Wt ≥ b}. Then (Tb ) b≥0 is an intrinsic Lévy process and subordinator for the filtration (F Tb ) b≥0 . Indeed, if 0 ≤ a < b < ∞ then Ta − Tb is independent of F Ta and distributed as Tb−a by the strong Markov property of W . We will see that T is a stable process with parameter α = 1/2. 1.4.3 Theorem. Let X be a Lévy process. Then f t (z) := E[e izX t ] = e−tΨ(z) for some continuous function Ψ = ΨX . Furthermore, M tz := e izX t / f t (z) is a martingale for all z ∈ R. PROOF: Fix z ∈ R. By the stationarity and independence of the increments, f t (z) = f t−s (z) fs (z) for all 0 ≤ s < t < ∞.

(1.1)

We would like to show that t 7→ f t (z) is right-continuous. By the multiplicative property (1.1), it suffices to show t 7→ f t (z) is right continuous at zero. Suppose that t n ↓ 0; we need to show that f t n (z) → 1. By definition, | f t n (z)| ≤ 1, so we are done if we can show that every convergent subsequence converges to 1. Suppose without loss of generality that f t n (z) → a ∈ C. Since X is continuous in probability, e izX t n → 1 in probability. Along a subsequence we have convergence almost surely, i.e. f t t n (z) → 1. Therefore a = 1 and we are done. k

The multiplicative property and right continuity imply that f t (z) = e−tΨ(z) for some number Ψ(z). Since f t (z) is a characteristic function, z 7→ f t (z) is continuous (use dominated convergence). Therefore Ψ must be continuous as well. In particular, f t (z) 6= 0 for all t and all z. Let 0 ≤ s < t < ∞.  izX t  e e izX s F = E[e iz(X t −X s ) |Fs ] independent increments E s f t (z) f t (z) =

e izX s f t (z)

f t−s (z)

stationary increments

1.4. Lévy processes

=

13 e izX s fs (z)

f is multiplicative.

Therefore M tz is a martingale.

ƒ

1.4.4 Theorem. If X is an additive process then X is Markov with transition function Ps,t (x, B) = P[X t − X s ∈ B − x]. In particular, X is spatially homogeneous, i.e. Ps,t (x, B) = Ps,t (0, B − x). 1.4.5 Corollary. If X is additive, " > 0, and T < ∞ then limu↓0 α",T (u) = 0, where α",T is defined in the next theorem. PROOF: Ps,t (x, B" (x)C ) = P(|X t − X s | ≥ ") → 0 as s → t uniformly on [0, T ] in probability. It is an exercise to show that the continuity in probability of X implies uniform continuity in probability on compact intervals. ƒ 1.4.6 Theorem. Let X be a Markov process on Rd with transition function Ps,t . If limu↓0 α",T (u) = 0, where α",T (u) = sup{Ps,t (x, B" (x)C ) : x ∈ Rd , s, t ∈ [0, T ], 0 ≤ t − s ≤ u}, then X has a càdlàg modification. If furthermore limu↓0 α",T (u)/u = 0 then X has a continuous modification. PROOF: See Lévy processes and infinitely divisible distributions by Kin-Iti Sato. The important steps are as follows. Fix " > 0 and ω ∈ Ω. Say that X (ω) has an "-oscillation n-times in M ⊆ [0, ∞) if there are t 0 < t 1 < · · · < t n all in M such that |X t j (ω) − X t j−1 (ω)| ≥ ". X has "-oscillation infinitely often in M if this holds for all n. Define Ω02 :=

∞ ∞ \ \ N =1 k=1

{ω : X (ω) does not have 1k -oscillation infinitely often in [0, N ] ∩ Q}

It can be shown that Ω02 ∈ F , and also that n o Ω02 ⊆ ω : lim X s exists for all t and lim X s exists for all t > 0 . s↓t,s∈Q

The hard part is to show that P[Ω02 ] = 1.

s↑t,s∈Q

ƒ

Remark. (i) If X is a Feller process then X has a càdlàg modification (see Revuz-Yor). (ii) From now on we assume that we are working with a càdlàg version of any Lévy process that appears. (iii) P[X t 6= X t− ] = 0 for any process X that is continuous in probability. Of course, this does not mean that X doesn’t jump, it means that X has no fixed times of discontinuity.

14

Preliminaries 1.4.7 Theorem. Let X be a Lévy process and N be the collection of P-null sets. If G t := F tX ∨ N for all t then G = (G t ) t≥0 is right continuous. PROOF: Let (s1 , . . . , sn ) and (u1 , . . . , un ) be arbitrary vectors in (R+ )n and Rn respectively. We first want to show that E[e i(u1 X s1 +···+un X sn ) |G t ] = E[e i(u1 X s1 +···+un X sn ) |G t+ ]. It is clear that it suffices to show this with s1 , . . . , sn > t. We take n = 2 for notational simplicity, and assume that s2 ≥ s1 > t. E[e i(u1 X s1 +u2 X s2 ) |G t+ ] = lim E[e i(u1 X s1 +u2 X s2 ) |Gw ] w↓t

exercise

= lim E[e iu1 X s1 Msu22 |Gw ] fs2 (u2 ) w↓t

= lim E[e iu1 X s1 Msu12 |Gw ] fs2 (u2 )

tower property

= lim E[e i(u1 +u2 )X s1 |Gw ] fs2 −s1 (u2 )

f is multiplicative

w↓t

w↓t

= lim w↓t

e i(u1 +u2 )X w f w (u1 + u2 )

fs1 (u1 + u2 ) fs2 −s1 (u2 )

= e i(u1 +u2 )X t fs1 −t (u1 + u2 ) fs2 −s1 (u2 )

M is a martingale X is càdlàg

By the same steps, we obtain E[e i(u1 X s1 +u2 X s2 ) |G t ] = e i(u1 +u2 )X t fs1 −t (u1 + u2 ) fs2 −s1 (u2 ). X By a monotone class argument, E[Z|G t ] = E[Z|G t+ ] for any bounded Z ∈ F∞ . It follows that G t+ \ G t consists only of sets from N . Since N ⊆ G t , they must be equal. ƒ

1.4.8 Corollary (Blumenthal 0-1 Law). Let X be a Lévy process. If A ∈ then P[A] = 0 or 1.

T

X t>0 F t

1.4.9 Theorem. If X is a Lévy process and T is a stopping time then, on {T < ∞}, Yt := X T +t − X T is a Lévy process with respect to (F t+T ) t≥0 and Y has the same finite dimensional distributions as X . 1.4.10 Corollary. If X is a Lévy process then X is a strong Markov process. 1.4.11 Theorem. If X is a Lévy process with bounded jumps then E[|X t |n ] < ∞ for all t and all n. PROOF: Suppose that sup t |∆X t | ≤ C. Define a sequence of random times Tn recursively as follows. T0 ≡ 0 and ¨ inf{t > Tn : |X t − X Tn | ≥ C} if Tn < ∞ Tn+1 = ∞ if Tn = ∞

1.5. Lévy measures

15

By definition of the Tn , sups |X sTn | ≤ 2nC. The 2 comes from the possibility that X could jump by C just before hitting the stopping level. Since X is càdlàg, the Tn are all stopping times. Further, since X is a strong Markov process on {Tn < ∞}, (i) Tn − Tn−1 is independent of F Tn−1 and (d)

(ii) Tn − Tn−1 = T1 . Therefore E[e−Tn ] = E

Y n

 e−(Tk −Tk−1 ) ≤ (E[e−T1 ])n =: αn ,

k=0

where 0 ≤ α < 1. The ≤ comes from the fact that e−∞ = 0 and some of the Tn may be ∞. (We interpret Tk − Tk−1 to be ∞ if both of them are ∞.) By Chebyshev’s inequality, E[e−Tn ] ≤ αn e t . P[|X t | > 2nC] ≤ P[Tn < t] ≤ e−t Finally, E[e

β|X t |

]≤1+ ≤1+

∞ X n=0 ∞ X

E[eβ|X t | 12nC 2nC]

n=0

≤ 1 + e2β C e t

∞ X (αe2β C )n n=0

Choosing an appropriately small, positive β shows that |X t | has an exponential moment, so it has polynomial moments of all orders. ƒ

1.5

Lévy measures

Suppose that Λ ∈ B(R) and 0 ∈ / Λ. Let X be a Lévy process (with càdlàg paths, as always) and define inductively a sequence of random times TΛ0 ≡ 0 and TΛn+1 := inf{t > TΛn : ∆X t ∈ Λ}. These times have the following properties. (i) {TΛn ≥ t} ∈ F t+ = F t by the usual conditions and (TΛn )n≥1 is an increasing sequence of (possibly ∞-valued) stopping times. (ii) TΛ1 > 0 a.s. since X 0 ≡ 0, X has càdlàg paths, and 0 ∈ / Λ. n (iii) limn→∞ TΛ = ∞ since X has càdlàg paths and 0 ∈ / Λ (see Homework 1, problem 10). Let N Λ be the number of jumps with size in Λ before time t. NtΛ

:=

X 0 α] < " for all k ≥ k0 . Then P[|I X t (H k )| > α] = P[|I X t (H k )| > α; Tn0 ≤ t] + P[|I(X Tn0 )t (H k )| > α; Tn0 > t] < 2" for all k ≥ k0 . Therefore I X t (H k ) → 0 in probability, so X is a semimartingale.

2.3

ƒ

Elementary examples of semimartingales

2.3.1 Theorem. Every adapted càdlàg process with paths of finite (total) variation is a (total) semimartingale. PROOF: Suppose that the R ∞total variation of X is finite. It is easy to see that, for all H ∈ S, |I X (H)| ≤ kHk∞ 0 d|X |s . ƒ 2.3.2 Theorem. Every càdlàg L 2 -martingale is a total semimartingale. PROOF: Without loss of generality X 0 = 0. Let H ∈ S have the canonical representation. 2   X n−1 2 E[(I X (H)) ] = E H i (X Ti+1 − X Ti ) i=1

=E

X n−1

H i2 (X Ti+1 − X Ti )2

 X is a martingale

i=1

 X n−1 2 (X Ti+1 − X Ti )

≤

kHk2u E

=

i=1 2 kHku E[X T2n ]

≤

2 kHk2u E[X ∞ ]

X is a martingale by Jensen’s inequality

Therefore if H k → H in Su then I X (H k ) → I X (H) in L 2 , so also in probability.

ƒ

2.3.3 Corollary. (i) A càdlàg locally square integrable martingale is a semimartingale. (ii) A càdlàg local martingale with bounded jumps is a semimartingale. (iii) A local martingale with continuous paths is a semimartingale. (iv) Brownian motion is a semimartingale. (v) If X = M + A is càdlàg, where M is a locally square integrable martingale and A is locally of finite variation, then X is a semimartingale. Such an X is called a decomposable process (vi) Any Lévy process is a semimartingale by the Lévy-Itô decomposition.

2.4. The stochastic integral as a process

2.4

27

The stochastic integral as a process

2.4.1 Definition. (i) D := {adapted, càdlàg processes} (ii) L := {adapted, càglàd processes} (iii) If C is a collection of processes then bC is the collection of bounded processes from C. 2.4.2 Definition. For processes H n and H we say that H n → H uniformly on compacts in probability (or u.c.p.) if h i lim P sup |Hsn − Hs | ≥ " = 0 n→∞

0≤s≤t

(p)

for all " > 0 and all t. Equivalently, if (H n − H)∗t − → 0 for all t. Remark. If we define a metric d(X , Y ) :=

∞ X 1 n=1

2n

E[(X − Y )∗n ∧ 1]

then u.c.p. convergence is compatible with this metric. We will denote by Ducp , Lucp , and Sucp the topological spaces D, L, and S endowed with the u.c.p. topology. The u.c.p. topology is weaker than the uniform topology. It is important to note that Ducp and Lucp are complete metric spaces. 2.4.3 Theorem. S is dense in Lucp . PROOF: Let Y ∈ L and define R n := inf{t : |Yt | > n}. Then Y R n → Y u.c.p. (exercise) and Y Rn is bounded by n because Y is left continuous. Thus bL is dense in Lucp , so it suffices to show that S is dense in bLucp . Assume that Y ∈ bL and define Z t := limu↓t Yn for all t ≥ 0, so that Z ∈ D. For each " > 0 define a sequence of stopping times T0" := 0 and " Tn+1 := inf{t > Tn" : |Z t − Z Tn" | > "}.

We have seen that the Tn" are stopping times because they are hitting times for the càdlàg process Z. Also because Z ∈ D, Tn" ↑ ∞ a.s. (this would not necessarily happen it T " were defined in the same way but with Y ). Let X " Z " := Z Tn" 1[Tn" ,Tn+1 ). n≥0

Then Z " is bounded and Z " → Z uniformly on compacts as " → 0. Let X " Z Tn" 1(Tn" ,Tn+1 U " := Y0 1{0} + ]. n≥0

28

Semimartingales and stochastic integration Then U " → Y0 1{0} + Z− uniformly on compacts as " → 0. If we define Y n," := Y0 1{0} +

n X

" Z Tk" 1(Tk" ,Tk+1 ]

k≥1

then Y n," → Y u.c.p. as " → 0 and n → ∞.

ƒ

2.4.4 Definition. Let H ∈ S and X be a càdlàg process. Pn The stochastic integral process of H with respect to X is JX (H) := H0 X 0 + i=1 H i (X Ti+1 − X Ti ) when Pn−1 H = H0 1{0} + i=1 H i 1(Ti ,Ti+1 ] . R Rt Notation. We write Hs dX s := H ·X := JX (H) and 0 Hs dX s := I X t (H) = (JX (H)) t R∞ and 0 Hs d X s := I X (H). 2.4.5 Theorem. If X is a semimartingale then JX : Sucp → Ducp is continuous. PROOF: We begin by proving the weaker statement, that JX : Su → Ducp is continuous. Suppose that kH k k∞ → 0. Let δ > 0 and define a sequence of stopping times T k := inf{t : |(H k · X ) t | ≥ δ}. Then H k 1[0,T k ] ∈ S and kH k 1[0,T k ] k∞ → 0 because this already happens for (H k )k≥1 . Let t ≥ 0 be given. P[(H k · X )∗t ≥ δ] ≤ P[(H k · X )∗t∧T k ≥ δ] = P[I X t (H k 1[0,T k ] ) ≥ δ] → 0 since X is a semimartingale. Therefore JX (H k ) → 0 u.c.p. Assume H k → 0 u.c.p. Let " > 0, t > 0, and δ > 0 be given. There is η such that kHk∞ < η implies P[JX (H)∗t ≥ δ] < ". Let Rk := inf{s : |Hsk | > η}. Then ˜ k := H k 1[0,Rk ] → 0 u.c.p., and kH ˜ k k ≤ η. Rk → ∞ a.s., H P[(H k · X )∗t ≥ δ] = P[(H k · X )∗t ≥ δ; Rk < t] + P[(H k · X )∗t ≥ δ; Rk ≥ t] ˜ k · X )∗ ≥ δ] < 2" ≤ P[Rk < t] + P[(H t for k large enough.

ƒ

If H ∈ L and (H n )n≥1 ⊆ S are such that H n → H u.c.p. then (H n )n≥1 has the Cauchy property for the u.c.p. metric. Since JX is continuous, (JX (H n ))n≥1 also has the Cauchy property. Since Ducp is complete there is JX (H) ∈ D such that JX (H n ) → JX (H) u.c.p. We say that JX (H) is the stochastic integral of H with respect to X . 2.4.6 Example. Let B be standard Brownian motion and let (Πn )n≥1 be a sequence of partitions of [0, t] with mesh size converging to zero. Define X B n := B t i 1(t i ,t i+1 ] , Πn

2.5. Properties of the stochastic integral

29

so that B n → B u.c.p. (check this as an exercise). X (JB (B n )) t = B t i (B t i+1 − B t i ) Πn

1X 2 1X = (B t i+1 − B 2t i ) − (B t i+1 − B t i )2 2 Π 2 Π n n 1 2 1X (B t i+1 − B t i )2 = Bt − 2 2 Π n

(p)

− →

1 2

B 2t −

1 2

t

Hence

Rt

2.5

Properties of the stochastic integral

0

Bs d Bs = 12 Bs − 21 t.

2.5.1 Theorem. Let H ∈ L and X be a semimartingale. (i) If T is a stopping time then (H · X ) T = (H1[0,T ] · X ) = H · (X T ). (ii) ∆(H · X ) is indistinguishable from H · (∆X ). (iii) If Q P P then H ·Q X is indistinguishable from H ·P X . P (iv) If Q = k≥1 λk Pk with λk ≥ 0 and k≥1 λk = 1 then H ·Q X is indistinguishable from H ·Pk X for any k for which λk > 0. (v) If X is both a P and Q semimartingale then there is H · X that is a version of both H ·P X and H ·Q X . (vi) If G is another filtration and H ∈ L(G) ∩ L(F) then H ·G X = H ·F X . (vii) If X has paths of finite variation on compacts then H · X is indistinguishable from the path-by-path Lebesgue-Stieltjes integral. (viii) Y := H · X is a semimartingale and K · Y = (K H) · X for all K ∈ L. PROOF (OF (VIII)): It can be shown using limiting arguments that K · (H · X ) = (K H) · X when H, K ∈ S. To prove that Y = H · X is a semimartingale when H ∈ L there are two steps. First we show that if K ∈ S then K · Y = (K H) · X (and this makes sense even without knowing that Y is a semimartingale). Fix t > 0 and choose H n ∈ S with H n → H u.c.p. We know H n · X → H · X u.c.p. because X is a semimartingale. Therefore there is a subsequence such that (Y nk − Y )∗t → 0 a.s., where Y nk := H nk · X . Because of the a.s. convergence, (K · Y ) t = limk→∞ (K · Y nk ) t a.s. Finally, K · Y nk = (K H nk ) · X since H n , K ∈ S, so K · Y = u.c.p.-lim (K H nk ) · X = (K H) · X k→∞

since K H nk → K H u.c.p. and X is a semimartingale. For the second step, proving that Y is a semimartingale, assume that Gn → G in Su . We must prove that (G n · Y ) t → (G · Y ) t in probability for all t. We have G n H → GH in Lucp , so lim G n · Y = lim (G n H) · X = (GH) · X = G · Y.

n→∞

n→∞

Since this convergence is u.c.p. this proves the result.

ƒ

30

Semimartingales and stochastic integration 2.5.2 Theorem. If X is a locally square integrable martingale and H ∈ L then H · X is a locally square integrable martingale. PROOF: It suffices to prove the result assuming X is a square integrable martingale, since if (Tn )n≥1 is a localizing sequence for X then (H · X ) Tn = H · X Tn , so we would have that H · X is a local locally square integrable martingale. Conclude with Theorem 1.6.2, since the collection of locally square integrable martingale is stable under stopping. Furthermore, we may assume that H is bounded because left continuous processes are locally bounded (a localizing sequence is R n := inf{t : |H t | > n}) and that X 0 = 0. Construct, as in Theorem 2.4.3, H n ∈ S such that H n → H u.c.p. By construction the H n are bounded by kHk∞ . Let t > 0 be given. It is easy to see that H n · X is a martingale and 2

E[(H · X ) ] = E n

 X n

H in

T X t i+1

2 

−

T
Xt i

i=1

≤ kHk2∞ E[X t2 ] < ∞. (The detail are as in the proof of Theorem 2.3.2.) It follows that ((H n · X ) t )n≥1 is u.i. Since we already know (H n · X ) t → (H · X ) t in probability, the convergence is actually in L 1 . This allows us to prove that H · X is a martingale, and by Fatou’s lemma E[(H · X )2t ] ≤ lim inf E[(H n · X )2t ] ≤ kHk2∞ E[X t2 ] < ∞ n

so H · X is a square integrable martingale.

ƒ

2.5.3 Theorem. Let X be a semimartingale. Suppose that (Σn )n≥1 is a sequence of random partitions, Σn : 0 = T0n ≤ T1n ≤ · · · ≤ Tkn where the Tkn are stopping n times such that (i) limn→∞ Tkn = ∞ a.s. and n n (ii) supk |Tk+1 − Tkn | → 0 a.s. R P n n Then if Y ∈ L (or D) then k YTkn (X Tk+1 − X Tk ) → Y− dX . Remark. The hard part of the proof of this theorem is that the approximations to Y do not necessarily converge u.c.p. to Y (if they did then the theorem would be a trivial consequence of the definition of semimartingale). 2.5.4 Example. Let M t = Nt − λt be a compensated Poisson process (a martingale) and let H = 1[0,T1 ) (a bounded process in D), where T1 is the first jump time Rt of the Poisson process. Then 0 Hs d Ms = −λ(t ∧ T1 ), which is not a local martingale. Examples of this kind are part of the reason that we want our integrands from L.

2.6. The quadratic variation of a semimartingale

2.6

The quadratic variation of a semimartingale

2.6.1 Definition. Let X and Y be semimartingales. The quadratic variation of X and Y is [X , Y ] = ([X , Y ] t ) t≥0 , defined by Z Z [X , Y ] := X Y −

X − dY −

Y− dX .

Write [X ] := [X , X ]. The polarization identity is sometimes useful. [X , Y ] =

1 2

([X + Y ] − [X ] − [Y ])

2.6.2 Theorem. (i) [X ]0 = X 02 and ∆[X ] = (∆X )2 . (ii) If (Σn )n≥1 are as in Theorem 2.5.3 then X n n u.c.p. (X Tk+1 − X Tk )2 −→ [X ]. X 02 + k

(iii) [X ] is càdlàg, adapted, and increasing. (iv) [X , Y ] is of finite variation, [X , Y ]0 = X 0 Y0 , ∆[X , Y ] = ∆X ∆Y , and X n n n n u.c.p. X 0 Y0 + (X Tk+1 − X Tk )(Y Tk+1 − Y Tk ) −→ [X , Y ]. k

(v) If T is any stopping time then [X T , Y ] = [X , Y T ] = [X T , Y T ] = [X , Y ] T . PROOF: R0 (i) Recall that X 0− := 0, so 0 X s− dX s = 0 and [X ]0 = (X 0 )2 . For any t > 0, 2 − 2X t X t− (∆X t )2 = (X t − X t− )2 = X t2 + X t− 2 = X t2 − X t− − 2X t− ∆X t

= (∆X 2 ) t − 2∆(X − · X ) t = ∆(X 2 − 2(X − · X )) t = ∆[X ] t (ii) Fix n and let R n := supk Tkn . Then (X 2 )Rn → X 2 u.c.p., so apply Theorem 2.5.3 and the summation-by-parts trick. (iii) Let’s see why [X ] is increasing. Fix s < t rational. If we can prove that [X ]s ≤ [X ] t a.s. then we are done since [X ] is càdlàg. Use partitions (Σn )n≥1 in Theorem 2.5.3 that include s and t. Excluding terms from the sum makes it smaller since all of the summands are squares (hence non-negative), so [X ]s ≤ [X ] t a.s. ƒ

31

32

Semimartingales and stochastic integration 2.6.3 Corollary. (i) If X is a continuous semimartingale of finite variation then [X ] is constant, i.e. [X ] t = X 02 for all t. (ii) [X , Y ] and X Y are semimartingales, so {semimartingales} form an algebra. 2.6.4 Theorem (Kunita-Watanabe identity). Let X and Y be semimartingales and H, K : Ω × [0, ∞) → R be F ⊗ B-measurable. Z∞ Z ∞ 2 Z ∞ 2 |Hs ||Ks |d|[X , Y ]|s ≤ Hs d[X ]s Ks2 d[Y ]s . 0

0

0

PROOF: Use the following lemma. 2.6.5 Lemma. Let α, β, γ : [0, ∞) → R be such that α(0) = β(0) = γ(0) = 0, α is of finite variation, β and γ are increasing, and for all s ≤ t, (α(t) − α(s))2 ≤ (β(t) − β(s))(γ(t) − γ(s)). Then for all measurable functions f and g and all s ≤ t, Z t Z t 2 Z t 2 | fu ||gu |d|α|u ≤ fu dβu gu2 dγu . s

s

s

The lemma can be proved with the following version of the monotone class theorem. 2.6.6 Theorem (Monotone class theorem II). Let C be an algebra of bounded real valued functions. Let H be a collection of bounded real valued functions closed under monotone and uniform convergence. If C ⊆ H then bσ(C) ⊆ H. Both are left as exercises.

ƒ

2.6.7 Definition. Let X be a semimartingale. The continuous part of quadratic variation, [X ]c , is defined by X [X ] t = [X ]ct + X 02 + (∆X s )2 . 0 n} ∧ n. p sup |MsTn | ≤ n + |∆M Tn | ≤ n + [M ]n ƒ 0≤s≤t Hence sups≤t |MsTn | ∈ L 2 ⊆ L 1 for all n and all t. By Doob’s maximal inequality. . . 2.6.15 Example (Inverse Bessel process). Let B be a three dimensional Brownian motion that starts at x 6= 0. It can be shown that M t := 1/kB t k is a local martingale. Furthermore, E[M t2 ] < ∞ for all t and lim t→∞ E[M t2 ] = 0. This implies that M is not a martingale (because if it were then M 2 would be a submartingale, which contradicts that lim t→∞ E[M t2 ] = 0). Moreover, E[[M ] t ] = ∞ for all t > 0. It can be shown that M satisfies the SDE d M t = −M t2 dB t . 2.6.16 Corollary. Let X be a continuous local martingale. Then X and [X ] have the same intervals of constancy a.s. (i.e. pathwise, cf. Corollary 2.6.11). 2.6.17 Theorem. Let X be a quadratic pure jump semimartingale. For any semimartingale Y , X [X , Y ] = X 0 Y0 + ∆X s ∆Ys . 0 0 there is M such that P[Q n ≥ M ] < α for all n). PROOF: For all n, let H n := −

P2n

j=1 S( j−1)2

−n

1(( j−1)2−n , j2n ] , a simple predictable pro-

cess. Recall that −a(b − a) = 12 (a − b)2 + 12 (a2 − b2 ). (H n · S) t = −

2n X

S t∧( j−1)2−n (S t∧ j2−n − S t∧( j−1)2−n )

j=1 n

=

2 1X 1 (S t∧ j2−n − S t∧( j−1)2−n )2 + (S02 − S t2 ) 2 j=1 2

3.2. Proofs of Theorems 3.1.7 and 3.1.8

47

1

1 Q n + (S01 − S t2 ) 2 2

=

Clearly kH n ku ≤ 1, and (H n · S) t ≥ − 12 for all t because kSku ≤ 1 and Q n is nonnegative. Assume for contradiction that (Q n )n≥1 is not bounded in L 0 . Then there is α > 0 such that for all m > 0 there is nm such that P[(H nm · S)1 ≥ m] ≥ α. Whence (H nm /m)m≥1 would be a FLVR+LI. ƒ For c > 0 define a sequence of stopping times σn (c) := inf



k 2n

:

 k X (S j2−n − S( j−1)2−n )2 ≥ c − 4 . j=1

Given " > 0 there is c1 such that P[σn (c1 ) < ∞] < "/2 by Lemma 3.2.3. 3.2.4 Lemma. Under the same assumptions as Proposition 3.2.2, the stopped n,σ (c ) martingales M n,σn (c1 ) satisfy kM1 n 1 k2L 2 ≤ c1 . PROOF: For n ≥ 1 and k = 1, . . . , 2n , since the An s are predictable and the M n s are martingales, σ (c )

σ (c )

σ (c )

σ (c )

σ (c )

σ (c )

2 2 2 n 1 n 1 n 1 n 1 n 1 E[(Sk2n−n1 − S(k−1)2 − M(k−1)2 − A(k−1)2 −n ) ] = E[(M k2−n −n ) ] + E[(A k2−n −n ) ] σ (c )

σ (c )

2 n 1 ≥ E[(Mk2n−n 1 )2 − (M(k−1)2 −n ) ] σ (c1 ) 2

) ] as a telescoping series and simplify to get X σ (c ) σ (c ) σn (c1 ) 2 2 E[(M1 n 1 )2 ] = E[(Sk2n−n1 − S(k−1)2 −n ) ] + E[(Sσn (c1 ) − Sσn (c1 )−2−n ) ]

Write E[(M1 n

k2−n ≤σn (c1 )

≤ (c1 − 4) + 22 = c1 .

ƒ

P2n (σ (c )∧1) n 3.2.5 Lemma. Let V n := TV(An,σn (c1 ) ) = i=1 n 1 |A j2−n − An( j−1)2−n |. Under the n assumptions of Proposition 3.2.2 the sequence (V )n≥1 is bounded in probability. PROOF: Assume for contradiction that (V n )n≥1 is not bounded in probability. Then there is α > 0 such that for all k there is nk such that P[V nk ≥ k] ≥ α. For n ≥ 1 define   n,σ (c ) n,σn (c1 ) b nj−1 := sign A j2−nn 1 − A( j−1)2 ∈ F( j−1)2−n −n and H n (t) :=

P2n j=1

b nj−1 1(( j−1)2n , j2−n ] (t). Then kH n ku ≤ 1 and

(H n,σn (c1 ) · S) t =

X j≤bt2n c

    σ (c ) σn (c1 ) σ (c ) σ (c ) n b nj2−n S j2n−n 1 − S( j−1)2 + bbt2 S t n 1 − Sbt2n n c1 nc −n

≥ (H n,σn (c1 ) · An )bt2n c2−n + (H n,σn (c1 ) · M n )bt2n c2−n − 2

48

The Bichteler-Dellacherie Theorem and its connexions to arbitrage and at time t = 1 we have (H n,σn (c1 ) · S)1 = V n + (H n,σn (c1 ) · M n )1 . But the second summand is bounded in L 2 (it is at most c1 by Lemma 3.2.4), so we conclude that (H n,σn (c1 ) · S)1 is not bounded in probability. Define a sequence of stopping times   j n,σn (c1 ) n ηn (c) := inf : |(H · M ) j2−n ≥ c . 2n Because E[(sup1≤ j≤2n ((hn,σn (c1 ) · M n ) j2−n )2 ] ≤ 4c1 by Doob’s sub-martingale inequality, (H n,σn (c1 ) · M n ) is bounded in probability. Therefore there is c 0 > 0 such 0 that P[ηn (c 0 ) < ∞] ≤ α/2. Note that H n,σn (c1 )∧ηn (c ) · S is (uniformly) bounded 0 n,σn (c1 )∧ηn (c 0 ) below by c . We claim (H · S)1 is not bounded in probability. Indeed, for any n and any k, α ≤ P[(H n,σn (c1 )∧ηn (c ) · S)1 ≥ k] 0

≤ P[(H n,σn (c1 ) · S)1 ≥ k, ηn (c 0 ) = ∞] + P[ηn (c 0 ) < ∞]. Since P[ηn (c 0 ) < ∞] ≤ α/2, the probability of the other event is at least α/2. This gives the desired contradiction because it is now easy to construct a FLVR+LI. ƒ PROOF (OF PROPOSITION 3.2.2): Define a sequence of stopping times τn (c) := inf



k 2n

:

k X

 |Anj2−n − An( j−1)2−n | ≥ c .

j=1

By Lemma 3.2.5 there is c2 such that P[τn (c2 ) < ∞] < "/2. Take C := c1 ∨ c2 and ρn := σn (c1 ) ∧ τn (c2 ). ƒ 3.2.6 Lemma. Let f , g : [0, 1] → R be measurable functions, where f is left PK continuous and takes finitely many values. Say f = k=1 f (sk )1(sk−1 ,sk ] . Define ( f · g)(t) :=

K X

f (sk−1 )(g(sk ) − g(sk−1 )) + f (sk(t) )(g(t) − g(sk(t) ))

k=1

where k(t) is the biggest of the k such that sk less than or equal to t. Then for all partitions 0 ≤ t 0 ≤ · · · ≤ t M ≤ 1, M X i=1

|( f · g)(t i ) − ( f · g)(t i−1 )| ≤ 2 TV( f )kgk∞ +

X M

 |g(t i ) − g(t i−1 )| k f k∞ .

i=1

3.2.7 Proposition. Let S = (S t )0≤t≤1 be càdlàg and adapted, with S0 = 0 and such that kSku ≤ 1 and S satisfies NFLVR+LI. For all " > 0 there is C and a [0, 1] ∪ {∞} valued stopping time α such that P[α < ∞] < " and sequences (Mn )n≥1 and (An )n≥1 of continuous time càdlàg processes such that, for all n, (i) A0n = M0n = 0

3.2. Proofs of Theorems 3.1.7 and 3.1.8

49

(ii) S α = An,α + Mn,α n,α (iii) Mn,α is a martingale with kM1 k2L 2 ≤ C P2n n,α n,α (iv) j=1 |A j2−n − A( j−1)2−n | ≤ C. PROOF: Let " > 0 be given. Let C, M n , An , and ρn be as in Proposition 3.2.2. Extend M n and An to all t ∈ [0, 1] by defining M tn := E[M1n |F t ] and Ant = S t − M t . Note that the extended An is no longer necessarily predictable, and currently we only have control of the total variation of An,ρn over Dn , i.e. 2n (ρ n ∧1) X

|Anj2−n − An( j−1)2−n | ≤ C.

j=1

Notice that, for t ∈ (( j − 1)2−n , j2−n ], Ant = S t − M tn n = S t − E[M j2 −n |F t ]

= S t − E[S j2−n − Anj2−n |F t ] = Anj2−n − (E[S j2−n |F t ] − S t ) From this and kSku ≤ 1 it follows that kAnt − Anj2−n k∞ ≤ 2, so kAn,ρn ku ≤ C + 2. How do we find the “limit” of sequence of stopping times (ρn )n≥1 ? The trick is to define Rn := 1[0,ρn ∧1] , a simple predictable process, and note that stopping at ρn is like integrating Rn , i.e. An,ρn = Rn · An and M n,ρn = Rn · M n . We have that 1 ≥ E[Rn1 ] = E[1ρn =∞ ] = 1 − P[ρn < ∞] ≥ 1 − ". Apply Komlós’ lemma to (Rn1 )n≥1 to obtain convex weights (µnn , . . . , µnNn ) such that Rn :=

∞ X

µni Ri1 → R1 a.s. as n → ∞

i=n

By the dominated convergence theorem, E[R1 ] ≥ 1 − ". Observe that Rn · S =

∞ X

µni (Ri · M i ) +

2

µni (Ri · Ai )

i=n

i=n

|

∞ X

{z

L norm ≤

p

C

}

|

{z

}

TV over Dn is ≤ C

Define αn := inf{t : Rnt ≤ 12 }. Each Rn is a left continuous, decreasing process. In particular, Rαn n ≥ 12 > 0, so we can divide by this quantity. We claim that P[αn < ∞] < ". Indeed, on the event [αn < ∞], R1n ≤ Rαn n + ≤ " ≥ E[1 − R1n ] ≥ E[(1 − Rαn n + )1αn 5 3 k=1   ∞ X 1 nk ≤ 3" + P |R1 − R1 | ≥ 15 k=1 ≤ 4" Therefore (Mn )n≥1 , (An )n≥1 , and α have the desired properties.

ƒ

Remark. One thing to take away from this is that if you need to take a “limit of stopping times” then one way to do it is to turn the stopping times into processes and take the limits of the processes. PROOF (OF (I) IMPLIES (II) IN THEOREM 3.1.7): We may assume the hypothesis of Proposition 3.2.7. Let " > 0 and take C, α, (Mn )n≥1 , and (An )n≥1 as in Proposition 3.2.7. Apply Komlós’ lemma to find convex weights (λnn , . . . , λnNn ) such that n,α

N ,α

λnn M1 + · · · + λnNn M1 n → M1 N ,α

n n λnn An,α → At t + · · · + λNn A t

3.2. Proofs of Theorems 3.1.7 and 3.1.8

51

for all t ∈ D, where the convergence is a.s. (and also in L 2 for M). For all n, 2n X

n,α

n,α

|A j2−n − A( j−1)2−n | ≤ C

j=1

so the total variation of A over D is bounded by C. Further, we have S α = M t + A t (where M t = E[M1 |F t ]). A is càdlàg on D, so define it on all of [0, 1] to make it càdlàg. M is an L 2 martingale so it has a càdlàg modification. Since P[α < ∞] < " and " > 0 was arbitrary, and the class of classical semimartingales is local, S must be a classical semimartingale. ƒ Remark. On the homework there is an example of a (not locally bounded) adapted càdlàg process that satisfies NFLVR and is not a classical semimartingale. PROOF (OF (I) IMPLIES (II) IN THEOREM 3.1.8): We no longer assume that S is locally bounded. The trick is to leverage the result for locally bounded processes by subtracting the “big”Pjumps from S. Assume without loss of generality that S0 = 0 and define J t := s≤t ∆Ss 1|∆Ss |≥1 . Then X := S − J is an adapted, càdlàg, locally bounded process. We will show that Theorem 3.1.8(i) for S implies NFLVR+LI for X , so that we may apply Theorem 3.1.7 to X . Then since J is of finite variation, this will then imply S is a classical semimartingale. Suppose H n ∈ S are such that kH n ku → 0 and k(H n · X )− ku → 0. We need to prove that (H n · X ) T → 0 in probability. First we will show that k(H n · S)− ku → 0. n n − sup (H n · S)− t ≤ sup (H · X ) t + sup |(H · J) t |

0≤t≤T

0≤t≤T

0≤t≤T

n ≤ sup (H n · X )− t + (kH k∞ · TV(J)) T 0≤t≤T

→ 0 by the assumptions on H n By (i), (H n · S) T → 0 in probability. Since (H n · J) → 0 in probability (as J is a semimartingale), we conclude that (H n · X ) T = (H n · S) T − (H n · J) T → 0 in probability. Therefore X satisfies NFLVR+LI. ƒ Remark. (i) S satisfies NFLVR if k(H n ·S)− ku → 0 implies (H n ·S) T → 0 in probability. This is equivalent to the statement that {(H · S) T : H ∈ S, H0 = 0, k(H · S)− ku ≤ 1} is bounded in probability. The latter is sometimes called no unbounded profit with bounded risk or NUPBR. (ii) It is important to note that this definition of NFLVR does not imply that there is no arbitrage. The usual definition of NFLVR in the literature is as follows. Let K := {(H · S) T : H ∈ S, H0 = 0} and let C := { f ∈ L ∞ : f ≤ g for some ∞ = {0} (the closure g ∈ K}. It is said that there is NFLVR if and only if C ∩ L+ ∞ is taken in the norm topology of L ). This is equivalent to NUPBR+NA, ∞ (where NA is no arbitrage, the condition that K ∩ L+ = {0}).

52

The Bichteler-Dellacherie Theorem and its connexions to arbitrage

3.3

A short proof of the Doob-Meyer theorem

3.3.1 Definition. An adapted process S is said to be of class D if the collection of random variables {Sτ : τ is a finite valued stopping time} is uniformly integrable.

càdlàg + predictable implies what? Note that *-martingale means càdlàg.

3.3.2 Theorem (Doob-Meyer for sub-martingales). Let S = (S t )0≤t≤T be a submartingale of class D. Then S can be written uniquely as S = S0 + M + A where M is a u.i. martingale with M0 = 0 and A is a càdlàg, increasing, predictable process with A0 = 0. We will require the following Komlós-type lemma to obtain a limit in the proof of the theorem. 3.3.3 Lemma. If ( f n )n≥1 is a u.i. sequence of random variables then there are g n ∈ conv( f n , f n+1 , . . . ) such that (g n )n≥1 converges in L 1 .

The details need to be filled in here.

PROOF: Define f nk = f n 1| f n |≤k for all n, k ∈ N. Then f nk ∈ L 2 for all n and k since they are bounded random variables. As noted before, there are convex weights (λnn , . . . , λnNn ) and f k ∈ L 2 such that λnn f nk + · · · + λnNn f Nkn → f k ∈ L 2 . Since ( f n )n≥1 is u.i., lim sup k(λnn f nk + · · · + λnNn f Nkn ) − (λnn f n + · · · + λnNn f Nn )k1 = 0

k→∞

n

which implies that (λnn f n + · · · + λnNn f Nn )n≥0 is Cauchy in L 1 .

ƒ

PROOF (OF THE DOOB -MEYER THEOREM FOR SUB -MARTINGALES): Assume without loss of generality that T = 1. Subtracting the u.i. martingale (E[S1 |F t ])0≤t≤1 from S we may further assume that S1 = 0 and S t ≤ 0 for all t. Define An0 = 0 and, for t ∈ Dn , Ant − Ant−2−n := E[S t − S t−2−n |F t−2−n ] M tn := S t − Ant . An = (Ant ) t∈Dn is predictable and it is increasing since S is a sub-martingale. Furthermore, M1n = −An1 since S1 = 0, and in fact M tn = − E[An1 |F t ] for all t ∈ Dn . Therefore for every Dn valued stopping time τ, Sτ = − E[An1 |Fτ ] + Anτ . We would like to show that (An1 )n≥1 is u.i. (and hence that (M1n )n≥1 is u.i.). For c > 0 define τn (c) := inf{( j − 1)2−n : Anj2−n > c} ∧ 1, which is a stopping time since An is predictable. Then Aτn (c) ≤ c and {An1 > c} = {τn (c) < 1}, so E[An1 1An1 >c ] = E[An1 1τn (c) n or |M t | > n} so that |A| Tn = |A| Tn − + |∆A Tn | ≤ n + |∆A Tn | and |∆A Tn | ≤ |∆M Tn | ≤ n + |M Tn |. Hence |A| Tn ≤ 2n + |M Tn | ∈ L 1 because M is u.i., so therefore |A| is locally integrable and Ap exists. Let B := A − Ap and N := M − B. Clearly B is of finite variation, and it is a local martingale by definition of Ap , and N is a local martingale because

57

58

The Bichteler-Dellacherie Theorem and its connexions to arbitrage it is a difference of local martingales. Define X t := ∆M t 1|∆M t | n} and note that (M ∗ ) Tn ≤ n ∨ |M Tn |, which is integrable.) If X is a special semimartingale then write X = M + A as in the definition and note that X ∗ ≤ M ∗ + A∗ ≤ M ∗ + |A|, so X t∗ is locally integrable for all t. (II) IMPLIES (III) For all s ≤ t, |∆X s | = |X s − X s− | ≤ 2X t∗ , which implies J ≤ 2X ∗ . (III) IMPLIES (IV) Homework. (IV) IMPLIES (V) Trivial. (V) IMPLIES (I) Write X = M + A where A is locally of integrable variation. Then by Corollary 3.5.4, Ap exists and X = (M + A − Ap ) + Ap . Note that the first summand is a local martingale and the second, Ap , is predictable and of finite variation. ƒ 3.6.5 Theorem. If X is a continuous semimartingale then X may be written X = M + A, where M is a local martingale and A is continuous and of finite variation. In particular, X is a special semimartingale. PROOF: By Theorem 3.6.4(iii) X is special, so write X = M + A in the unique way. Let’s see that A is continuous. We have A ∈ P , so pA = A and p(∆A) = ∆A. But ∆A = p(∆A) = p(∆X ) − p(∆M ) = p(∆X )

p

(∆M ) = 0 by Theorem 3.5.7

=0

X is continuous by assumption.

ƒ

3.6.6 Definition. Two local martingales M and N are orthogonal if M N is a local martingale. In this case we write M ⊥ N . A local martingale M is purely discontinuous if M0 = 0 and M ⊥ N for all continuous local martingales N . ˜ t := Nt − λt is purely discontinRemark. If N is a Poisson process of rate λ then N ˜ uous. However, N is not the sum of its jumps, i.e. not locally constant pathwise. 3.6.7 Theorem. Let H2 be the space of L 2 martingales. (i) For all M ∈ H2 , E[[M ]∞ ] < ∞ and 〈M 〉 exists.

59

60

The Bichteler-Dellacherie Theorem and its connexions to arbitrage (ii) (M , N )H2 := E[〈M , N 〉∞ ] + E[M0 N0 ] defines an inner product on H2 , and H2 is a Hilbert space with this inner product. (iii) M ⊥ N if and only if 〈M , N 〉 ≡ 0, which happens if and only if M T ⊥ N − N0 for all stopping times T . (iv) Take H2,c to be the collection of martingales in H2 with continuous paths and H2,d to be the collection of purely discontinuous martingales in H2 . Then H2,c is closed in H2 and (H2,c )⊥ = H2,d . In particular, H2 = H2,c ⊕ H2,d . 3.6.8 Theorem. Any local martingale M has a unique decomposition M = M0 + M c + M d , where M c is a continuous local martingale, M d is a purely discontinuous local martingale, and M0c = M0d = 0. PROOF (SKETCH): Uniqueness follows from the fact that a continuous local martingale orthogonal to itself must be constant. For existence, write M = M0 + M 0 + M 00 as in the fundamental theorem of local martingales, where M 0 has bounded jumps and M 00 is of finite variation. Then M 00 is purely discontinuous and M 0 is locally in H2 . But then M 0 = M c + N where M c is a continuous local martingale and N ∈ H2,d is purely discontinuous. Hence we may take M d = M 00 + N . ƒ Suppose that h : R → R and h(x) = x in a neighbourhood (−b, b) of zero, e.g. h(x) = x1|x| n or |A| t > n} and define Y = X Tn − . Yt∗ = (X Tn − )∗t ≤ (M Tn − )∗t + (ATn − )∗t ≤ 2n It follows that Y is a special semimartingale. We may assume, by further localization, that [Y ] is bounded. (Indeed, take R n = inf{t : [Y ] t > n} and note that [Y Rn ]∗ ≤ n+(∆YRn )∗ ≤ 3n.) Suppose that Y = N +B where N is a local martingale and B is predictable. We have seen that |B| is locally bounded (homework), so we may assume that B ∈ H2 by localizing once more. Again from the homework, E[[Y ]∞ ] = E[[N ]∞ ] + E[[B]∞ ], so E[[N ]∞ ] < ∞ and hence N ∈ H2 as well. ƒ 4.1.13 Definition. Let X ∈ H2 with canonical decomposition X = M + A. We say that H ∈ P is (H2 , X )-integrable if Z ∞   Z ∞ 2  Hs2 d[M ]s + E

E 0

|Hs |d|A|s

< ∞.

0

4.1.14 Theorem. If X ∈ H2 and H ∈ P is (H2 , X )-integrable then the sequence ((H1|H|≤n ) · X )n≥1 is a Cauchy sequence in H2 . PROOF: Let H n := H1|H|≤n . Clearly H n ∈ bP , so H n · X is well-defined. kH n · X − H m · X kH2 = dX (H n , H m ) = k((H n − H m )2 · [M ])1/2 k L 2 + k|(H n − H m )2 · A|∞ k L 2 → 0 by the dominated convergence theorem.

ƒ

4.1.15 Definition. If X ∈ H2 and H ∈ P is (H2 , X )-integrable then define H · X to be the limit in H2 of ((H1|H|≤n ) · X )n≥1 , which exists by Theorem 4.1.14. Yet otherwise said, H · X := H2 -lim(H1|H|≤n ) · X . n→∞

For any semimartingale X and H ∈ P , we say that H ·X exists if there is a sequence of stopping times (Tn )n≥1 that witnesses X is prelocally in H2 and such that H is (H2 , X Tn − )-integrable for all n. In this case we define H · X to be H · (X Tn − ) on [0, Tn ) and we say that H ∈ L(X ), the set of X -integrable processes.

4.1. Stochastic integrals with respect to predictable processes

69

Remark. (i) H · X is well-defined because H · (X Tm − ) = (H · X Tn − ) Tm − for n > m (cf. Theorem 4.1.9). By similar reasoning, H · X does not depend on the prelocalizing sequence chosen. (ii) If H ∈ (bP )loc then H ∈ L(X ) for any semimartingale X . (iii) Notice that if H ∈ L(X ) then H · X is a semimartingale. (Prove as an exercise that if C is the collection of semimartingales then C = Cloc = Cpreloc .) (iv) Warning: We have not defined H · X = H · M + H · A when X = M + A. If it Is it true? is true we will need to prove it as a theorem. 4.1.16 Theorem. Let X and Y be semimartingales. (i) L(X ) is a vector space and Λ : L(X ) → {special semimartingales} : H 7→ H ·X is a linear map. (ii) ∆(H · X ) = H∆X for H ∈ L(X ). (iii) (H · X ) T = (H1[0,T ] ) · X = H · (X T ) for H ∈ L(X ) and stopping times T . Rt (iv) If X is of finite variation and 0 |Hs |d|X |s < ∞ then H · X coincides with the Lebesgue-Stieltjes integral for all H ∈ L(X ). (v) Let H, K ∈ P . If K ∈ L(X ) then H ∈ L(K · X ) if and only if H K ∈ L(X ), and in this case (H K) · X = H · (K · X ). (vi) [H · X , K · Y ] = (H K) · [X , Y ] for H ∈ L(X ) and K ∈ L(Y ). (vii) X ∈ H2 if and only if supkHku ≤1 k(H · X )∗∞ k L 2 < ∞. In this case, 2 2 kX kH2 ≤ sup k(H · X )∗∞ k L 2 + 2k[X ]1/2 ∞ k L ≤ 5kX kH

kHku ≤1

and kX kH2 ≤ 3 sup k(H · X )∗∞ k L 2 ≤ 9kX kH2 . kHku ≤1

PROOF (OF (VII)): To be added.

ƒ

4.1.17 Exercise. Suppose that H ∈ P and X is of finite variation and also that Rt |Hs |d|X |s < ∞ a.s. for all t (this is the Lebesgue-Stieltjes integral). Is it true 0 that H ∈ L(X )? Note that this is not the same statement as Theorem 4.1.16(iv). 4.1.18 Theorem. Suppose that X is a semimartingale and H ∈ L(X ) under P. If Q  P then H ∈ L(X ) under Q and H ·P X = H ·Q X . PROOF (SKETCH): Let (Tn )n≥1 witness that H ∈ L(X ) under P, so that Tn ↑ ∞ P-a.s. dQ and H is (H2 , X Tn − )-integrable for all n. Let Z t := EP [ d P |F t ], Sn := inf{t : |Z t | > n} and R n := Tn ∧ Sn . Since |Z Rn − | ≤ n one can show that 1/2 kX kH2 (Q) ≤ sup k(H · X )∗∞ k L 2 (Q) + 2 EQ [[X Rn − ]∞ ] kHku ≤1

= sup EP [ZRn ((H · X )∗∞ )2 ]1/2 + 2 EP [ZRn [X Rn − ]1/2 ∞ ] kHku ≤1

70

General stochastic integration p ≤ 5 nkX kH2 (P) (The last inequality is an exercise. There was a hint but I missed it.) This implies that X ∈ H2 (Q). The proof that H is (H2 , X Rn − )-integrable under Q is similar. ƒ 4.1.19 Theorem. Let M be a local martingale and H ∈ P be locally bounded. Then H ∈ L(M ) and H · M is a local martingale. PROOF: We need the following lemma. 4.1.20 Lemma. Suppose that X = M + A, where M ∈ H2 and A is of integrable variation. If H ∈ P then there is a sequence (H n )n≥1 ∈ bL such that H n ·M → H ·M in H2 and E[|H n − H| · |A|∞ ] → 0. ˜X (H, J) := E[(H − J)2 · [M ]∞ ]1/2 + E[|H − J| · |A|∞ ] for all H, J ∈ P . PROOF: Let δ ˜X (H, J) < "}. One can Let H = {H ∈ P : for all " > 0 there is J ∈ bL such that δ see that H is a monotone vector space that contains bL. It follows that bP ⊆ H.ƒ By localization we may assume that H ∈ bP and M = M 0 + A, where M 0 ∈ H2 and A is of integrable variation (this follows from the fundamental theorem of local martingales and a problem from the exam). By the lemma there is (H n )n≥1 ∈ bL ˜X (H n , H) → 0. We know that H n · M = H n · M 0 + H n · A is a martingale such that δ for each n. Finally, H n · M 0 → H · M 0 in H2 and E[|H n − H| · |A|∞ ] → 0 imply that H · M 0 and H · A are martingales. ƒ 4.1.21 Theorem (Ansel-Stricker). Let M be a local martingale and H ∈ L(M ). Then H · M is a local martingale if and only if there stopping times Tn ↑ ∞ and T (θn )n≥1 nonpositive integrable random variables such that (H∆M ) t n ≥ θn for all t and all n. Remark. If H is locally bounded and Tn is such that M T∗n is integrable and H Tn is bounded by n then one can take θn := −2nM T∗n in the theorem to show that H · M is a local martingale. PROOF: If H · M is a local martingale then there is are Tn ↑ ∞ such that (H · M )∗Tn is integrable, and one may take θn := −2(H · M )∗Tn to see that the condition is necessary. Proof of sufficiency may be found in the textbook. ƒ 4.1.22 Theorem. Let X = M + A be a special semimartingale and H ∈ L(X ). If H · X is a special semimartingale then H ∈ L(M )∩ L(A), H · M is a local martingale, and H · A is predictable and of finite variation. In this case H · X = H · M + H · A is the canonical decomposition of H · X . 4.1.23 Theorem (Dominated convergence). Let X be a semimartingale, G ∈ L(X ), H m ∈ P such that |H m | ≤ G, and assume that H n → H pointwise. Then H m , H ∈ L(X ) for all m and H n · X → H · X u.c.p.

4.1. Stochastic integrals with respect to predictable processes PROOF: Suppose that Tn ↑ ∞ are such that G is (H2 , X Tn − )-integrable for all n. If X Tn − = M + A is the canonical decomposition then E[(H m )2 · [M ]∞ ]1/2 + E[|H m | · |A|2∞ ]1/2 ≤ E[G 2 · [M ]∞ ]1/2 + E[|G| · |A|2∞ ]1/2 < ∞ Therefore H m are all (H2 , X Tn − )-integrable and hence so is H by the dominated convergence theorem. For t 0 fixed k sup |(H m − H) · X Tn − |k2L 2 ≤ 8k(H m − H) · X Tn − k2H2 → 0 t≤t 0

by the dominated convergence theorem. Whence H m · X Tn − → H · X Tn − uniformly on [0, t 0 ] in probability. Given " > 0 let n and m be such that P[Tn < t 0 ] < " and P[((H m − H) · X Tn − )∗t 0 > δ] < ". Whence P[((H m − H) · X )∗t 0 > δ] < 2". ƒ 4.1.24 Theorem. Let X be a semimartingale. There exists Q ∼ P such that X t ∈ dQ H2 (Q) for all t and d P is bounded. 4.1.25 Example (Emery). Suppose that T ∼ Exponential(1), P[U = 1] = P[U = −1] = 12 , and U and T are independent. Let X := U1 t≥T and let the filtration be the natural filtration of X , so that X is an H2 martingale of finite variation and [X ] t = 1 t≥T . Let H t := 1t 1 t>0 , which is predictable because it is left continuous. But H is not (H2 , X )-integrable.   1 2 =∞ E[H · [X ]∞ ] = E T2 Is H ∈ L(X )? We do know that H · X exists as a pathwise Lebesgue-Stieltjes integral, namely (H · X ) t = UT 1 t≥T . We have E[|H · X | t ] = ∞ for all t (because the problem is around zero and not at ∞). It follows in particular that H · X is not a martingale. Claim. If S is a finite stopping time with P[S > 0] > 0 then E[|(H · X )S |] = ∞. Indeed, we have |H · X )S | = T1 1S≥T . The hard part is to show that there is " > 0 such that “{T ≤ "} implies {S ≥ T }”. If this is the case then E[|(H · X )S |] = E[ T1 1 T ≤" ] = ∞. (Perhaps use {T ≤ "} = {T ≤ ", S < T } ∪ {T ≤ ", S ≥ T } and 1S