Crossings Problems in Random Processes Theory and Their Applications in Aviation 1527527956, 9781527527959

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Crossings Problems in Random Processes Theory and Their Applications in Aviation



Crossings Problems in Random Processes Theory and Their Applications in Aviation By

Sergei L. Semakov

Crossings Problems in Random Processes Theory and Their Applications in Aviation By Sergei L. Semakov This book first published 2019 Cambridge Scholars Publishing Lady Stephenson Library, Newcastle upon Tyne, NE6 2PA, UK British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Copyright © 2019 by Sergei L. Semakov All rights for this book reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. ISBN (10): 1-5275-2795-6 ISBN (13): 978-1-5275-2795-9

To My Mother

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Main Classes and Characteristics of Random Processes 1.1 Intuitive Prerequisites of the Theory . . . . . . . . . . . . . . . . . . . . . 4 1.2 Fundamental Concepts and Results Underlying the Construction of the Mathematical Theory . . . . . . . . . . . . . . . 6 1.3 Mathematical Expectation, Variance, and Correlation Function of a Random Process . . . . . . . . . . . . . . . . . . . . . . . . . .10 1.4 Types of Convergence in a Probabilistic Space and Characteristics of Smoothness of a Random Process . . . . 14 1.5 Stationary Random Processes . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.6 Gaussian Random Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.7 Markov Random Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.8 Continuity and Not Differentiability of the Wiener Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 1.9 Integration of Random Processes . . . . . . . . . . . . . . . . . . . . . . . 47 1.10 Spectral Density of a Random Process . . . . . . . . . . . . . . . . . 56 Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .66 Crossings Problems 2.1 Problem Formulation and Examples from Application Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.2 Average Number of Crossings . . . . . . . . . . . . . . . . . . . . . . . . . . .69 2.3 Finiteness of a Number of Crossings and Absence of Touchings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 2.4 Characteristics of Extreme Values of a Random Process 76 2.5 Problem Formulation about the First Achievement of Boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 2.6 The First Achievement of Boundaries by the Markov Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 2.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 2.8 The First Achievement of Boundaries by Non-Markovian Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 2.8.1 Probability Estimates in One-Dimensional Case . . 102 2.8.2 Limit of Sequence of Estimates . . . . . . . . . . . . . . . . . . .109

viii

Contents

2.8.3 Application to Gaussian Processes . . . . . . . . . . . . . . . 112 2.8.4 Probability Estimates in Multidimensional Case . . 121 2.9 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Applications in Aviation when Landing Safety Study 3.1 Problem Formulation of Estimating the Probability of Safe Landing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 3.2 Scheme of Estimation Method . . . . . . . . . . . . . . . . . . . . . . . . . 130 3.3 Implementation of the Method when Landing on a Ship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 3.3.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 3.3.2 Coordinate Systems and the Nominal Landing Trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 3.3.3 Linearization of Equations of Motion . . . . . . . . . . . . . 138 3.3.4 Perturbation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 3.3.5 Control Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 3.3.6 Calculation System of Equations and Calculation Formulas for Estimating the Probability . . . . . . . . . 150 3.3.7 Analysis of Calculations Results and Estimate of Efficiency of the Method . . . . . . . . . . . . . . . . . . . . . . 156 3.4 Implementation of the Method for Overland Landing Variant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .176 3.4.1 Equations of Motion, Perturbation Model, and Control Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 3.4.2 Calculation System of Equations and Calculation Formulas for Estimating the Probability . . . . . . . . . 179 3.4.3 Results of Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 3.4.4 Using the Method for Solving Problems of Civil Aviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .184 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

Introduction

This monograph presents the author’s results that have only been previously partially published in the form of journal articles.1 These results relate to the special section on the theory of random processes and their application in solving aviation problems.2 This section is known in literature as outliers of random processes, or crossings problems.3 The book consists of three chapters. In the first chapter, the basic concepts in the theory of probability and random processes are stated at an elementary level in order to prepare the reader for the second and the third chapters. Specialists in the theory of probability and random processes can skip the first chapter. In this chapter, we have introduced the concepts of probabilistic space, random variables, random processes, correlation function, and the spectral density of random process, and other important concepts. We have defined types of convergence in a probabilistic space and we have formulated the corresponding characteristics in the smoothness of random processes. Stationary, Gaussian, and Markov processes are considered. Questions of the integration of random processes are discussed. 1 See, for example, the following articles: (a) S.L. Semakov, First Arrival of a Stochastic Process at the Boundary, Autom. Remote Control, 1988, vol. 49, no. 6, pp. 757-764; (b) S.L. Semakov, The Probability of the First Hiting of a Level by a Component of a Multidimensional Process on a Prescribed Interval under Restrictions of the Remaining Components, Theor. Prob. App., 1989, vol. 34, no. 2, pp. 357-361; (c) S.L. Semakov, The Application of the Known Solution of a Problem of Attaining the Boundaries by Non-Markovian Process to Estimation of Probability of Safe Airplane Landing, J. Comput. Syst. Sci., 1996. vol. 35, no. 2, pp. 302-308; (d) S.L. Semakov, Estimating the Probability That a Multidimensional Random Process Reaches the Boundary of Region, Autom. Remote Control, 2015, vol. 76, no. 4, pp. 613-626. 2 The formulations of the main results are listed at the end of the book in the ”Conclusion”. 3 See, for example, the book: H. Cramer and M.R. Leadbetter, Stationary and Related Stochastic Processes, New York-London-Sydney: Wiley, 1967.

2

Introduction

The behavior of any real system is a process which is, to a greater or lesser degree, probabilistic. As a rule, it is impossible to specify exactly what external influences and internal mechanisms in the interaction of the system components will be decisive in the future. As a consequence, we cannot accurately predict the behavior of the system. We can only talk about a probability that the system will come to a particular state in the future. If we pose the problem of the probabilistic description of all possible future states of the system, then this problem will be very difficult. Fortunately, for research purposes, it is often enough to get answers to simpler questions; for example, the question “How long will on average the system operate in a given mode?” or the question “What is the probability that the process of functioning of the system will come out of a given mode to a specific point in time?” Problems of this type are called problems about outliers of random processes, or problems about the crossing of a level. In the second chapter, we have stated some of the most important fundamental results, which are related to crossings problems. Known problems about attaining boundaries in a random process will be discussed in detail. We have stated the well-known solution of diffusion Markov processes for these problems and presented the author’s results for arbitrary continuous processes. In the third chapter, we have applied the mathematical results of the second chapter to an investigation on the safety of airplane landings. We have not considered simple examples and model problems, whose purpose is only to illustrate the theory, but real concrete problems posed by practice. We have shown how the results of the second chapter can be used to calculate the probability of an airplane’s safe landing. A safe landing is defined as an event when the airplane touches down the landing surface on a given segment for the first time, and, at this moment, the coordinates of the airplane (elevation angle, banking angle, vertical velocity and so on) remain inside admissible ranges. These restrictions are established by flight standards and their violation leads to an accident. The scheme for this calculation is described and the implementation of this scheme is given for overland and ship-landing options. The results of the numerical calculations are discussed in detail. The first chapter, and part of the second chapter are used to carrying out of lessons with students of the Moscow Institute of Physics and Technology and the Moscow Aviation Institute. The author’s results in the second chapter will be of interest for mathematicians who

Introduction

3

study crossings problems. The third chapter is addressed to specialists in the field of aviation, as well as engineers and scientists who are interested in the application of random processes theory and use its methods. Some results of the second and third chapters were presented in the author’s report at the 57th IEEE Conference on Decision and Control (Miami Beach, FL, USA, December 17-19, 2018).1 This monograph was recommended for publication by the Division of Mathematical Sciences at the Russian Academy of Sciences.

1 S.L. Semakov and I.S. Semakov, Estimating the Probability That a Random Process First Reaches the Boundary of a Region on a Given Time Interval, Proceedings of the 57th IEEE Conference on Decision and Control (CDC), Miami Beach, USA, 2018, pp. 256-261.

Chapter 1 Main Classes and Characteristics of Random Processes

1.1 Intuitive Prerequisites of the Theory Let us consider the practical situation connected with an experiment E when each outcome of E is defined by a casual mechanism and the influence cannot be predicted in advance. In any event we are interested in A, which is connected with E; this is in the sense that the event A can either occur or not as a result of E. In many life situations there is a necessity to predict the possible degree of realization of the event A as a result of carrying out E. This degree of possibility is characterized by the number P {A}. This number is called the probability of A and is defined by the frequency of occurrence of A in numerous repetitions of the experiment E or, more precisely, P {A} = lim

N →∞

NA , N

where N is the number of experiments carried out, NA = NA (N ) is the number of experiments within which the event A was observed. We will put aside the question about the existence of the above limit and the possibility of its practical calculation. As a rule, certain quantitative characteristics can be connected with the experiment E. These characteristics accept numerical values that change in a random way from one experiment to another. These characteristics are called random values. For example, if the experiment E consists in n tossings of a coin, the number of dropped-out coats of arms, the maximum number of coats of arms which dropped out in a row, the number of coats of arms dropped out at even throwing, and so on will be random values. It is convenient to describe the problematic behavior of a random value X by means of the function F (x) = P {X< x},

Main Classes and Characteristics of Random Processes

5

which is called a distribution function of X and is the probability of an event concluded in braces: i.e., the event consisting of the experiment value of X less the real number x. It turns out that knowledge of F (x) provides a chance to receive the probability that a random value of X belongs to the preset subset of real numbers. This problem often arises in the solution to various practical tasks. It is possible to consider a random vector value (X1 , . . . , Xn ), where every Xi has a scalar random value. It is important from the practical point of view to be able to determine the probability that a casual point (X1 , . . . , Xn ) in the n-dimensional space Rn as a result of E will get to the preset subset from Rn . For the big class of such subsets in the case of a scalar random value, this probability is unambiguously defined by the joint distribution function of the random values X1 , . . . , Xn : F (x1 , . . . , xn ) = P {X1 < x1 , . . . , Xn < xn }, which is the probability that each of Xi appeared less than the corresponding value of xi as a result of E. A vector random value (X1 , . . . , Xn ) can be considered and defined as a family of random values {Xt }, where t runs some set of indexes T ; in this case T = {1, . . . , n}. If T represents a set of integers, then the family {Xt } is called a random process with a discrete parameter. If T is an interval of a real axis, then a family of random values {Xt } is called a random process with a continuous parameter. In this case, an outcome of the experiment is a set of values Xt , where, for example, t ∈ [a, b]; in other words, an outcome of this experiment gives a function of the variable t, where t ∈ [a, b]. The parameter t is called the argument of a random process. Most often the argument of a random process is time (from here and the chosen designation of t); however, this is optional. We will consider, for example, a process of change in the flying height H of a trip plane in its flight from point A to point B. This is dependent on the flying range l; l ∈ [0, L], where L is a distance between points A and B. Here the experiment E is the flight of the plane from A to B. A result of E is the concrete (corresponding to this concrete flight) height change function of the variable l. It is clear that this type of function can change from flight to flight and depends on many random factors: weather conditions, the technical condition of the plane, the health of the pilot, indications of the land services of air traffic control, and other factors. At every fixed value of the l value, the Hl from experiment to experiment (from flight to

6

Chapter 1

flight) will change in an unpredicatable fashion and can, therefore, be considered a random value, and the family {Hl } can be considered as a random process. Therefore, in a definite sense, the random process combines the features of a random value and a function; at the fixed argument it turns into a random value, and at each implementation of an experiment it turns out to be a determined (not random) function of this argument. As a rule, we will further designate the random process as X(t); specifying that if it is necessary then the possible range of change in the argument t, and the physical sense of magnitudes in X and t can be various.

1.2 Fundamental Concepts and Results Underlying the Construction of the Mathematical Theory 1. We consider the experiment E without being interested in its concrete type. An event A is connected with E and is called observable if we can say unambiguously that event A has occurred or that event A has not occurred as a result of E. An event is called a persistent event and denoted by Ω if this event occurs every time when carrying out experiment E. An event is called an impossible event and denoted by 0 if this event never occurs when carrying out experiment E. We consider that events, Ω and 0, are observable. Let A and B be observable events that are connected with carrying out experiment E. We define: 1) the additional event concerning A; this event is denoted by A¯ and consists in the fact that A does not occur; 2) the sum, or crowd of events A and B; this event is denoted by A + B or A ∪ B and consists in the fact that at least one of events A or B occurs; 3) the product, or intersection of events A and B; this event is denoted by AB or A ∩ B and consists in the fact that both events A and B occur. We consider that if events A and B are observable, then events ¯ B, ¯ A+B, and AB are also observable. The family of all observable A, events connected with E forms a field of events F0 : i.e., the class of ¯ B, ¯ A+B, AB ∈ F0 . events such that Ω ∈ F0 and if A, B ∈ F0 , then A,

Main Classes and Characteristics of Random Processes

7

It can easily be checked out that events of field F0 satisfy the following relations: A + A = AA = A, A + B = B + A, (A + B) + C = A + (B + C), AB = BA, (AB)C = A(BC), A(B + C) = AB + AC, A + A¯ = Ω, AA¯ = 0, A + Ω = Ω, AΩ = A, A + 0 = A, A0 = 0. Let A , B  , . . . be sets of points ω of any space Ω . The sets + B  , A B  are defined in the elementary theory of sets. These definitions (which are assumed known for the reader) show that all ratios given above for events remain fair when A, B, . . . are sets. A family of sets from Ω is called a field of sets in Ω if this family includes the entire space Ω and is closed relative to operations A , A + B  , and A B  . It turns out that the following result takes place: for any field F0 of events satisfying the listed above ratios it is possible to find some space Ω of points ω and field F0 of ω-sets. This means that a biunique correspondence exists between events A taking from F0 and sets A taking from F0 . If the event A corresponds to set A and event B corresponds to set B  , then event A¯ corresponds to set A , event A + B corresponds to set A + B  , and event AB corresponds to set A B  . Points ω correspond to some elementary events which can or cannot be observed separately (i.e., can enter or cannot enter F0 separately); a persistent event corresponds to the whole space Ω , and an impossible event corresponds to the empty set ∅. The formulated result allows us to use the technique of the theory of sets for an analysis of fields of events. Further strokes are omitted and the same designations F0 , A, B, . . . are used for events and ωsets. A set from F0 is called an observable event, or ω-set, and Ω is considered as a persistent event or as the whole space. We now assume that the number P0 {A} is placed according to each event A from F0 . This number is called the probability of event A. We consider that P0 {A} is a function, so that P0 {A} is determined for all A ∈ F0 and the following conditions are satisfied: A , B  , A

1) 0 ≤ P0 {A} ≤ 1; 2) P0 {Ω} = 1; 3) if A = A1 + . . . + An , where Ai ∈ F0 , i = 1, . . . , n, Aj Ak = ∅ for j = k, then P0 {A} = P0 {A1 } + . . . + P0 {An }.

8

Chapter 1

The statement about the existence of the function P0 {A} with these properties is accepted as an axiom, i.e., as a statement which does not demand proof. Let A1 , A2 , . . . be any countable sequence of sets from F0 . Sets of a type A1 + A2 + . . . and A1 A2 . . . (i.e., sets that are denumerable number of crowds and intersections of sets from F0 ) may not be elements of field F0 . A field of ω-sets is called a borelevsky field or σ-field if this field includes all countable (finite or infinite) crowds and intersections of elements of this field. It turns out that for any field F0 of observable events there is a minimum σ-field containing F0 . This minimum σ-field is denoted by F. The field F, as well as the field F0 , may also contain unobservable events. One of the main results is that there is only a unique extension of the function P0 {A}, which is defined for all sets A ∈ F0 , to the function P {A}, which is defined for all sets A ∈ F: if A ∈ F0 , then P {A} = P0 {A}. The function P {A} possesses all properties of the function P0 {A}. In particular, 0 ≤ P {A} ≤ 1 for any A ∈ F and if A = A1 + A2 +. . . , where all Ai ∈ F and Ai Aj = ∅ for i = j, then P {A} = P {A1 } + P {A2 } + . . . . The space Ω of points ω, the σ-field F of sets from Ω, and the probability P {A} defined for sets A from F form the probabilistic space (Ω, F, P ). Sets from F are called measurable. 2. Let X = X(ω) be a function defined for all ω. If for any x ∈ (−∞, ∞) the set {ω : X(ω) < x} is an element of σ-field F, then we say that function X is measurable and we call this function a random value. If X = X(ω) is a random value, then the probability F (x) = P {ω : X(ω) < x} (or briefly F (x) = P {X < x}) represents the nondecreasing function of the variable x. It is easy to prove that this function is continuous at the left, lim F (x) = 0, and lim F (x) = 1. The function x→−∞

x→∞

F (x) is called a distribution function of the random value X. If function f (x) exists such that x F (x) =

f (t)dt, −∞

then f (x) is called a distribution density of random value X. For a system of random values X1 , . . . , Xn , a function of joint distribution

Main Classes and Characteristics of Random Processes

9

of these random values is defined as F (x1 , . . . , xn ) = P {ω : X1 (ω) < x1 , . . . , Xn (ω) < xn }. The density of joint distribution of these random values (if this density exists) is defined as a function f (x1 , . . . , xn ) such that x1 F (x1 , . . . , xn ) =

xn ...

−∞

f (t1 , . . . , tn )dt1 . . . , dtn .

−∞

Let (Ω, F, P ) be some probabilistic space and T be a set of values from the parameter t. The function X(t, ω), where t ∈ T and ω ∈ Ω, is called a random process on a probabilistic space (Ω, F, P ) if the following condition is satisfied: for each fixed t = t˜ from T , the X(t˜, ω) is some random value on this probabilistic space (Ω, F, P ). The records X(t) or Xt instead of X(t, ω) are used for a brief designation of this random process. Let X(t) be a random process. For each fixed t = t1 , the random value X(t1 ) = X(t1 , ω) has the distribution function F (x1 , t1 ) = P {ω : X(t1 , ω) < x1 }. Let t1 , . . . , tn be any finite set of values t. The random values X(t1 ), . . . , X(tn ) have the function of joint distribution F (x1 , . . . , xn ; t1 , . . . , tn ) = P {ω : X(t1 , ω) < x1 , . . . , X(tn , ω) < xn }. The family of all such joint distributions for n = 1, 2, . . . and for all possible values tj ∈ T , where j = 1, 2, . . . , n, is called a family of finitedimensional distributions of the process X(t). As it will become clear from further sections, many properties of random processes are defined by the properties of their finite-dimensional distributions. Two random processes X(t), t ∈ T , and Y (t), t ∈ T , are called equivalent if for each fixed t ∈ T the random values X(t) and Y (t) are equivalent random values, i.e., P {ω : X(t, ω) = Y (t, ω)} = 1. It is easy to prove that families of finite-dimensional distributions for equivalent processes coincide. From the definition of the random process, it follows that X(t, ω) becomes a function of variable t ∈ T for every fixed elementary event ω. In other words, a nonrandom function of the variable t corresponds to each possible outcome of the experiment. Each such function x(t) is called a realization or a trajectory or a sample function of process X(t).

10

Chapter 1

1.3 Mathematical Expectation, Variance, and Correlation Function of a Random Process Let X = X(ω) be a discrete random value defined on a probabilistic space (Ω, F, P ). Possible values of X are given by countable (finite or infinite) numerical sequence x1 , x2 , . . . . We will assume that a set Ai ∈ Fis formed by those and only those ω for the X(ω) = xi . If the xi P {Ai } converges absolutely, then its sum is called a mathseries i

ematical expectation of random value X and is denoted by M {X}:  M {X} = xi P {Ai }. i

If X is a continuous random value having a distribution density f (x), then by definition ∞ xf (x)dx M {X} = −∞

given that this integral converges absolutely. In the case of divergence of this integral (or in the case of divergence of the above series if X is a discrete random value), we say that the corresponding random value has no mathematical expectation. By definition, the mathematical expectation of random process X(t) is a nonrandom function m(t) = M {X(t)}. The right part of this equality is a mathematical expectation of the random value X(t). We interpret this random value as the random process cross-section corresponding to the argument t. By definition, the variance of random process X(t) is a nonrandom function D(t) = M {(X(t) − m(t))2 }. For each fixed t, the number D(t) gives the variance of random value X(t). Random values X1 , . . . , Xn are called independent if F (x1 , . . . , xn ) = F1 (x1 ) . . . Fn (xn ) or (if f (x1 , . . . , xn ) exists) f (x1 , . . . , xn ) = f1 (x1 ) . . . fn (xn ),

Main Classes and Characteristics of Random Processes

11

where Fi (x) is a distribution function of random value Xi , i = 1, . . . , n, and fi (x) is a distribution density of random value Xi , i = 1, . . . , n. To characterize a degree of dependence between various process crosssections, i.e., between random values X(t1 ) and X(t2 ), we define a nonrandom function of two variables as K(t1 , t2 ) = M {(X(t1 ) − m(t1 ))(X(t2 ) − m(t2 ))}. This function is called a correlation function of the random process X(t). If we introduce a centered random process as ◦

X (t) = X(t) − m(t), then we obtain

◦

◦

K(t1 , t2 ) = M {X (t1 ) X (t2 )}. Notice that K(t, t) = D(t). Suppose D(t) = 0 ∀t ∈ T . Then we can introduce the function k(t1 , t2 ) =

K(t1 , t2 ) , σ(t1 )σ(t2 )

  where σ(t1 ) = D(t1 ) and σ(t2 ) = D(t2 ) are so-called mean square deviations. The function k(t1 , t2 ) is called a normalized correlation function of the random process X(t). A convenience from the introduction of k(t1 , t2 ) consists of the fact that k(t1 , t2 ) is a nondimensional quantity and |k(t1 , t2 )| ≤ 1 for any values t1 and t2 . It is easy to prove that k(t1 , t2 ) = 0 if X(t1 ) and X(t2 ) are independent random values. But an independence by the process of cross-sections X(t1 ) and X(t2 ) does not follow from the condition k(t1 , t2 ) = 0. Now we will consider some examples. 1. Let X(t) be a random process of following type X(t) = U cos ωt + V sin ωt, where ω > 0 is a constant number, and U and V are two independent random values with mathematical expectations M {U } = M {V } = 0 and variances D{U } = D{V } = D. Then for each t m(t) = cos ωtM {U } + sin ωtM {V } = 0, so K(t1 , t2 ) = M {(U cos ωt1 + V sin ωt1 )(U cos ωt2 + V sin ωt2 )} =

12

Chapter 1

= cos ωt1 cos ωt2 M {U 2 } + sin ωt1 sin ωt2 M {V 2 } + + (cos ωt1 sin ωt2 + sin ωt1 cos ωt2 )M {U V }. Using the independence of U from V , we have M {U V } = M {U } · M {V } = 0. Therefore, K(t1 , t2 ) = D cos ωt1 cos ωt2 + D sin ωt1 sin ωt2 = D cos ω(t1 − t2 ). 2. We will now generalize the previous example. We will consider the random process X(t) = U cos Ωt + V sin Ωt, where U, V , and Ω are independent random values, M {U }=M {V }=0, M {U 2 } = M {V 2 } = D, and a random value Ω is characterized by a distribution density f (ω). We find m(t) = M {U cos Ωt} + M {V sin Ωt}. Taking into account an independence of U from Ω and the independence of V from Ω, we obtain m(t) = M {U }M {cos Ωt} + M {V }M {sin Ωt} = 0. To simplify the calculation of K(t1 , t2 ) we note that from the previous example, it follows that the conditional correlation function K(t1 , t2 |Ω = ω) is equal to K(t1 , t2 |Ω = ω) = M {X(t1 )X(t2 )|Ω = ω} = D cos ω(t1 − t2 ). To find the correlation function K(t1 , t2 ) it is necessary to multiply this expression using an element of probability f (ω)dω and to integrate all possible values of frequency ω.1 Thus, ∞ f (ω) cos ω(t1 − t2 )dω.

K(t1 , t2 ) = D 0

1 The simplification for the calculation of K(t , t ), of course, needs a justifica1 2 tion. Lowering the level of proof, we note that this simplification follows on from the mathematical expectation properties and from the total probability formula.

Main Classes and Characteristics of Random Processes

i.e.,

13

For example, if the random value Ω has the Cauchy distribution,  2λ 1 if ω ≥ 0, π λ2 +ω 2 f (ω) = 0 if ω < 0,

where λ is some positive number, then ∞ cos ω(t1 − t2 ) 2Dλ K(t1 , t2 ) = dω = D exp{−λ|t1 − t2 |}. π λ2 + ω 2 0

3. We will now consider one more example. Let λ be a constant positive number, and t1 , t2 , . . . be a random sequence of points on axis t such that 1) the probability Pn (T ) that a time interval duration T contains exactly n points is equal to (λT )n exp{−λT } Pn (T ) = n! and does not depend on the provision of this interval on a timebase; 2) if the intervals do not intersect, the corresponding numbers of points are independent random values. In this case, let us say that this sequence of points t1 , t2 , . . . forms the Poisson stream with a constant density λ. We suppose that a random process X(t) is defined by its realizations as follows: ⎧ 0 at −∞ < t < t1 , ⎪ ⎪ ⎪ ⎪ at t1 ≤ t < t2 , ⎨ x1 at t2 ≤ t < t3 , x2 x(t) = ⎪ ⎪ at t3 ≤ t < t4 , x ⎪ 3 ⎪ ⎩ and so on, where numbers x1 , x2 , x3 , . . . are realizations of independent random values X1 , X2 , X3 , . . . with zero mathematical expectations and with identical variances D. It is clear that M {X(t)} ≡ 0 because M {X1 } = M {X2 } = = M {X3 } = . . . = 0. We find the correlation function K(t, t ): K(t, t ) = M {X(t)X(t )} = = M {X(t)X(t )|A}P {A} + M {X(t)X(t )|B}P {B},

14

Chapter 1

where the event A means that the interval (t, t ) contains at least one of the points t1 , t2 , t3 , . . . , and the event B means that the interval ¯ Taking into (t, t ) does not contain points t1 , t2 , t3 , . . . , i.e., B = A. account the independence of random values X1 , X2 , X3 , . . . , we have M {Xi Xj } = M {Xi }M {Xj } = 0 for i = j, so M {X(t)X(t )|A} = 0. If there was the event B, the value X(t) coincides with the value X(t ) and, therefore, M {X(t)X(t )|B} = M {X 2 (t)} = D. Since the probability P {B} is Pn (T ) when n = 0 and T = |t − t |, we obtain K(t, t ) = D exp{−λ|t − t |}. Note that the correlation function from example 3 is equal to the correlation function from example 2. At the same time, the realizations of random processes have a different nature in these examples: realizations are sinusoids in example 2, and realizations are step functions in example 3. Therefore, the same correlation function can correspond to random processes having a various nature of realizations.

1.4 Types of Convergence in a Probabilistic Space and Characteristics of Smoothness in a Random Process Several types of convergence are considered in the theory of random processes and respectively various definitions are introduced for a continuity and a differentiability of random processes. Let X1 , X2 , . . . be a sequence of random values defined on some probabilistic space; let X be one more random value defined on this space. We define the following three main types of convergence of sequence Xn , n = 1, 2, . . . , to X as n → ∞. 1. Convergence with probability 1 (other names are “almost everywhere convergence” and “almost sure convergence”). A sequence of random values Xn , n = 1, 2, . . . , converges to a random value X with probability 1 if P {ω : lim Xn (ω) = X(ω)} = 1. n→∞

15

Main Classes and Characteristics of Random Processes

This requirement is shorter when it is written down as follows: a.e. a.s. P {Xn → X} = 1, or Xn −→ X, or Xn −→ X, where reductions a.e. or a.s. mean, respectively, almost everywhere or almost sure. 2. Convergence in the mean-square. A sequence of random values Xn , n = 1, 2, . . . , converges to a random value X in the mean-square if lim M {(Xn − X)2 } = 0. n→∞

m.s.

Write Xn −→ X. 3. Convergence on probability. A sequence of random values Xn , n = 1, 2, . . . , converges to a random value X on probability if for any given ε > 0 lim δn = 0,

n→∞

δn = P {|Xn − X| > ε}.

where

P

Write Xn −→ X. A common feature of all three definitions can be formulated as follows: for enough big n the random value Xn and the limit random value X are close in a certain probabilistic sense. We show that a convergence on probability is the weakest type of convergence, i.e., both a convergence in the mean-square and a convergence with probability 1 involve a convergence on probability. The first of these statements follows from Chebyshev’s inequality: if for a random value Y there exists M {|Y |k }, where k > 0, then for any fixed ε > 0 M {|Y |k } . P {|Y | ≥ ε} ≤ εk We prove this inequality, for example, when Y is a continuous random value with a distribution density f (y). In this case, ∞ M {|Y | } =

−ε |y| f (y)dy ≥

k

−∞

−∞

≥

= εk ⎝

ε

∞ k

εk f (y)dy =

ε f (y)dy + −∞

−ε

ε

∞ f (y)dy +

−∞

|y|k f (y)dy ≥

k

−ε ⎛

∞ |y| f (y)dy +

k

ε

⎞

f (y)dy ⎠ = εk P {|Y | ≥ ε},

16

Chapter 1

and Chebyshev’s inequality is proved. Now having put k = 2 and Y = Xn − X, we obtain P {|Xn − X| ≥ ε} ≤

M {(Xn − X)2 } . ε2

Therefore, if M {(Xn − X)2 } → 0 as n → ∞, then P {|Xn − X| > > ε} → 0 as n → ∞, i.e., a convergence on probability follows from a convergence in the mean-square. Now we prove that a convergence with probability 1 also involves a convergence on probability. We will choose any ε > 0 and we will consider the event Aε = {ω :

∃N = N (ω)

∀n ≥ N

|Xn (ω) − X(ω)| < ε}.

Clearly, if a convergence of sequence of random values Xn , n = 1, 2, . . . , to a random value X takes place with probability 1, then P {Aε } = 1. Opposite to the event Aε , the event Aε is Aε = {ω :

∀N

∃n ≥ N

|Xn (ω) − X(ω)| ≥ ε};

P {Aε } = 0.

We will now introduce the event Bε,N = {ω :

∃n ≥ N

|Xn (ω) − X(ω)| ≥ ε}.

Then Aε = Bε,1 Bε,2 Bε,3 . . . =

∞ 

Bε,k ,

k=1

and

Bε,1 ⊃ Bε,2 ⊃ . . . ⊃ Bε,k ⊃ Bε,k+1 ⊃ . . .

.

In this situation, obviously, the following presentation takes place: Bε,1 = Aε + Bε,1 Bε,2 + Bε,2 Bε,3 + . . . + Bε,k Bε,k+1 + . . . = = Aε +

∞ 

Bε,k Bε,k+1 .

k=1

Since any two summands in the right part of this equality are non-joint events, we have P {Bε,1 } = P {Aε } +

∞  k=1

P {Bε,k Bε,k+1 }.

17

Main Classes and Characteristics of Random Processes

By definition, put SN =

N −1

P {Bε,k Bε,k+1 }. Then

k=1

P {Bε,1 } = P {Aε } + lim SN . N →∞

Since Bε,k+1 ⊂ Bε,k , we get P {Bε,k Bε,k+1 } = P {Bε,k } − P {Bε,k+1 } for every

k = 1, 2, . . . .

Therefore, SN = P {Bε,1 } − P {Bε,N } and P {Bε,1 } = P {Aε } + (P {Bε,1 } − lim P {Bε,N }). N →∞

From this it follows that lim P {Bε,N } = P {Aε } = 0.

N →∞

Now we introduce the event Cε,N = {ω :

|XN (ω) − X(ω)| ≥ ε}.

Then Cε,N ⊂ Bε,N and 0 ≤ P {Cε,N } ≤ P {Bε,N }. Passing in the last double inequality to a limit as N → ∞, we obtain limN →∞ P {Cε,N } = limN →∞ P {|XN − X| ≥ ε} = 0. This completes the proof. Thus, a convergence with probability 1 as well as convergence in the mean-square involves a convergence on probability. Various determinations of convergence in probabilistic space lead to various understandings of the continuity and differentiability of a random process. According to the determinations of convergence, as well as the continuity and differentiability of a random process, it is possible to understand it as follows: a) a continuity and a differentiability with probability 1, b) a continuity and a differentiability in the mean-square, c) a continuity and a differentiability on probability. We give, for example, definitions of continuity and differentiability in the mean-square (m.s.). If a random process X(t) satisfies the condition lim M {(X(t) − X(t0 ))2 } = 0,

t→t0

18

Chapter 1

then we say that X(t) is continuous in the mean-square at point t = t0 . If this condition is satisfied for all points of some interval (a, b), then process X(t) is called continuous in the mean-square on the interval (a, b). If a random process X(t) and a random value Y satisfy the condition  2  X(t) − X(t0 ) −Y = 0, lim M t→t0 t − t0 then process X(t) is called differentiable in the mean-square at point t0 , and random value Y is called a derivative in the mean-square of process X(t) at point t0 , and in this case Y is denoted by X  (t0 ). Let K(t, u) be a correlation function of process X(t). We find the sufficient condition of continuity in the mean-square of process X(t) in terms of correlation function. We have M {(X(t0 + h) − X(t0 ))2 } = M {X(t0 + h)X(t0 + h)}− −2M {X(t0 + h)X(t0 )} + M {X(t0 )X(t0 )}. Let m(t) = M {X(t)}. Since for any t and t K(t , t ) = M {X(t )X(t )} − m(t )m(t ), we obtain M {(X(t0 + h) − X(t0 ))2 } = K(t0 + h, t0 + h) + m(t0 + h)m(t0 + h) − − 2K(t0 + h, t0 ) − 2m(t0 + h)m(t0 ) + K(t0 , t0 ) + m(t0 )m(t0 ). From this it follows that if m(t) is continuous at point t0 and K(t, u) is continuous at point t = u = t0 , then the process X(t) is continuous in the mean-square at point t0 . It is possible to show that this condition is also necessary: a continuity of mathematical expectation m(t) at point t0 and a continuity of the correlation function K(t, u) at point t = u = t0 follow from a continuity in the mean-square of the process X(t) at point t0 . We note that a continuity in the mean-square of process X(t) does not mean a continuity with probability 1. For example, sample functions of process from example 3 from the previous section are step functions and, therefore, are discontinuous functions with probability 1. Nevertheless, this process is continuous in the mean-square

Main Classes and Characteristics of Random Processes

19

at any point because its mathematical expectation m(t) = 0 and correlation function K(t, u) = D exp{−λ|t − u|} are continuous. A necessary and sufficient condition for differentiability in the mean-square of the process X(t) at point t0 can also be formulated in terms of mathematical expectation m(t) and correlation function K(t, u) of process X(t). It is possible to show that such condition is an existence of derivatives   ∂K 2 (t, u)  dm(t)  and . dt t=t0 ∂t∂u t=u=t0 As well as in the case of continuity in the mean-square, an existence of derivative in the mean-square does not mean that the sample functions of the process are differentiated in the usual sense. At last, we will formulate one more result. Let the random process X(t) have a mathematical expectation m(t) and a correlation function K(t1 , t2 ). If the process X(t) is differentiated in the meansquare and its derivative in the mean-square is equal to X  (t), then formulas m1 (t) =

dm(t) dt

and

K1 (t1 , t2 ) =

∂ 2 K(t1 , t2 ) ∂t1 ∂t2

define a mathematical expectation m1 (t) and a correlation function K1 (t1 , t2 ) of the process X  (t). Formally this result can be obtained if we change a sequence of performance for operations of mathematical expectation and differentiability: 

m1 (t) = M {X (t)} = M



d X(t) dt

 =

d dm(t) M {X(t)} = , dt dt

K1 (t1 , t2 ) = M {(X  (t1 ) − m1 (t1 ))(X  (t2 ) − m1 (t2 ))} =       d d (X(t) − m(t)) (X(t) − m(t)) =M = dt t=t1 dt t=t2   ∂2 (X(t1 ) − m(t1 ))(X(t2 ) − m(t2 )) = =M ∂t1 ∂t2 =

∂ 2 K(t1 , t2 ) ∂2 M {(X(t1 ) − m(t1 ))(X(t2 ) − m(t2 ))} = . ∂t1 ∂t2 ∂t1 ∂t2

20

Chapter 1

If we are interested in a mutual correlation function K01 (t1 , t2 ) of processes X(t) and X  (t), where by definition K01 (t1 , t2 ) = M {(X(t1 ) − m(t1 ))(X  (t2 ) − m1 (t2 ))}, then the same formala leads to the result ∂K(t1 , t2 ) K01 (t1 , t2 ) = . ∂t2 This result is also fair and can be proved mathematically.

1.5 Stationary Random Processes In many appendices it is necessary to study functions determined by casual factors when the behavior of these factors are more or less constant throughout an observation period. Considering these functions, x(t), as realizations of a random process, X(t), we ask what is the definition of the process property which characterizes this situation. Such property is defined as the property of invariancy from all the finite-dimensional distributions of process relative to shifts of time t: i.e., when for any n an joint n-dimensional distribution of random values X(t1 + τ ), . . . , X(tn + τ ) does not depend on τ for any finite sequence of points t1 , . . . , tn . Such a process is called a stationary procces. Let us consider the case when densities of distributions exist. Let f (x1 , . . . , xn ; t1 , . . . , tn ) be a density of joint distribution of random values X(t1 ), . . . , X(tn ), and let f (x1 , . . . , xn ; t1 + τ, . . . , tn + τ ) be a density of joint distribution of random values X(t1 +τ ), . . . , X(tn +τ ). If process X(t) is stationary, then f (x1 , . . . , xn ; t1 , . . . , tn ) = f (x1 , . . . , xn ; t1 + τ, . . . , tn + τ ) for any τ . In particular, for n = 1 f (x1 ; t1 ) = f (x1 ; 0), i.e., a distribution density of random value X(t1 ) does not depend on t1 . In this case, m(t) = M {X(t)} = M {X(0)} = const if a mathematical expectation of process X(t) exists.

Main Classes and Characteristics of Random Processes

21

Furthermore, for n = 2 f (x1 , x2 ; t1 , t2 ) = f (x1 , x2 ; 0, t2 − t1 ), i.e., a distribution density of system of random values X(t1 ), X(t2 ) depends only on the difference t2 − t1 . Therefore, all characteristics of system {X(t1 ), X(t2 )}, in particular, the correlation function K(t1 , t2 ), do not separately depend on t1 and t2 . These characteristics are completely defined (if, of course, they exist) by the difference t2 − t1 . In particular, K(t1 , t2 ) = K(τ ), where τ = t2 − t1 . A stationary random process is also called strictly stationary, or stationary in a narrow sense. Besides, there is the concept of stationarity in a wide sense: a random process is called wide-sense stationary if its mathematical expectation is constant and its correlation function depends only on the difference of arguments. For example, all processes considered in examples from section 1.3 are known as widesense stationary. As shown above, a wide-sense stationarity follows from a strict stationarity, if both the mathematical expectation and the correlation function of the random process exist. We claim that the function K(τ ) is even. Indeed, if K(t1 , t2 ) = = K(t2 − t1 ) = K(τ ), then K(t2 , t1 ) = K(t1 − t2 ) = K(−τ ). It is obvious that K(t1 , t2 ) = K(t2 , t1 ). Therefore, K(τ ) = K(−τ ). Thus for a stationary process it is possible to write K(t1 , t2 ) = K(τ ), where τ = |t2 − t1 |. If a random process is stationary, the conditions of continuity and differentiability in the mean-square become simpler. A continuity in function K(τ ) at τ = 0 is required for a continuity in the meansquare and an existence of the second derivative K  (0) is required for differentiability in the mean-square. For example, the process from example 1 of section 1.3 has the correlation function K(τ ) = = D cos ωτ and this process is continuous and differentiate in the mean-square. The processes from examples 2 and 3 have the correlation function K(τ ) = D exp{−λ|τ |} and these processes are continuous in the mean-square, but are not differentiated in the mean-square. Stationary random processes are encountered in practice quite often. By means of such processes, it is possible to model, for example, a random noise in radio sets; fluctuations of tension in a lighting network; pitching in a ship; and fluctuations in the height of airplane at cruising horizontal flight. As a rule, the change of phase coordinates of any stochastic system is described by a stationary random process, if this system functions in steady mode.

22

Chapter 1

Let us consider an example from a bank activity. We will assume that the flow of deposits to the bank is described by the Poisson process with a density of λ (see example 3 from section 1.3). The value λ is the mathematical expectation of a number of deposits that arrive during a unit interval of time. We denote by F (x) the distribution function of the duration of the contribution. Let r be the percentage rate on deposits. We use the formula of continuous percents: the capital M after t units of time is equal to M exp{rt}. For simplicity, all contributions have the same size m. We find the total capital at moment t, if this capital is equal to zero at the initial moment t0 = 0. For this purpose, we will consider a partition of the interval (t0 , t) by intermediate moments si : t0 = s0 < s1 < s2 < . . . < si−1 < si < . . . < sn−1 < sn = t. Let n be rather great and all Δsi = si − si−1 be rather small. The average contribution made during the period [si−1 , si ) is equal to mλΔsi . At the moment t this contribution will be more than mλΔsi exp{r(t − si )} and less than mλΔsi exp{r(t − si−1 )}. There exists a point ξi ∈ (si−1 , si ) such that this contribution is equal to mλΔsi exp{r(t − ξi )}. Let Xi be the capital that remained on the account at the moment t from the contribution mλΔsi exp{r(t − ξi )}, which arrived during the interval [si−1 , si ). Then the total capital on the account at the moment t is n  Xi , X(t) = i=1

and the mathematical expectation of this capital is M {X(t)} =

n 

M {Xi }.

i=1

For the calculation M {Xi } at small Δsi , we consider approximately that the entire contribution mλΔsi exp{r(t − ξi )} is made at moment ξi and Xi is equal to mλΔsi exp{r(t − ξi )} with the probability pi or Xi is equal to 0 with probability 1 − pi , where pi is the probability that a duration of contribution made at moment ξi is equal to not less than (t − ξi ) time units: pi = P {duration ≥ t − ξi } = = 1 − P {duration < t − ξi } = 1 − F (t − ξi ).

Main Classes and Characteristics of Random Processes

23

Thus, M {Xi } ≈ mλΔsi exp{r(t − ξi )}(1 − F (t − ξi )) and M {X(t)} ≈

n 

mλΔsi exp{r(t − ξi )}(1 − F (t − ξi )),

i=1

and an approximation error decreases with decreasing Δsi . The exact result is obtained as max Δsi → 0, n → ∞ and is given by the integral i

t M {X(t)} = λm

exp{r(t − s)}(1 − F (t − s))ds. 0

To find lim M {X(t)} as t → ∞, we will transform this integral by means of the replacement u = t − s: t

0 exp{r(t − s)}(1 − F (t − s))ds = −

0

exp{ru}(1 − F (u))du = t

t exp{ru}(1 − F (u))du.

= 0

Therefore, ∞ lim M {X(t)} = λm

exp{ru}(1 − F (u))du

t→∞

0

if the integral converges. The obtained result means the following: if we wait for some characteristic time T , a change of M {X(t)} will practically stop and we can consider that M {X(t)} = const. For example, if the duration of the contribution is distributed in accordance with the exponential law: F (u) = 1 − exp{−μu}, then

∞

μ > 0, ∞

exp{ru}(1 − F (u))du = λm

λm 0

exp{(r − μ)u}du. 0

24

Chapter 1

The integral converges if μ > r. Note that the average duration of contribution is ∞ 1 uF  (u)du = . μ 0

Therefore, the integral converges if the average duration of contribution is less than 1r . In this case, ∞ λm . λm exp{(r − μ)u}du = μ−r 0

Note that this value differs from the value   T λm λm exp{(r − μ)u}du = 1 − exp{−(μ − r)T } μ−r 0

1 ln 100. less than for one percent if exp{−(μ − r)T } < 0.01, or T > μ−r

1.6 Gaussian Random Processes The random process X(t), t ∈ T , is called normal (Gaussian) if for any n and any t1 , t2 , . . . , tn ∈ T there exists a joint distribution density of random values Xi = X(ti ), i = 1, 2, . . . , n, and this density is defined by the formula1 f (x1 ,⎧ x2 , . . . , x n ) = ⎫ n ⎨  1 (xi − mi )(xj − mj ) ⎬ 1 exp − D , = ij ⎩ 2D ⎭ σi σj (2π)n/2 σ1 σ2 . . . σn D1/2 i,j=1 where mi = M {Xi } is a mathematical expectation of random value Xi , σi2 = M {(Xi − mi )2 } is a variance of random value Xi , and Dij is an algebraic addition of element kij of determinant D, where by definition    1 k12 . . . k1n    k21 1 . . . k2n   D= . .. ..  , kij = kji , kii = 1, ..  .. . . .    kn1 kn2 . . . 1  1 This formula describes nondegenerate case, when the D = 0; general definition of the joint normal distribution of the system of the random values including a degenerate D = 0 case: see, for example, W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd ed. New York: Wiley, 1971.

25

Main Classes and Characteristics of Random Processes

kij = σi1σj M {(Xi − mi )(Xj − mj )} is a correlation cofficient of random values Xi and Xj . In special cases, n = 1 and n = 2, we obtain the one-dimensional and the two-dimensional distribution densities of normal process:   1 (x1 − m1 )2 f (x1 ) = √ exp − 2σ12 2πσ1 and

1 √

· 2πσ1 σ2 1 − k 2    (x1 −m1 )2 1 (x1 −m1 )(x2 −m2 ) (x2 −m2 )2 · exp − −2k + , 2(1−k 2 ) σ12 σ1 σ2 σ22 f (x1 , x2 ) =

where k = k12 = k21 = σ11σ2 M {(X1 − m1 )(X2 − m2 )} is a correlation coefficient. The following result is well known: if a normal process X(t) is differentiable, then its derivative X  (t) is also a normal process.1 Let a normal process X(t) have a mathematical expectation m(t), a correlation function K(t1 , t2 ), and a variance σ 2 (t)=K(t, t). It can be proved that the density fXY (x, y) of the joint distribution of random values X(t) and Y (t) ≡ X  (t) is normal and is given by the formula fXY (x, y) =  · exp −

1 2(1 − k12 )



2πσσ1

(x − m)2 (x − m)(y − m1 ) (y − m1 )2 −2k + 1 σ2 σσ1 σ12

where m1 = m1 (t) = σ12

=

σ12 (t)

1  · 1 − k12

  = K1 (t1 , t2 )

 ,

dm(t) , dt

 ∂ 2 K(t1 , t2 )  =  ∂t1 ∂t2  t1 =t2 =t

, t1 =t2 =t

    ∂K(t , t ) 1 1  1 2 k1 = k1 (t) = K01 (t1 , t2 ) =   σσ1 σσ ∂t 1 2 t1 =t2 =t

. t1 =t2 =t

1 This statement is a consequence of a more general result: any linear transformation of normal process keeps the property of normality.

26

Chapter 1

These formulas become simpler for a stationary process X(t). In this case, M {X(t)} = m = const, D{X(t)} = σ 2 = const, K(t1 , t2 ) = = σ 2 k(τ ), where τ = |t1 −t2 |, k(τ ) is a correlation coefficient of random values X(t1 ) and X(t2 ). Then m1 = 0 and we obtain for k1 and σ12 :  2  2 2d k = −σ 2 k  (0). k1 = 0, σ1 = −σ  dτ 2  τ =0

For fXY (x, y) we have

   1 y2 1 (x − m)2 fXY (x, y) = fX (x)fY (y) = + 2 . exp − 2πσσ1 2 σ2 σ1

Note that random values X(t) and Y (t) ≡ X  (t) are independent in this case. We consider one important example of a normal process which is often used in appendices. A random process W (t), 0 ≤ t < ∞, is called the Wiener process if the following conditions are satisfied: 1) W (0) = w0 = const; 2) the random values W (t1 ) − W (t0 ), W (t2 ) − W (t1 ), . . . , W (tn ) − W (tn−1 ) are independent for any 0 ≤ t0 < t1 < t2 < . . . < tn ; 3) the random value W (t) − W (s), 0 ≤ s < t, has the normal distribution with zero mathematical expectation and variance t − s. It is not hard to prove that all finite-dimensional distributions of the process W (t) are normal and the correlation function of the process W (t) is K(s, t) = min(s, t). We prove, for example, the formula for K(s, t). Considering t ≥ s, we have K(s, t) = M {(W (s) − w0 )(W (t) − w0 )} = = M {(W (s) − W (0))(W (t) − W (s) + W (s) − W (0))} = = M {(W (s) − W (0))(W (t) − W (s)) + (W (s) − W (0))2 } = = M {(W (s) − W (0))(W (t) − W (s))} + M {(W (s) − W (0))2 }. From the definition of W (t), it follows that M {(W (s) − W (0))(W (t) − W (s))} =

Main Classes and Characteristics of Random Processes

27

= M {W (s) − W (0)} · M {W (t) − W (s)} = 0 · 0 = 0, and M {(W (s) − W (0))2 } = s. Therefore, K(s, t) = s if t ≥ s. Similarly, K(s, t) = t if t ≤ s. The formula for K(s, t) is proved. Studying the evolution of the system leads to the Wiener process when slow fluctuations of the system are interpreted as a result of a huge number of consecutive small changes caused by random influences. Such a situation can take place in the description of many phenomena in physics, economy, and other areas of knowledge. One example of such phenomena is the well-known Brownian motion of particles when a motion of a particle weighed in liquid is considered as a result of chaotic collisions with molecules. Another example is the process of the change of the price in the market: the price changes as a result of the aggregate effect of many random impulses, which are a consequence of obtaining new information. The given two examples are the most known examples of the Wiener process. For this reason the latter is still called the process of Brownian motion, or the Bachelier1 process, or the Wiener-Bachelier process. We will show how a consideration of random walks leads to the Wiener process if a length of separate steps is small, but the steps quickly follow one by one so that the movement is represented as continuous. We suppose that this particle is currently at zero and the step number n brings it into position Sn = X 1 + . . . + X n , where Xi , i = 1, 2, . . . , are independent random values, each of which has the same distribution: P {Xi = 1} = p, P {Xi = −1} = q, p + q = 1. Then n  M {Xi } = p − q, M {Sn } = M {Xi } = (p − q)n; i=1

D{Xi } = (1 − (p − q))2 p + (−1 − (p − q))2 q = 4pq, n  D{Sn } = D{Xi } = 4pqn. i=1

If a length of separate steps is not equal to 1 and is equal to δ, then the shift of particle after n steps is the random value Sn δ. Note that 1 French researcher L. Bachelier was the first who proposed to consider the process of change to the price as the Wiener process. See: L. Bachelier, Th´ eorie de la Sp´ eculation, Ann. Ecole Norm. Sup., 1900, vol. 17, pp. 21-86.

28

Chapter 1

the mathematical expectation is M {Sn δ} = δM {Sn } = (p−q)nδ, and the variance is D{Sn δ} = δ 2 D{Sn } = 4pqnδ 2 . We have assumed that the average shift of the particle and the average variance of the shift per unit of time can be found from an observation of the particle motion. If r is the number of collisions per unit of time, then (p − q)rδ = c,

4pqrδ 2 = d.

(6.1)

We have studied random walks when the length δ of separate steps is small, the number r of steps per unit of time is great, the value p − q is small, and equalities (6.1) take place, where c ≥0 and d > 0 are fixed constants. We consider the limit case δ → 0, r → ∞, p → 12 , and (p − q)rδ → c, 4pqrδ 2 → d. We introduce the probability vk,n = P {Sn = k}, i.e., the probability that the particle is in the position Sn δ = kδ after n steps. We find the probability that the particle is in the vicinity of point x at the moment t. For this purpose it is necessary to investigate the behavior of vk,n when k → ∞, n → ∞, nr → t, and kδ → x. If m steps from the first n steps are made to the right, then n − m steps are made to the left. The event {Sn = k} means that m − (n−m) = k, or m = n+k 2 . Therefore, the probability, vk,n , can be successful outcomes occurred considered as the probability that n+k 2 in a series from n independent trials with two possible outcomes. This probability is given by the binomial distribution: n+k

vk,n = Cn 2 p

n+k 2

q n−

n+k 2

.

Passing the limit k → ∞, n → ∞ and using the Moivre-Laplace local theorem,1 we obtain  2 (n + k)/2 − np 1 x 1 √ vk,n ∼ √ ϕ(x), where x = , ϕ(x) = exp − , √ npq npq 2 2π or vk,n

∼√

  1 (k − n(p − q))2 exp − , 8npq 2πnpq

1 See, for example: W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. New York: Wiley, 1968.

Main Classes and Characteristics of Random Processes

29

where sign ∼ between two values means that the ratio of these values tends to 1. Furthermore, √

1 = 2πnpq 2π ·

1 n r

· pqr · 4δ 2 ·

1 4δ 2

=

2δ 2δ , ∼√ 2 2π · · 4pqrδ 2πtd n r

 2 kδ · 1δ − (p − q)δ · r · nr · 1δ (k − (p − q)n)2 = = 8npq 2 · 4pqrδ 2 · δ12 · nr  2 1 kδ − (p − q)δr · nr (x − ct)2 2 , ∼ = δ 1 n 2 2td δ 2 · 2 · 4pqrδ · r so vk,n ∼ √

  (x − ct)2 ) 2δ exp − . 2td 2πtd

In a limit the random value Sn δ changes continuously, and the probability vk,n can be considered as the probability that Sn δ is between (k − 1)δ and (k + 1)δ. Therefore, the ratio vk,n /2δ can be considered as the distribution density. From above it follows that vk,n /2δ tends to v(t, x) = √

  1 (x − ct)2 ) exp − . 2td 2πtd

If W (t), t ≥ 0, is the Wiener process such that W (0) = 0, then at c = 0 and d = 1 this formula gives the distribution density of random value W (t).

1.7 Markov Random Processes Another important class of random processes is the class of Markov processes. A common property of Markov processes can be formulated as follows: if the value of the process X(t) is known at the moment of t = s, then a probabilistic description of the behavior of process X(t) at t > s does not depend on values of X(t) at t < s. We will consider only the Markov processes that have the distribution density. To define the Markov process, it is necessary to introduce the concept of conditional distribution density. Let a system of random values X1 and X2 have a continuous density of joint

30

Chapter 1

distribution f2 (x1 , x2 ) and let the random value X1 have a distribution density f1 (x1 ). The function f (x2 |x1 ) =

f2 (x1 , x2 ) f1 (x1 )

is called the conditional distribution density of the random value X2 provided that X1 = x1 . This expression does not make sense if f1 (x1 ) = 0, but we will consider the situation when a set of such points x1 has zero probability. We will assume that f (x2 |x1 ) = 0 if f1 (x1 ) = 0. Let us find out the probabilistic sense of the function f (x2 |x1 ). For this purpose, we will consider the conditional probability of event {X2 < x2 } provided that the event {x1 ≤ X1 < x1 + h} has occurred: x1 +h

x2

dξ P {X2 < x2 |x1 ≤ X1 < x1 + h} =

f2 (ξ, η)dη

−∞

x1

x1 +h

.

f1 (ξ)dξ x1

From this it follows that

x2 lim P {X2 < x2 |x1 ≤ X1 < x1 + h} =

h→0

−∞

f2 (x1 , η) dη. f1 (x1 )

By definition, put P {X2 < x2 |X1 = x1 } = lim P {X2 < x2 |x1 ≤ X1 < x1 + h}. h→0

Then

x2 P {X2 < x2 |X1 = x1 } =

f (η|x1 )dη, −∞

i.e., an integral from a conditional density of distribution gives a conditional probability. Similarly, we will introduce the conditional distribution density f (x3 |x2 , x1 ) =

f3 (x1 , x2 , x3 ) f2 (x1 , x2 )

Main Classes and Characteristics of Random Processes

31

of the random value X3 provided that X2 = x2 , X1 = x1 , and so on. Now we can formulate the definition of the Markov process. A random process X(t) is called Markov if for any natural n and any sequence of moments t1 < t2 < . . . < tn−1 < tn the conditional distribution density f (xn |xn−1 , xn−2 , . . . , x2 , x1 ) of a system of random values X1 = X(t1 ), X2 = X(t2 ), . . . , Xn−1 = X(tn−1 ), Xn = X(tn ) coincides with the conditional distribution density f (xn |xn−1 ): f (xn |xn−1 , xn−2 , . . . , x2 , x1 ) ≡ f (xn |xn−1 ). We claim that any finite-dimensional distribution density of the Markov process is expressed through the two-dimensional distribution density of this process. Indeed, f (x1 , . . . , xn ) = f (xn |xn−1 , xn−2 , . . . , x2 , x1 ) · f (x1 , . . . , xn−1 ) = = f (xn |xn−1 ) · f (x1 , . . . , xn−1 ) = . . . = = f (xn |xn−1 ) · f (xn−1 |xn−2 ) . . . f (x3 |x2 ) · f (x2 |x1 ) · f (x1 ). Let f2 (x1 , x2 ) be a two-dimensional distribution density of the random process X(t) corresponding to moments t1 and t2 > t1 : i.e., f2 (x1 , x2 ) is the distribution density of the random values X(t1 ) and X(t2 ). Similarly, let f3 (x1 , x, x2 ) be a three-dimensional distribution density corresponding to moments t1 , t, and t2 , where t1 < t < t2 . Then ∞ f (x1 , x2 ) = f3 (x1 , x, x2 )dx. −∞

If X(t) is a Markov process, f3 (x1 , x, x2 )=f (x2 |x)f (x|x1 )f1 (x1 ). Using the equality f2 (x1 , x2 ) = f (x2 |x1 )f1 (x1 ), we obtain ∞ f (x2 |x1 ) =

f (x2 |x)f (x|x1 )dx. −∞

32

Chapter 1

It is clear that the function f (x2 |x1 ) depends on moments t1 and t2 . To reflect this fact we write f (x2 , t2 |x1 , t1 ) and ∞ f (x2 , t2 |x1 , t1 ) =

f (x2 , t2 |x, t)f (x, t|x1 , t1 )dx.

(7.1)

−∞

This equation is called the Kolmogorov-Chapman equation. We will consider equation (7.1) and we put t = t1 + Δ, where 0 < Δ < t2 − t1 . Then ∞ f (x2 , t2 |x1 , t1 ) =

f (x2 , t2 |x, t1 + Δ)f (x, t1 + Δ|x1 , t1 )dx. (7.2) −∞

We assume that the conditional distribution density f (x2 , t2 |x, t1+Δ) can be decomposed by the Taylor formula as a function of the scalar argument x in the vicinity of point x = x1 : f (x2 , t2 |x, t1 + Δ) = f (x2 , t2 |x1 , t1 + Δ) +  ∂f (x2 , t2 |ξ, t1 + Δ)  (x − x1 ) + +   ∂ξ ξ=x1

 1 ∂ 2 f (x2 , t2 |ξ, t1 + Δ)  + (x − x1 )2 +   2 ∂ξ 2 ξ=x1   3 1 ∂ f (x2 , t2 |ξ, t1 + Δ)  + (x − x1 )3 ,   6 ∂ξ 3 ξ=x1 +ε(x−x1 )

where 0 < ε < 1. Introducing the notation  ∂f (x2 , t2 |ξ, t1 + Δ)    ∂ξ  ∂ 2 f (x2 , t2 |ξ, t1 + Δ)    ∂ξ 2

=

∂f (x2 , t2 |x1 , t1 + Δ) , ∂x1

=

∂ 2 f (x2 , t2 |x1 , t1 + Δ) , ∂x21

ξ=x1

ξ=x1

33

Main Classes and Characteristics of Random Processes

 ∂ 3 f (x2 , t2 |ξ, t1 + Δ)    ∂ξ 3

ξ=x1 +ε(x−x1 )

∂ 3 f (x2 , t2 |x1 + ε(x − x1 ), t1 + Δ) = ∂x31

and taking into account equality (7.2), we obtain ∞ f (x2 , t2 |x1 , t1 ) = f (x2 , t2 |x1 , t1 + Δ) f (x, t1 + Δ|x1 , t1 )dx + −∞

+

∂f (x2 , t2 |x1 , t1 + Δ) ∂x1

∞ (x − x1 )f (x, t1 + Δ|x1 , t1 )dx + −∞

1 ∂ 2 f (x2 , t2 |x1 , t1 + Δ) + 2 ∂x21 +

1 6

∞

−∞

∞

(7.3) (x − x1 )2 f (x, t1 + Δ|x1 , t1 )dx +

−∞

3

∂ f (x2 , t2 |x1 + ε(x−x1 ), t1 +Δ) (x−x1 )3 f (x, t1 +Δ|x1 , t1 )dx. ∂x31

Since

∞ f (x, t1 + Δ|x1 , t1 )dx = 1, −∞

from (7.3) we get f (x2 , t2 |x1 , t1 + Δ) − f (x2 , t2 |x1 , t1 ) = Δ ∞ ∂f (x2 , t2 |x1 , t1 + Δ) 1 =− (x − x1 )f (x, t1 + Δ|x1 , t1 )dx − ∂x1 Δ −∞ (7.4) 1 ∂ 2 f (x2 , t2 |x1 , t1 + Δ) 1 − 2 ∂x21 Δ −

1 1 6Δ

∞ −∞

∞

(x − x1 )2 f (x, t1 + Δ|x1 , t1 )dx −

−∞

3

∂ f (x2 , t2 |x1 + ε(x−x1 ), t1 +Δ) (x−x1 )3 f (x, t1+Δ|x1 , t1 )dx. ∂x31

By definition, put 1 M {X(t + Δ) − X(t)|X(t) = x} = Δ→0+ Δ

a(t, x) = lim

(7.5)

34

Chapter 1

1 = lim Δ→0+ Δ

∞ (z − x)f (z, t + Δ|x, t)dz, −∞

1 M {(X(t + Δ) − X(t))2 |X(t) = x} = Δ ∞ 1 = lim (z − x)2 f (z, t + Δ|x, t)dz. Δ→0+ Δ

b(t, x) = lim

Δ→0+

(7.6)

−∞

We assume that  ∞ 1 ∂ 3 f (x2 , t2 |x1 + ε(x − x1 ), t1 + Δ) (x − x1 )3 · lim Δ→0+ Δ ∂x31 −∞  (7.7) ·f (x, t1 + Δ|x1 , t1 )dx

= 0.

The Markov process is called diffusion, if assumption (7.7) holds. Assumption (7.7) means that the probability of large deviations |X(t + Δ) − X(t)| decreases with the decreasing Δ and all moments of random value |X(t + Δ) − X(t)|, starting with the 3-th moment, have a higher order of smallness in comparison with Δ. In this case, passing the limit Δ → 0+ in equality (7.4), we obtain   ∂f 1 ∂2f ∂f = − a(t1 , x1 ) + b(t1 , x1 ) 2 . ∂t1 ∂x1 2 ∂x1

(7.8)

Similarly, it can be proved that if assumption (7.7) is satisfied, then ∂ 1 ∂2 ∂f =− (a(t2 , x2 )f ) + (b(t2 , x2 )f ). ∂t2 ∂x2 2 ∂x22

(7.9)

Equations (7.8) and (7.9) are called the first and the second of Kolmogorov’s equations, respectively. The function a(t, x) introduced by formula (7.5) is called the drift coefficient of Markov process X(t) and the function b(t, x) introduced by formula (7.6) is called the diffusion coefficient of Markov process X(t). The properties of diffusion Markov processes allow us to solve a number of problems that are not able to be solved in arbitrary continuous processes. Examples of such problems are given in Chapter 2.

35

Main Classes and Characteristics of Random Processes

Now we will consider the example of the Markov process. We will prove that the Wiener process is Markov. Let W (t), t ≥ t0 , be the Wiener process such that W (t0 ) = 0. We will consider any sequence of moments t0 < t1 < t2 < . . . < tn and the random values X1 = W (t1 ) − W (t0 ) = W (t1 ), X(t2 ) = W (t2 ) − W (t1 ), . . . , Xn = W (tn ) − W (tn−1 ). The random values X1 , X2 , . . . , Xn are independent; the random value Xi has the distribution density   (2π)−1/2 x2 exp − fi (x) = √ , ti − ti−1 2(ti − ti−1 ) i = 1, 2, . . . , n, and the random values X1 , X2 , . . . , Xn have the density of joint distribution f (x1 , . . . , xn ) = f1 (x1 ) . . . fn (xn ) =   n   (2π)−1/2 x2k √ exp − = . tk − tk−1 2(tk − tk−1 ) k=1

We will introduce the random values Y1 = W (t1 ), Y2 = W (t2 ), . . . , Yn = W (tn ). Then X1 = Y1 ,

X2 = Y 2 − Y 1 ,

...

,

Xn = Yn − Yn−1 .

(7.10)

Denote by g(y1 , y2 , . . . , yn ) the density of the joint distribution of random values Y1 , Y2 , . . . , Yn . Values of the random vector X = (X1 , X2 , . . . , Xn ) can be interpretated as points in n-dimensional space Ox1 x2 . . . xn , and values of random vector Y = (Y1 , Y2 , . . . , Yn ) can be interpretated as points in n-dimensional space Oy1 y2 . . . yn . From (7.10) it follows that if X is equal to x0 = (x01 , x02 , . . . , x0n ), then Y is equal to y0 = (y10 , y20 , . . . , yn0 ), where y10 = x01 ,

y20 = x01 + x02 ,

...

,

yn0 = x01 + x02 + . . . + x0n .

36

Chapter 1

The point Y is in some domains Q of space Oy1 y2 . . . yn if, and only if, the point X is in the corresponding domain Q of space Ox1 x2 . . . xn . Therefore, P {X ∈ Q } = P {Y ∈ Q}, or  

  f (x1 , . . . , xn )dx1 . . . dxn = ... Q

g(y1 , . . . , yn )dy1 . . . dyn . ... Q

(7.11)

We will transform the integral in the left part of this equality by means of replacing the variables; we will pass from variables x1 , . . . , xn to variables y1 , . . . , yn using the formulas: x1 = h1 (y1 , . . . , yn ) ≡ y1 ,

x2 = h2 (y1 , . . . , yn ) ≡ y2 − y1 ,

...,

xn = hn (y1 , . . . , yn ) ≡ yn − yn−1 .  

Then

f (x1 , . . . , xn )dx1 . . . dxn =

  = ... Q

(7.12)    ∂(h1 , . . . , hn )   dy1 . . . dyn , f (y1 , y2 − y1 , . . . , yn − yn−1 )  ∂(y1 , . . . , yn )  ... Q

where       ∂(h1 , . . . , hn )  = ∂(y1 , . . . , yn )     

∂h1 ∂y1

∂h1 ∂y2

∂h1 ∂y3

...

∂h1 ∂yn−1

∂h1 ∂yn

∂h2 ∂y1

∂h2 ∂y2

∂h2 ∂y3

...

∂h2 ∂yn−1

∂h2 ∂yn

∂h3 ∂y1

∂h3 ∂y2

∂h3 ∂y3

∂h3 ∂yn−1

∂h3 ∂yn

∂hn ∂y1

∂hn ∂y2

∂hn ∂y3

... .. . ...

∂hn ∂yn−1

∂hn ∂yn

.. .

.. .

.. .

  1 0 0 ...   −1 1 0 ...   =  0 −1 1 . . .  .. .. .. ..  . . . .   0 0 0 ... Combining (7.11) and (7.12), we obtain

.. .

.. .

       =      

 0 0  0 0  0 0  = 1. .. ..  . .  −1 1 

37

Main Classes and Characteristics of Random Processes

  g(y1 ,. . . , yn )dy1 . . . dyn = ... Q

  f (y1 , y2 − y1 ,. . . , yn − yn−1 )dy1 . . . dyn .

= ... Q

Due to arbitrariness of domain Q, from the last equality we get the following formula for g(y1 , y2 , . . . , yn ): g(y1 , y2 , . . . , yn ) = f (y1 , y2 − y1 , . . . , yn − yn−1 ) =   n   (2π)−1/2 (yk − yk−1 )2 √ exp − = , tk − tk−1 2(tk − tk−1 )

(7.13)

k=1

where y0 = 0. From formula (7.13) it follows that g(yn |yn−1 , yn−2 , . . . , y1 ) =

g(y1 , y2 , . . . , yn ) = g(y1 , y2 , . . . , yn−1 )

  (2π)−1/2 (yn − yn−1 )2 =√ , exp − tn − tn−1 2(tn − tn−1 )   (2π)−1/2 g(yn−1 , yn ) (yn − yn−1 )2 =√ exp − , g(yn |yn−1 ) = g(yn−1 ) tn − tn−1 2(tn − tn−1 ) i.e., g(yn |yn−1 , yn−2 , . . . , y1 ) = g(yn |yn−1 ). This completes the proof.

1.8 Continuity and Not Differentiability of the Wiener Process The Wiener process takes a noticeable place in the theory of random processes and its appendices. In particular, it is often used for the mathematical modeling of real stochastic processes of economic and financial contents. Therefore, it is reasonable to study its properties. We will show that the Wiener process is continuous in the following sense: for the Wiener process W (t), there exists an equivalent1 1 We are reminded that two processes X(t) and Y (t) are called equivalent if P {X(t) = Y (t)} = 1 at any fixed t.

38

Chapter 1

process Y (t) such that sample functions of this process are continuous with probability 1. For this purpose we will prove the following theorem.1 T h e o r e m 1. Let X(t) be a random process, 0 ≤ t ≤ 1. We will assume that for all t, t + h from the segment [0, 1] P {|X(t + h) − X(t)| ≥ g(h)} ≤ q(h), where g and q are even functions that are not increasing and are aspiring to zero as h ↓ 0 and such that ∞ 

g(2−n ) < ∞,

n=1

∞ 

2n q(2−n ) < ∞.

n=1

Then, for X(t), there is an equivalent random process Y (t) such that sample functions of this process are continuous with probability 1 on the segment 0 ≤ t ≤ 1. P r o o f. We assign the sequence of moments tn,r =

r , 2n

r = 0, 1, 2, 3, . . . , 2n ,

to each natural n. We will approximate the process X(t) by a process Xn (t) such that a trajectory of Xn (t) consists of rectilinear segments (see Fig. 1–1): ...................................... ........................P ............... ... PP ............................. ........... . . . . PP ..................  . . . . . .. PP ............... x(t)..................  . . . x . . PP ..............  . n+1 (t) . . . . . . PP.............  ... ..... . . . P............. xn+1 (t) .. (( ..... . ( . .. ....  ( . . . ( . . . ( . .. ..  (( ( .... ( . . .. ( . . (..( ...  ( .. . ( . ( . x (t) . . ( n . ( .. . . .. ...  .( (((( . ( ... . .. . . . .. .. .. ... ... .. ... .. .. .. .. .. .. .. .. .. .. .. .. -t . . tn+1,2r+1 tn+1,2r = tn,r tn+1,2r+2 = tn,r+1 Fig. 1–1 1 See: H. Cramer and M.R. Leadbetter, Stationary and Related Stochastic Processes, New York-London-Sydney: Wiley, 1967.

Main Classes and Characteristics of Random Processes

Xn (t) = X(tn,r ) +

39

X(tn,r+1 ) − X(tn,r ) (t − tn,r ), 1/2n

where tn,r ≤ t ≤ tn,r+1 . Then for realizations xn (t), xn+1 (t), x(t) of processes Xn (t), Xn+1 (t), X(t) at any t ∈ [tn,r ; tn,r+1 ] |xn+1 (t) − xn (t)| ≤ |x(tn+1,2r+1 ) − xn (tn+1,2r+1 )| =     1 = x(tn+1,2r+1 ) − [x(tn+1,2r ) + x(tn+1,2r+2 )] = 2    1 1 1  = [x(tn+1,2r+1 )−x(tn+1,2r )]+[x(tn+1,2r+1 )−x(tn+1,2r+2 )] ≤ a+ b, 2 2 2 where, by definition, a = |x(tn+1,2r+1 ) − x(tn+1,2r )|,

b = |x(tn+1,2r+1 ) − x(tn+1,2r+2 )|,

i.e., a and b do not depend on t, so max

tn,r ≤t≤tn,r+1

Thus random values

|xn+1 (t) − xn (t)| ≤

max

tn,r ≤t≤tn,r+1

1 1 a + b. 2 2

|Xn+1 (t)−Xn (t)|, A = |X(tn+1,2r+1 )−

−X(tn+1,2r )|, and B = |X(tn+1,2r+1 ) − X(tn+1,2r+2 )| satisfy the inequality 1 1 max |Xn+1 (t) − Xn (t)| ≤ A + B. tn,r ≤t≤tn,r+1 2 2 Therefore,   −n−1 P max |Xn+1 (t) − Xn (t)| ≥ g(2 ) ≤ tn,r ≤t≤tn,r+1

≤P In turn,



 1 1 −n−1 A + B ≥ g(2 ) . 2 2

(8.1)



 1 1 −n−1 A + B ≥ g(2 P ) ≤ 2 2    1 1 1 1 −n−1 −n−1 ≤P A ≥ g(2 B ≥ g(2 ) ) ≤ 2 2 2 2 ≤ P {A ≥ g(2−n−1 )} + P {B ≥ g(2−n−1 )}.

(8.2)

40

Chapter 1

By assumption of the Theorem, P {A ≥ g(2−n−1 )} ≡ ≡ P {|X(tn+1,2r+1 ) − X(tn+1,2r )| ≥ g(2−n−1 )} ≤ q(2−n−1 ). Similarly,

P {B ≥ g(2−n−1 )} ≤ q(2−n−1 ).

Considering (8.1) and (8.2), we get  P

max

tn,r ≤t≤tn,r+1

and

 P ≤P

r=0

≤ 2q(2−n−1 ),

max |Xn+1 (t) − Xn (t)| ≥ g(2−n−1 )

max

tn,r ≤t≤tn,r+1

r=0

≤





0≤t≤1

2n −1   n 2 −1

|Xn+1 (t) − Xn (t)| ≥ g(2−n−1 )

|Xn+1 (t) − Xn (t)| ≥ g(2

−n−1

|Xn+1 (t) − Xn (t)| ≥ g(2

−n−1

 P

max

tn,r ≤t≤tn,r+1

≤  ≤

)  )

≤

≤ 2n+1 q(2−n−1 ). Furthermore, the proof of the Theorem uses the following results. L e m m a. Let Y1 , Y2 , . . . be sequence of random values, and δ1 , δ2 , . . . and ε1 , ε2 , . . . be sequences of positive numbers such that the ∞ ∞   δi and the series εi converge. If for all n series i=1

i=1

P {|Yn+1 − Yn | ≥ δn } ≤ εn , then there exists a random value Y such that Yn → Y with probability 1. P r o o f of the Lemma. We will show that P {|Yn+1 − Yn | ≥ δn for infinitely many n} = 0.

(8.3)

Main Classes and Characteristics of Random Processes

41

In contradiction, we will assume that this probability is ε˜ > 0. Owing ∞  εi there is m such that to the convergence of the series i=1

εm+1 + εm+2 + . . . =

∞ 

εi < ε˜.

i=m+1

Further, P {|Yn+1 − Yn | ≥ δn for infinitely many n} ≤ ≤ P {|Yn+1 − Yn | ≥ δn at least for one n > m} ≤ ∞  εi < ε˜. ≤ εm+1 + εm+2 + . . . = i=m+1

The obtained contradiction proves (8.3). Therefore, everywhere beyond the exception of perhaps ω-sets of P -measures zero, it is possible to find a finite number N = N (ω) such that |yn+1 (ω) − yn (ω)| < δn for all n > N (ω). For all such points ω, the sequence y1 (ω), y2 (ω), . . . converges1 to some finite limit y(ω). By definition, put y(ω) = 0 on zero ω-set, where this limit does not exist. Thus yn (ω) → y(ω) for all ω behind an exception perhaps ω-sets measures zero, i.e., P {Yn → Y } = = P {ω : yn (ω) → y(ω)} = 1. The Lemma is proved. We will return to the proof of the Theorem now. It is shown above that   −n−1 P max |Xn+1 (t) − Xn (t)| ≥ g(2 ) ≤ 2n+1 q(2−n−1 ). 0≤t≤1

Since

∞ 

g(2−n−1 ) < ∞ and

n=1

∞ 

2n+1 q(2−n−1 ) < ∞, we see that the

n=1

following statement follows from the Lemma: almost for all ω (i.e., for all beyond the exception ω-sets of a zero measure) the sequence of functions xn (t, ω), n = 1, 2, . . . , converges uniformly on t ∈ [0, 1]. By definition, put xn (t, ω) → y(t, ω) as n → ∞. The limit function y(t, ω) is a continuous function of t on [0, 1] because xn (t, ω) is a continuous 1

This fact follows from Cauchy’s criterion of convergence of sequence and con-

vergence of the series

∞ 

i=1

δi .

42

Chapter 1

function of t on [0, 1] at each n=1, 2, . . . . Functions y(t, ω) can be considered as realizations of some random process Y (t). A continuity of almost all sample functions y(t, ω) on [0, 1] means a continuity of process Y (t) on [0, 1] with probability 1. By construction of functions xn (t, ω), for all ω xn (t, ω) = x(t, ω) at t = tn,r , xn+p (t, ω) = xn (t, ω) at t = tn,r for any p = 1, 2, . . . . Therefore, for all ω xn+p (t, ω) = x(t, ω) at t = tn,r for any p = 1, 2, . . . . It follows that Y (t) = X(t) with probability 1 at points t = tn,r . Let now t = tn,r for all n and r. We consistently split segment [0, 1] in half, every time taking the half where the point t is. We choose sequence rn so that the sequence of points tn,rn coincides with a sequence of left ends from such system of enclosed segments. Then 0 < t − tn,rn < 2−n , t = lim tn,rn and, according to the assumption n→∞ of the Theorem, we obtain P {|X(tn,rn ) − X(t)| ≥ g(2−n )} ≤ ≤ P {|X(tn,rn ) − X(t)| ≥ g(t − tn,rn )} ≤ q(t − tn,rn ) ≤ q(2−n ). From here, using the Lemma, we conclude that the sequence of random values X(tn,rn ) converges to the random value X(t) with probability 1. Since Y (t) has continuous (with probability 1) sample functions, we see that Y (tn,rn ) → Y (t) with probability 1. But, as it is established above, P {X(tn,rn ) = Y (tn,rn )} = 1. Therefore, P {X(t) = Y (t)} = 1. The Theorem is proved. Let now W (t) be the Wiener process, so W (t + h) − W (t) is for each t a normal random value with zero mathematical expectation and the variance |h|. Then ε √

P {|W (t + h) − W (t)| ≥ ε} = 1 − −ε

  1 x2 exp − dx. 2|h| 2π|h|1/2

By definition, put 1 Φ(x) = √ 2π

x −∞

exp{−v 2 /2}dv

and

1 ϕ(x) = √ exp{−x2 /2}. 2π

43

Main Classes and Characteristics of Random Processes

Then

ε √

1− ⎛ =1−⎝

−ε

0

−ε

  1 x2 exp − dx = 2|h| 2π|h|1/2

  1 x2 √ exp − dx + 2|h| 2π|h|1/2 ε + 0

ε =1−2 ⎛ = 1 − 2⎝

0

ε

−∞

⎞   1 x2 √ exp − dx⎠ = 2|h| 2π|h|1/2

  1 x2 √ exp − dx = 2|h| 2π|h|1/2

  1 x2 √ exp − dx − 2|h| 2π|h|1/2 

0 √

−   =1−2 Φ

−∞

ε |h|1/2



−

1 2

2

1 x exp − 2|h| 2π|h|1/2



  =2 1−Φ

ε |h|1/2



⎞ dx⎠ =

 .

We claim that 1 − Φ(x)
0.

Indeed, the difference Δ(x) ≡ ϕ(x)/x − (1 − Φ(x)) monotonously and strictly decreases on (0, ∞) because Δ (x) ≡ −ϕ(x)/x2 < 0. Since lim Δ(x) = 0, we have Δ(x) > 0 for any x > 0. Therefore,

x→+∞

  ε 2|h|1/2 ϕ P {|W (t + h) − W (t)| ≥ ε} ≤ . ε |h|1/2 Let ε = |h|a , where a ∈ (0, 1/2). Then P {|W (t + h) − W (t)| ≥ |h|a } ≤ 2|h|1/2−a ϕ(|h|a−1/2 ).

44

Chapter 1

By definition, put q(h) = 2|h|1/2−a ϕ(|h|a−1/2 ).

g(h) = |h|a , Then the series

∞ 

g(2−n ) =

n=1

∞ 

2−na

n=1

converges (for example, by Cauchy’s test of convergence) and for the series   ∞ ∞   n −n n −n 1/2−a −n a−1/2 2 q(2 ) = 2 2 (2 ) ϕ (2 ) = n=1

=2

∞ 

n=1

2n 2na−n/2 ϕ(2n/2−na ) = 2

n=1

∞ 

2n/2+na ϕ(2n/2−na )

n=1

it is convenient to use Dalamber’s test of convergence: if bn is the n-member of this series, then  1 (1−2a)(n+1) ·2 bn+1 1/2+a exp − 2 1 =2 · = bn exp − 2 · 2(1−2a)n    1 = 21/2+a · exp − 2(1−2a)n · 21−2a − 2(1−2a)n = 2    1 → 0 as n → ∞, = 21/2+a · exp − · 2(1−2a)n 21−2a − 1 2 and the series

∞ 

2n q(2−n ) coverges by Dalamber’s test of conver-

n=1

gence. Therefore, the conditions of Theorem 1 are satisfied for the specified functions g(h) and q(h), and for the process W (t) there exists an equivalent process Y (t) with continuous (with probability 1) sample functions. Theorem 1 is proved. We will now show that the Wiener process W (t) is not differentiable and we will prove that the following result takes place. T h e o r e m 2. If for any t W (t + h) − W (t) → Z(t) h

as

h→0

45

Main Classes and Characteristics of Random Processes

in any sense of convergence (with probability 1, in the mean-square, on probability), then the equality P {z1 ≤ Z(t) ≤ z2 } = 0 holds for any finite z1 and any finite z2 , i.e., the random value Z(t) is equal to ±∞ with probability 1. P r o o f. Besides the three previously introduced convergence types (with probability 1, in the mean-square, on probability) we will still consider one type of convergence, namely a convergence on distribution. This fourth type of convergence will be required to prove Theorem 2. By definition, a sequence of random values Yn , n = 1, 2, . . . , converges to a random value Y on distribution if the sequence of distribution functions FYn (x) converges to the distribution function FY (x) at all points x where FY (x) is continuous. We will prove that a convergence on distribution follows from a convergence on probability: if P {|Yn − Y | > ε} → 0 as n → ∞ for any ε > 0, then FYn (x) → FY (x) as n → ∞ if only FY (x) is continuous at point x. By the formula of total probability, we have ! ! P {Yn < x} = P {(Yn < x) (Y < x+ε)}+P {(Yn < x) (Y ≥ x+ε)} ≤ ≤ P {Y < x + ε} + P {|Yn − Y | > ε}, and similarly, P {Y < x−ε} = P {(Y < x−ε)

!

(Yn < x)}+P {(Y < x−ε)

!

(Yn ≥ x)} ≤

≤ P {Yn < x} + P {|Yn − Y | > ε}. By definition, put δn (ε) = P {|Yn − Y | > ε}. Then we get FY (x − ε) − δn (ε) ≤ FYn (x) ≤ FY (x + ε) + δn (ε).

(8.4)

We will take any as much as small Δ > 0. By a continuity of FY (x) at the point x, we can specify ε such that |FY (x + ε) − FY (x − ε)| < Δ/2. Furthermore, there is a natural N such that δn (ε) < Δ/4 for any n ≥ N . Therefore, due to (8.4) the inequality |FYn1 (x) − FYn2 (x)| < Δ

46

Chapter 1

holds for any n1 ≥ N and any n2 ≥ N , and by Cauchy’s criterion there exists lim FYn (x) as n → ∞. From (8.4) it follows that this limit is equal to FY (x). Previously it was shown (see section 1.4) that a convergence on probability is the weakest from three convergence types (with probability 1, in the mean-square, on probability). Therefore, it is enough to prove the statment of Theorem 2 for a convergence on probability. So, we suppose that a convergence on probability takes place: Zh ≡

W (t + h) − W (t) P −→ Z h

as

h → 0.

Then a convergence Zh to Z on distribution takes place: FZh (x) → FZ (x) as h → 0 if the function FZ (x) is continuous at point x. According to the definition of the Wiener process, the random value W (t + h) − W (t) has the Gaussian distribution with zero mathematical expectation and the variance |h|. Therefore, the random value (t) has the Gaussian distribution with zero matheZh ≡ W (t+h)−W h matical expectation and the variance |h|/h2 = 1/|h|. We have   x 1 v2 √  lim FZh (x) = lim P {Zh < x} = lim exp − dv = h→0 h→0 h→0 2/|h| 2π 1/|h| −∞

√   2  x  |h|x 2 |h| 1 |h|v u √ √ exp − exp − dv = lim du = = lim h→0 h→0 2 2 2π 2π −∞ −∞ ⎧ at x = −∞, ⎨ 0 1/2 at −∞ < x < ∞, = ⎩ 1 at x = ∞. By definition, put F (x) =

⎧ ⎨

0 1/2 ⎩ 1

at at at

x = −∞, −∞ < x < ∞, x = ∞.

Then we see that FZ (x) = F (x) if the function FZ (x) is continuous at point x. We claim that any function of distribution cannot have more than a denumerable number of jumps. Indeed, all jumps can be counted if we renumber them in the following order: at first all jumps exceeding 21 (such jump there can be only one), then all jumps exceeding 13

Main Classes and Characteristics of Random Processes

47

(such jumps there can be no more than two), then all jumps exceeding 14 (such jumps there can be no more than three), and so on. In particular, this means that a distribution function cannot have more than a denumerable number of points of discontinuity. Therefore, one of the following cases is necessarily realized for the function FZ (x). 1) FZ (x) has no points of discntinuity at x ∈ (−∞; ∞). Then FZ (x) coincides with F (x) and 1 1 P {z1 ≤ Z < z2 } = FZ (z2 )−FZ (z1 ) = F (z2 )−F (z1 ) = − = 0 2 2 at any finite z1 and any finite z2 . This means that the random value Z is equal to ±∞ with probability 1. 2) FZ (x) has a finite or denumerable number of points of discontinuity. Then for any z1 , z2 ∈ (−∞; ∞) there exist z1∗ ∈ (−∞; z1 ) and z2∗ ∈ (z2 ; ∞) such that FZ (x) is continuous at points z1∗ and z2∗ and we have P {z1 ≤ Z < z2 } ≤ P {z1∗ ≤ Z < z2∗ } = 1 1 = FZ (z2∗ ) − FZ (z1∗ ) = F (z2∗ ) − F (z1∗ ) = − = 0, 2 2 i.e., we have the same situation as in case 1). Theorem 2 is proved.

1.9 Integration of Random Processes Let W (t) be the Wiener process, let X(t) be continuous in the mean-square random process such that a random vector with components X(t1 ), . . . , X(tn ) does not depend on W (s)−W (t) for any n and any values t1 < . . . < tn ≤ t < s. Let a and b > a be two fixed time moments. We consider the random value Yn,τ =

n−1 

X(ti )(W (ti+1 ) − W (ti ))

i=0

for any n and any partition τ a = t0 < t1 < . . . < tn−1 < tn = b of the interval (a, b). By definition, put δτ = max (ti − ti−1 ). i=1,...,n

48

Chapter 1

If there is a random value Y such that M (Y − Yn,τ )2 → 0

as

n → ∞, δτ → 0,

(9.1)

then it is called Ito’s integral on interval (a, b) from random process X(t) on process W (t); in this case, we write b X(t)dW (t).

Y = a

The symbol, l.i.m., is often used for the designation of mean-square convergence. Condition (9.1) can be rewritten in the following equivalent form: Y = l.i.m. Yn,τ , or, in more detail, b X(t)dW (t) = l.i.m.

n−1 

X(ti )(W (ti+1 ) − W (ti )).

i=0

a

It is possible to show that a necessary and sufficient condition of existence in Ito’s integral, is an existence of the integral b M {X(t)}2 dt. a

Lowering the proof of this statement, we will now provide some examples. Let W (t) be the Wiener process that is coming out zero. We calculate T W (t)dW (t). 0

For this we introduce the partition τ : 0 = t0 < t1 < . . . < tn−1 < tn = T . We present       W (T ) = W (t1 )−W (t0 ) + W (t2 )−W (t1 ) +. . .+ W (tn )−W (tn−1 ) . We find W 2 (T ) =

n−1  i=0

(W (ti+1 ) − W (ti ))2 +

49

Main Classes and Characteristics of Random Processes

+ 2(W (tn ) − W (tn−1 ))

n−2 

(W (ti+1 ) − W (ti )) +

i=0 n−3 

+ 2(W (tn−1 ) − W (tn−2 ))

(W (ti+1 ) − W (ti )) +

i=0

+ . . . + 2(W (t2 ) − W (t1 ))(W (t1 ) − W (t0 )) = =

n−1 

n−1 

2

(W (ti+1 ) − W (ti )) + 2

i=0

(W (ti+1 ) − W (ti ))W (ti ).

i=0

It follows from this that n−1 

W (ti )(W (ti+1 ) − W (ti )) =

i=0

n−1 1 2 1 W (T ) − (W (ti+1 ) − W (ti ))2 . 2 2 i=0

Since the first summand in the right part does not depend on the partition τ , we have T

 1 2 1 W (T ) − l.i.m. (W (ti+1 ) − W (ti ))2 . 2 2 i=0 n−1

W (t)dW (t) = 0

We prove that the last limit is equal to T , i.e., n−1 2  2 lim M (W (ti+1 ) − W (ti )) − T = 0, n→∞,δτ →0

(9.2)

i=0

where δτ = max (ti −ti−1 ). For this purpose we find the mathematii=1,...,n

cal n−1 

expectation

and

the

variance

of

sum

(W (ti+1 ) − W (ti ))2 . We obtain

i=0

M

n−1 

 (W (ti+1 ) − W (ti ))

2

=

i=0

=

n−1  i=0

n−1 

D{W (ti+1 ) − W (ti )} =

i=0

and D

n−1  i=0

M {W (ti+1 ) − W (ti )}2 = (9.3) (ti+1 − ti ) = T

i=0

 (W (ti+1 ) − W (ti ))

n−1 

2

=

n−1  i=0

 D (W (ti+1 ) − W (ti ))2 =

50

Chapter 1

=

n−1 





4

M (W (ti+1 ) − W (ti )) − M (W (ti+1 ) − W (ti ))

i=0

=

n−1 



2

2  =

M (W (ti+1 ) − W (ti ))4 − (ti+1 − ti )2 .

i=0

We find M (W (ti+1 ) − W (ti ))4 . We have 4

∞

M (W (ti+1 ) − W (ti )) = −∞

  1 x2 √ √ exp − x4 dx. 2(ti+1 − ti ) 2π ti+1 − ti

The last integral is the integral, ∞ √ −∞

  1 x2 exp − 2 x4 dx; 2σ 2πσ

it is found by a consecutive application of the formula of integration in parts: ∞ −∞

     ∞ 1 x2 1 x2 4 √ exp − 2 x dx = √ x3 d −σ 2 exp − 2 = 2σ 2σ 2πσ 2πσ −∞

   ∞  ∞ 2 x 1 x2 1  3 2 2 x (−σ ) exp − 2  + √ σ exp − 2 3x2 dx = =√ 2σ  2σ 2πσ 2πσ −∞

3σ =√ 2π 3σ 3 +√ 2π

−∞

∞ −∞

Therefore,

−∞

    ∞ x2 x2  3σ 2 xd −σ exp − 2 = √ x(−σ ) exp − 2  + 2σ 2σ  2π

∞



2

−∞

    ∞ 1 x2 x2 4 √ exp − 2 dx = 3σ 4 . exp − 2 dx = 3σ 2σ 2σ 2πσ −∞

M (W (ti+1 ) − W (ti ))4 = 3(ti+1 − ti )2 ,

and for required variance we obtain  n−1 n−1   2 3(ti+1 − ti )2 − (ti+1 − ti )2 = (W (ti+1 ) − W (ti )) = D i=0

i=0

51

Main Classes and Characteristics of Random Processes

=2

n−1 

(ti+1 − ti )2 ≤ 2

i=0

max

i=0,...,n−1

= 2T δτ → 0

as

(ti+1 − ti )

n−1 

(ti+1 − ti ) =

(9.4)

i=0

n → ∞, δτ → 0.

However, by a variance definition  n−1  2 (W (ti+1 ) − W (ti )) = D i=0 n−1 2 n−1   2 2 (W (ti+1 ) − W (ti )) − M (W (ti+1 ) − W (ti )) , =M i=0

i=0

or, considering (9.3), 2  n−1 n−1   2 2 (W (ti+1 ) − W (ti )) (W (ti+1 ) − W (ti )) − T . =M D i=0

i=0

Equality (9.2) now follows from this and from result (9.4). Therefore, T W (t)dW (t) = 0

1 2 1 W (T ) − T. 2 2

This example shows that with Ito’s integrals it is impossible to handle it in the same way as the usual integrals; in particular, the usual formula of the replacement of variables in an integral does not take place. Now we will determine an integral from a random process G(t) on segment [t0 , T ] T G(t)dt t0

as a limit in the mean-square of the corresponding integral sums: T G(t)dt = t0

l.i.m.

max(tk+1 −tk )→0 k

n−1 

G(τk )(tk+1 − tk ),

k=0

where t0 < t1 < . . . < tn−1 < tn = T , τk ∈ [tk , tk+1 ]. The last equal"T ity means that there is a random value Z (designated as G(t)dt) t0

such that

52

Chapter 1

lim

max(tk+1 −tk )→0

 2 n−1  M Z− G(τk )(tk+1 − tk ) = 0 k=0

k

for any choice of points τk ∈ [tk , tk+1 ]; or, in more detail, for any number ε > 0 there exists a number δ = δ(ε) > 0 such that for any partition t0 < t1 < . . . < tn−1 < tn = T and for any choice of points τk ∈ [tk , tk+1 ] the inequality 2  n−1  G(τk )(tk+1 − tk ) < ε M Z− k=0

holds if only max(tk+1 − tk ) < δ. k

It is possible to show that a necessary and sufficient condition "T for an existence of integral G(t)dt is an integrability of correlation t0

function KG (u, v) of process G(t) on the square t0 ≤ u, v ≤ T , i.e., an existence of multiple integral T T KG (u, v)dudv t0 t0

from nonrandom function KG (u, v). For example, the integral T W (t)dt,

(9.5)

0

where W (t) is the Wiener process, exists for any segment [0, T ] because the correlation function KW (u, v) is equal to min(u, v) and T T

T T KW (u, v)dudv =

0

0

u

T du

= 0

min(u, v)dudv = 0



T vdv +

0

If the integral

udv u

"t t0

0

T  = 0

 u2 1 + (T − u)u du = T 3 . 2 3

G(τ )dτ from a random process G(τ ) exists for

any t ∈ [t0 , T ], then we speak about the random process

Main Classes and Characteristics of Random Processes

53

t Y (t) =

G(τ )dτ,

t0 ≤ t ≤ T.

t0

It is possible to show that mathematical expectations mY (t), mG (t) and correlation functions KY (t1 , t2 ), KG (t1 , t2 ) of processes Y (t), G(t) are connected by ratios: t mY (t) =

t1 t2 mG (τ )dτ,

KY (t1 , t2 ) =

t0

KG (τ1 , τ2 )dτ1 dτ2 . t 0 t0

In particular,

t t DY (t) =

KG (τ1 , τ2 )dτ1 dτ2 t0 t0

is the variance of process Y (t). It is possible to show that an integral, as well as a derivative (in case of their existence), from a normal random process is again a normal process.1 For example, (9.5) is a normal random value with zero mathematical expectation and the variance T 3 /3. The process Zt ≡ Z(t) can be presented in the form t Zt = Z0 +

t G(t)dt +

0

X(t)dW (t), 0

where Wt ≡ W (t) is the Wiener process, and G(t) and X(t) belong to the class of random processes for which the corresponding integrals exist. In this case, we say that the process Zt has the stochastic differential dZt and we write dZt = Gt dt + Xt dWt . The last equality is called also a stochastic differential equation for an unknown random process Zt ; random processes Gt and Xt generally can depend on Zt . T h e o r e m. Let F (t, x) be a nonrandom function that is continuously differentiable on t ∈ [0, T ] and twice continuously differentiable 1 This statement is a consequence of a more general result that any linear transformation of normal process keeps the property of normality.

54

Chapter 1

on x ∈ (−∞, ∞). Then the random process F (t, Zt ) has a stochastic differential, and     ∂F (t, x)  ∂F (t, x)  dF (t, Zt ) = +Gt dt + ∂t x=Zt ∂x x=Zt   (9.6) ∂F (t, x)  1 ∂ 2 F (t, x)  2 + X dW + X dt. t t t ∂x  2 ∂x2  x=Zt

x=Zt

Formula (9.6) is called Ito’s formula. It establishes the rule of the replacement of variables in a stochastic integral. We will not provide the proof of this theorem. However, we will show how result (9.6) can be obtained at an intuitive level or, so to speak, at a physical level of severity. For this purpose we will write Taylor’s decomposition for the expression F (t + dt, Zt + dZt ) as function of two variables to within members of an order of no more than dt: F (t+dt, Zt +dZt ) ≈ F (t, Zt )+

∂F ∂F dt+ dZt + ∂t ∂x

1 ∂2F (dt)2 2 2 ∂t # $% &

+

it is possible to neglect

1 ∂2F ∂F ∂2F ∂F + dtdZt ≈ F (t, Zt )+ dt+ (Gt dt+Xt dWt ) + (dZt )2 + 2 2 ∂x ∂t∂x ∂t ∂x   1 ∂2F 2 2 2 2 + (dt) + 2G X dtdW +X (dW ) G + t t t t t $% & 2 ∂x2 # t  it is possible to neglect 2  ∂ F 2 + Gt (dt) + Xt dtdWt ≈ ∂t∂x $% & # it is possible to neglect

∂F ∂F 1 ∂2F 2 dt + (Gt dt + Xt dWt ) + X (dWt )2 , (9.7) ≈ F (t, Zt ) + ∂t ∂x 2 ∂x2 t where all partial derivatives of function F (t, x) undertake at point (t, Zt ). Since, T 0

(dWt )2 =

l.i.m.

max(tk −tk−1 )→0 k

n 

(Wtk − Wtk−1 )2 = T,

k=1

we can write (dWt )2 = dt and from (9.7) it follows that equality (9.6) holds.

Main Classes and Characteristics of Random Processes

55

We will consider the following example: if we suppose F (t, x) ≡ xn ,

Gt ≡ 0,

Xt ≡ 1.

then by formula (9.6) we get 1 d(Wt )n = nWtn−1 dWt + n(n − 1)Wtn−2 dt, 2 i.e., t Wtn

−

W0n

t nWτn−1 dWτ

= 0

+ 0

n(n − 1) n−2 Wτ dτ. 2

We will consider one more example. Let Zt be the random process of change of the market share price. Analyzing the 1912 data on the Parisian securities market, L.Bashelye came to the conclusion that M {ΔZt } = 0 and M {ΔZt }2 is proportional to Δt for small Δt, ˜ 2 Δt, where σ ˜ 2 is a coefficient of proportionality. i.e., M {ΔZt }2 = σ These conditions will be satisfied if ˜ dWt , dZt = σ

or

Zt = Z0 + σ ˜ Wt ,

where Wt is the Wiener process. Later, in 1965, P. Samuelson modified Bashelye’s model as follows1 dZt = μdt + σdWt . (9.8) Zt In other words, P. Samuelson assumed that a relative change in price of an action is described by a process like the Brownian motion. The first term on the right side of (9.8) is interpreted as a not random (and, therefore, predicted) tendency in the change of Zt , and the second term as a random component. We will find the solution of stochastic differential equation (9.8). For this purpose we have copied this equation in the form dZt = μZt dt + σZt dWt , 1 See: P.A. Samuelson, Rational Theory of Warrant Pricing, Industrial Management Review, 1965, vol. 6, pp. 13-31.

56

Chapter 1

and we use Ito’s formula (9.6). If F (t, x) ≡ ln x, then we have for G(t) ≡ μZt and X(t) ≡ σZt :       1  1  d(ln Zt ) = μZt dt + σZt dWt +   x x  1  1 − 2 + 2 x  

 x=Zt

x=Zt

x=Zt

  1 1 σ 2 Zt2 dt = μdt+σdWt − σ 2 dt = μ− σ 2 dt+σdWt , 2 2

i.e.,

t  ln Zt − ln Z0 = 0

 t 1 2 μ − σ dt + σdWt . 2 0

If μ and σ are constants, then from this equality it follows that   1 Z(t) = Z0 exp{μt} exp σWt − σ 2 t . 2 For the accepted model, the price of an action behaves as the bank   1 2 account Z0 exp{μt} with the random multiplier exp σWt − 2 σ t .

1.10 Spectral Density of a Random Process Let us consider a random process X(t) that can be represented in the form n  Vi ϕi (t), (10.1) X(t) = mx (t) + i=1

where mx (t), ϕ1 (t), . . . , ϕn (t) are nonrandom functions, and V1 , . . . , Vn are uncorrelated random variables with zero mathematical expectations: M {Vi } = 0, M {Vi Vj } = 0 ∀i ∀j = i. The function mx (t) has a sense of mathematical expectation of process X(t). Representation (10.1) is called a canonical expansion of process X(t). It is easily proved that the correlation function of process X(t) is calculated as follows: n  ϕi (t)ϕi (t )Di , (10.2) Kx (t, t ) = i=1

M {Vi2 }

is the variance of random variable Vi . The exwhere Di = pression (10.2) is called a canonical expansion of correlation function.

57

Main Classes and Characteristics of Random Processes

It is easily proved that the return situation too is fair: if the canonical expansion of correlation function for a random process X(t) is given by expression (10.2), then this process X(t) has the canonical expansion of view (10.1) with functions ϕi (t) (these functions are called coordinate functions) and random coefficients Vi such that Di = M {Vi2 }. Now let Y (t) be a stationary (in a broad sense) random process that is observed on an interval (0, T ); without the restriction of a community in further reasonings it is possible to consider M {Y (t)} ≡ 0. Let KY (t, t + τ ) = KY (τ ) be the correlation function of process Y (t); KY (τ ) is an even function, τ ∈ (−T, T ). It is known that such function can be expanded in Fourier series on cosines: ∞  Dk cos ωk τ, (10.3) KY (τ ) = k=0

where ωk = kω1 ,

ω1 =

π 2π = , 2T T

and coefficients Dk are determined by formulas 1 D0 = 2T

Dk =

1 T

T −T

1 KY (τ )dτ = T

T KY (τ ) cos ωk τ dτ = −T

2 T

T KY (τ )dτ, 0

T KY (τ ) cos ωk τ dτ

at

k = 0.

0

We have passed the expression (10.3) for the correlation function from argument τ to two arguments t and t by formula τ = t − t. Then KY (t, t ) = =

∞ 

∞ 

Dk cos ωk (t − t) =

k=0 

(Dk cos ωk t cos ωk t + Dk sin ωk t sin ωk t).

(10.4)

k=0

The expression (10.4) is the canonical expansion of the correlation function KY (t, t ). Coordinate functions from this canonical expansion are cosines and sines of frequencies that are multiple to ω1 : cos ωk t,

sin ωk t

(k = 0, 1, . . . ).

58

Chapter 1

Therefore, the random function Y (t) can be presented in the form of the canonical expansion: Y (t) =

∞ 

(Uk cos ωk t + Vk sin ωk t),

(10.5)

k=0

where Uk , Vk are uncorrelated random values with zero mathematical expectations and variances D{Uk } = D{Vk } = Dk . The expansion (10.5) is called a spectral expansion of the stationary random process. By (10.5), it follows that DY = D{Y (t)} =

∞ 

(Dk cos2 ωk t + Dk sin2 ωk t) =

k=0

∞ 

Dk ,

k=0

i.e., the variance of stationary random process is equal to the sum of variances for all harmonicas from its spectral expansion. We find a limit as T →∞ for a spectral expansion. We see that 2π 2π → 0 as T →∞ and the distance Δω = ωk − ωk−1 = 2T = ω1 ω1=2T between the frequencies ωk−1 and ωk decreases. By definition, put (T )

SY (ωk ) =

Dk . Δω

By the previous ratios, it follows that DY =

∞ 

(T )

SY (ωk )Δω,

k=0

KY (τ ) =

(T )

SY (ωk ) =

∞ 

2 π

T KY (τ ) cos ωk τ dτ, 0

(T ) SY (ωk ) cos ωk τ Δω.

k=0

We assume that there is the function SY (ω) such that lim

Δω→0 (T →∞)

∞ 

(T ) SY (ωk )Δω

k=0

In this case,

∞ =

SY (ω)dω. 0

∞ DY =

SY (ω)dω, 0

(10.6)

Main Classes and Characteristics of Random Processes

2 SY (ω) = π

59

∞ KY (τ ) cos ωτ dτ,

(10.7)

0

∞ KY (τ ) =

SY (ω) cos ωτ dω.

(10.8)

0

The function SY (ω) is called the spectral density of the stationary random process Y (t). Its sense is clear from formula (10.6); as a small Δω, the product SY (ω)Δω provides a contribution to the variance DY of the process Y (t) from the harmonious components of the spectral expansion of process Y (t) with frequencies that are in the interval (ω, ω + Δω). An expression of type (10.8) is known in mathematical analysis as Fourier’s integral. Fourier’s integral is a generalization of the Fourier series in the case of acyclic functions. We will also use the concept of normalized spectral density sY (ω); its definition is SY (ω) . sY (ω) = DY We will consider the following example. We will suppose that the normalized spectral density sY (ω) of the process Y (t) is equal to a constant at some interval of the frequencies, (ω1 , ω2 ), and the density sY (ω) is zero outside this interval:  s at ω ∈ (ω1 , ω2 ), sY (ω) = 0 at ω ∈ / (ω1 , ω2 ). We will find the normalized correlation function kY (τ ) of the random process Y (t). Using (10.6) we will define the value s as follows: ∞ 1 sY (ω)dω = 1, s · (ω2 − ω1 ) = 1, s = . ω2 − ω1 0

Then from (10.8) we obtain: ω2 kY (τ ) = ω1

=

1 sY (ω) cos ωτ dω = ω2 − ω1

ω2 cos ωτ dω = ω1

1 (sin ω2 τ − sin ω1 τ ) = τ (ω2 − ω1 )

60

Chapter 1

=

2 cos τ (ω2 − ω1 )



ω1 + ω2 τ 2



 sin

ω2 − ω 1 τ 2

 .

The function kY (τ ) has a character of harmonic oscillations with a 2 . Theamplitude frequency of ω1 +ω 2  of these oscillations changes un2 1 , i.e., also periodically, but with a τ der the law τ (ω2 −ω1 ) sin ω2 −ω 2 smaller frequency. The limiting form of function kY (τ ) is of interest as ω1 → ω2 : lim kY (τ ) = cos ωτ,

ω1 →ω2

where ω is a general value ω1 = ω2 = ω. This limit case corresponds to the situation when a spectrum of random process turns from continuous to discrete with the unique value ω. In this case, KY (τ ) = DY cos ωτ and (as stated above) the spectral decomposition of process Y (t) is Y (t) = U cos ωt + V sin ωt, where U and V are uncorrelated random values with zero mathematical expectations and variances D{U } = D{V } = DY . We will now consider a dynamic system, so that a operation of this system is described by the linear differential equation with the constant coefficients an y (n) (t) + an−1 y (n−1) (t) + . . . + a1 y  (t) + a0 y(t) = = bm x(m) (t) + bm−1 x(m−1) (t) + . . . + b1 x (t) + b0 x(t),

(10.9)

where y(t) is the reaction of system, and x(t) is the external influence. We have introduced the symbol pk as the operator of differentiation: dk pk = dt Then the equation (10.9) can be copied as k , k = 1, 2, . . . . follows: An (p)y(t) = Bm (p)x(t), where

An (p) = an pn + an−1 pn−1 + . . . a1 p + a0 , Bm (p) = bm pm + bm−1 pm−1 + . . . b1 p + b0 .

(10.10)

Main Classes and Characteristics of Random Processes

61

By (10.10), it follows that y(t) =

Bm (p) x(t). An (p)

The relation Bm (p)/An (p) is called a transfer function of the system. If an external influence is a random process X(t), then the reaction of the system is also some random process Y (t). If the process X(t) is a stationary process, then the process Y (t) is also a stationary process provided that the coefficients, aj , bj , i = 0, 1, . . . , n, j = = 0, 1, . . . , m, are constants. In this case, the system is called a stationary system. We will suppose that a stationary random process X(t) with a mathematical expectation of mx and a correlation function of KX (τ ) arrives at the entrance of the stationary linear system that is described by differential equation (10.9). For this case, we will find the mathematical expectation my and the correlation function KY (τ ) of the random process Y (t) on the exit of the system. From the theory of differential equations it is known that the solution y(t) to the equation (10.9) consists of two summands: y(t) = = y0 (t)+ yˆ(t). The summand y0 (t) represents the solution of equation (10.9) with zero right part (i.e., when x(t) ≡ 0) and this summand defines the proper motion of system. This motion is made by the system in the absence of the entrance if the system was removed from an equilibrium state at the initial moment t = 0. For so-called steady systems, the proper motion eventually fades. We will consider that system (10.9) is such. In this case, if we consider big values t from the initial moment t = 0, then we can reject the first summand y0 (t) and only study the second summand yˆ(t). This summand defines the forced system’s motion, which is a reaction to the external entrance influence x(t). It is known that if an external influence x(t) is the harmonic oscillation of a certain frequency, then the reaction of system y(t) is also the harmonic oscillation of the same frequency, but changed on an amplitude and a phase. The coordinate functions of the spectral decomposition of the process X(t) are harmonic oscillations. Therefore, if we define the reaction of system on the harmonic oscillation of the set frequency, then we can find the coordinate functions of the spectral decomposition of process Y (t) and all others characteristics of this process. So, let the harmonic oscillation

62

Chapter 1

x∗ (t) = cos ωt arrive at the entrance of thesystem. We find the reaction of system y ∗ (t) = C(ω) cos(ωt + α(ω)), where C(ω) and α(ω) are unknown functions. We can simplify a technical aspect of the problem if we consider x∗ (t) as the valid part Re{x(t)} of a complex-valued function x(t) = eiωt ,

(10.11)

where eiωt = cos ωt + i sin ωt. If the complex-valued reaction y(t) corresponds to the complex-valued entrance influence x(t), then y ∗ (t) = = Re{y(t)}. We find the reaction y(t) as y(t) = Φ(iω)eiωt ,

(10.12)

where Φ(iω) = C(ω)eiα(ω) is a unknown complex multiplier. To find Φ(iω) we need to substitute functions (10.11) and (10.12) in the equation (10.9). Since at any k dk iωt e = (iω)k eiωt , dtk

dk  Φ(iω)eiωt = (iω)k eiωt Φ(iω), dtk

we have   Φ(iω) an ·(iω)n + an−1 ·(iω)n−1 + . . . + a1 ·(iω) + a0 = = bm ·(iω)m + bm−1 ·(iω)m−1 + . . . + b1 ·(iω) + b0 , or Φ(iω) =

Bm (iω) . An (iω)

The function Φ(iω) is called an amplitude-frequency characteristic of the linear system. From the last equality, it follows that to find the amplitude-frequency characteristic we can substitute iω instead of the operator of the differentiation p in the transfer function of a system. If the harmonic oscillation eiωt arrives at the entrance of a linear system with constant parameters, then the reaction of the system is

63

Main Classes and Characteristics of Random Processes

presented by the product with the same harmonic oscillation and the amplitude-frequency characteristic of system Φ(iω). If the influence x(t) = U1 eiω1 t + U2 eiω2 t arrives at the entrance of the system, then, owing to the linearity of the system, the reaction y(t) = U1 Φ(iω1 )eiω1 t + U2 Φ(iω2 )eiω2 t . Now let the stationary random process ◦

X(t) = mx + X(t) arrive at the entrance of the system. Here, mx = M {X(t)} = const is ◦

the mathematical expectation of the process X(t), X(t) = X(t) − mx . The mathematical expectation mx can be considered as a harmonic oscillation of zero frequency ω = 0. Therefore, the mathematical expectation M {Y (t)} of system’s reaction is   b0 iωt  = mx . my = mx Φ(iω)e  a0 ω=0 ◦

We now find a transformation in the second summand X(t). For ◦ this we present the function X(t) at the interval (0, T ) as the spectral decomposition ∞  ◦ (Uk cos ωk t + Vk sin ωk t), (10.13) (t) = X k=0

where ω0 = 0, ωk = kω1 , ω1 = π/T ; Uk , Vk are uncorrelated random values, and D{Uk } = D{Vk } = Dk , M {Uk } = M {Vk } = 0. It is easily proved that decomposition (10.13) can be written down in the following complex form: ∞  ◦ Wk eiωk t , X(t) = where

ωk = kω1 ,

k=−∞

⎧ at ⎨ U0 at (Uk − iVk )/2 Wk = ⎩ (U|k| + iV|k| )/2 at k = 0, ±1, ±2, . . . .

k = 0; k = 1, 2, . . . ; k = −1, −2, . . . ;

64

Chapter 1

It is easily proved that Wk are uncorrelated random values with zero mathematical expectations: M {Wk } = 0, M {Wi Wj } = 0 at i = j. By definition, the variance D{Wk } of the complex random value Wk is D{Wk } = M {|Wk − M {Wk }|2 }. From this it follows that D{W0 } = D0 ,

D{Wk } =

1 D|k| at k = 0. 2

(10.14)

◦

If Y (t) = my + Y (t) is the reaction of system (i.e., the process on an exit of system), then ∞ 

◦

Y (t) =

Wk Φ(iωk )eiωk t .

k=−∞

˜ k = Wk Φ(iωk ). Then By definition, put W ◦

Y (t) =

∞ 

˜ k eiωk t , W

(10.15)

k=−∞

˜ k are uncorrelated complex random values with zero mathwhere W ematical expectations. We find the variance of the complex random ˜ k in decomposition (10.15). We have, value W ˜ k } = M {|Wk Φ(iωk )|2 } = M {|Wk |2 |Φ(iωk )|2 } = D{W = |Φ(iωk )|2 M {|Wk |2 } = |Φ(iωk )|2 D{Wk }.

(10.16)

If we pass from a complex form of decomposition (10.15) to a valid form, we obtain ◦

Y (t) =

∞ 

˜k cos ωk t + V˜k sin ωk t), (U

k=0

˜ 0, U ˜k = W ˜k + W ˜ −k , V˜k = i(W ˜k − W ˜ −k ), k = 1, 2, . . . , ˜0 = W where U ˜k } = ˜k } = M {V˜k } = 0, D{U are uncorrelated valid random values, M {U = D{V˜k } = Dk ; at the same time ˜ 0} = D ˜ 0, D{W

˜ k} = D{W

1˜ D|k| at k = 0. 2

(10.17)

Combining (10.14), (10.17) and (10.16), we get ˜ k = |Φ(iωk )|2 Dk . D

(10.18)

Main Classes and Characteristics of Random Processes

65

As above, we will pass in a spectral representation of random processes to a limit as T → ∞, and from a discrete range to a spectral density. From (10.18) it follows that such transition will yield the following result: SY (ω) = |Φ(iω)|2 SX (ω), (10.19) where SX (ω) is a spectral density of the process X(t) and SY (ω) is a spectral density of the process Y (t). Formula (10.19) is called Hinchin’s formula. Hinchin’s formula allows us to solve the problem about finding the correlation function KY (τ ) of the random process Y (t) at the exit of the stationary linear system if a process X(t) with known correlation function KX (τ ) arrives at an entrance of this system. For this purpose, we first find the spectral density SX (ω) of process X(t): 2 SX (ω) = π

∞ KX (τ ) cos ωτ dτ. 0

Then we determine the transfer function Φ(p) of the linear system and by the formula (10.19) we find the spectral density SY (ω) of the process Y (t). At last, using SY (ω), we obtain the correlation function in question by the formula ∞ KY (τ ) =

SY (ω) cos ωτ dω. 0

Chapter 2 Crossings Problems

2.1 Problem Formulation and Examples from Application Domains We consider a realization or, to use different terminology, a sample function x(t) of the one-dimensional random process X(t) of the continuous argument t, where the argument t is changed on any finite interval; for example, [0, T ]. The sample function x(t) is represented in Fig. 2–1. We will assume it is continuous. By x0 we denote some fixed number. A relative positioning of this realization x(t) and the level x0 can be described by means of the following parameters (see Fig. 2–1): the moment t∗ from the first achievement of level x0 by the realization of x(t); a number of crossings N + of level x0 by the realization x(t) from below to top (in Fig. 2–1 N + = 3); a number of crossings N − of level x0 by the realization x(t) from top to down (in Fig. 2–1 N −= 2); an interval τ + between two consecutive crossings of level x0 from below to top and from top to down (i.e., at any intermediate moment t from this interval the condition x(t) ≥ x0 is satisfied); an interval τ − between two consecutive crossings of level x0 from top to down and from below to top (i.e., at any intermediate moment t from this interval the condition x(t) ≤ x0 is satisfied); a number of local maxima n exceeding the level x0 ; a height h of a local maximum exceeding the level x0 ; a height hmax of the greatest of local maxima; and a moment tmax of the achievement of hmax : x(tmax ) = hmax . This list can be expanded if we consider similar parameters related to the local minima of the realization x(t). The parameters h, τ + , τ − can take several meanings for one realization.1 These parameters together with the parameters t∗, N + , N − , n, tmax , and hmax change randomly 1 This depends on the chosen level x , the considered length T , and other 0 properties of the realization x(t).

67

Crossings Problems

from one realization to another: i.e., are random values. Statistical characteristics of these random values and probabilities of related events are the subject of many studies. x(t) 6 hmax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............... . . ... ..... ... . .. ... .. .. .. .... . . . . . . . . . . .. ..... . ... . ..... . . . . . . . . . . ... . ... .. . .. ... ... ... ... .. . .. ... .. .. .... .. ... .. . . . .... . ... ... . . .. . . . ... . .. ..... .. .. . . . . ..... . . . . . . . . . − + . . x0 . . . . . . . . . . ..... τ -..... . . . . . . . . . . . . . . . . ..... . . . .... . . ..... τ -.... . . . .... .. ... . . .... . .. .. .... ... .. .. ... ... .. . .. .. ... ... .. . . . ... . . . . . .... .. . . . ... ... .. .. .. ... ...... .. ... . .. .. . . . . . . . . . ... ... ... ... . . .. .. .. ... . ... . ... ... . .... . ..... .. ... .. .. ... ... . .. .. ... ... . . . . . . . . . ... .. ... .. ... .. .. . ... .. .. . . . . .... ..... . . . . . ... .. ... .. ... .. .. .. ............ .. ..... ........ ... ... .. .. .. ......... .. .. .. .. .. .. . . -t . ∗ t tmax T Fig. 2–1

We will give a number of examples from various application domains. These examples show a range of applicability and confirm the great practical significance of the studies on crossings problems. 1. An electronic relay is often applied in various radio engineering devices. Therefore, the processing of useful signals is always carried out in the presence of various hindrances. The impact on the relay from useful signals and hindrances depends on the ratio of the threshold tension in the operation of the relay and the hinderances’ intensity. There are false operations in the relay when the hindrances’ intensity is comparable with the threshold tension. These operations create errors in the functions of the contained relay devices. A number of false operations will be defined by the number N + , where N + is the number of crossings of the threshold tension by the hindrance from below to top. 2. A number of problems from waiting theory is reduced to that of crossing problems. The crossing of a level from below to top can be interpreted, for example, as a receipt of a random demand for a

68

Chapter 2

service, while the interval τ + can be interpreted as a holding time and the interval τ − can be interpreted as an idle time between servicings. 3. Problems related to the outliers of random processes arise in the research into various economic problems; for example, consider the well-known problem about a ruin: when an economic agent is ruined if the agent debt described by a random process surpasses some critical size. 4. Problems of reliability theory are one more example of problems when there is a need for research into the outliers from random processes. In this case, the number N + can characterize, for example, the number of refusals of a considered system during T ; the value τ + can characterize the time that is taken away by the elimination of a malfunction; and the value τ − can characterize the time of a no-failure operation after the elimination of the next malfunction. 5. Many problems related to estimating the stability of various systems are reduced to finding the probabilistic characteristics of the moment t∗ of the first achievement by the random process of a given level. As an example, consider the well-known problem about tracking failure. Its essence is as follows: measuring instruments are used in many technical systems and the purpose of these measuring instruments consists in the implementation of the automatic tracking of given parameters with an acceptable margin for error. In the presence of random hindrances, the tracking error will be random and can reach such a great value that further tracking will practically stop: i.e., there will be a tracking failure characterized by moment t∗ of the first achievements by an error from given borders. 6. The problem about estimating the accuracy of an airplane landing is another example of when the probabilistic properties of moment t∗ are used. This is required to estimate the probability that landing (i.e., the initial contact between an airplane and a landing surface) will be on a given interval (l1 , l2 ), where l is the distance counted from the forward edge of the landing strip. If the random process H(l) is the height of flight, this problem is reduced to estimating the probability that the moment l∗ from the first achievement of level h = 0 by the process H(l) will belong to the interval (l1 , l2 ). 7. We now suppose that the random process X(t) describes a number of individuals in a biological population. Such a process is often considered in studies of the dynamics of various biological systems. The nature of the functioning of such a system can qualitatively change at the moment of achievement by the process X(t) from some

Crossings Problems

69

critical borders. For example, properties of this system can be such that if X(t) falls below some critical level x0 , then the population is doomed to extinction. Therefore, in order to find the population lifetime it is necessary to estimate the probability of the achievement of level x0 by the process X(t) prior to a given moment T . 8. In many application domains there are problems related to the study of the extreme values of random processes when they describe changes of various physical magnitudes (temperatures, accelerations, speeds, pressure, electric and mechanical tension, and so on). For example, distributions of the number of maxima n and their heights h are used: 1) in calculations of the indicators of safety and risk in the creation of cars and designs; 2) in statistical studies of floods and droughts; 3) in calculations of the maximum loadings in power supply networks of industrial enterprise; 4) in studies of sea waves and rolling of courts; 5) in calculations of durability materials; 6) for the quantitative assessment of the roughnesses of rough surfaces. These examples give an idea of the range of the application and practical importance of the research into crossings problems.

2.2 Average Number of Crossings The decisions on the applied problems listed above are based on fundamental mathematical results that have been obtained in the last 50 − 60 years. Many specialists studied problems about crossings of a level by a random process. S.O. Rise was the first person to begin the systematic study of these problems.1 Using heuristic methods, he obtained a number of major results; in particular, a formula for the average number of crossings of a fixed level. Decades and efforts of many mathematicians were required for the identification of the weakest conditions of justice in these results and the mathematically faultless formulations and proof of the corresponding statements and their further generalization. Some of the most well known and important for applications results will be provided in this and a number of the subsequent sections. Firstly, as a rule, the essence of the result will be formulated; then a derivation of this result with a physical level of strictness will be given; and finally, an appropriate mathematically rigorous statement will be formulated. 1 S.O. Rice, Mathematical Analysis of Random Noise, Bell System Tech. J., 1945, vol. 24, no. 1, pp. 46-156.

70

Chapter 2

Let X(t) be a random process and u be a fixed number. We suppose that X(t) is continuous with probability 1 and differentiated in the mean-square on the interval (t0 , T ). We assign Nu+ (t0 , T ) and Nu− (t0 , T ) to each realization x(t) of the process X(t), where Nu+ (t0 , T ) is the number of crossings by this realization of the level u from below to top on the interval (t0 , T ) and Nu− (t0 , T ) is the number of crossings by this realization of the level u from top down on the interval (t0 , T ). The numbers Nu+ (t0 , T ) and Nu− (t0 , T ) are random values. Suppose that the mathematical expectations Nu+ (t0 , T ) = M {Nu+ (t0 , T )} and Nu− (t0 , T ) = M {Nu− (t0 , T )} exist. Numbers Nu+ (t0 , T ) and Nu− (t0 , T ) are called the average numbers of crossings of the level u respectively from bottom to top and from top to bottom on the interval, (t0 , T ). We will show that Nu+ (t0 , T )   Here, ft (u, y) = ft (x, y)

T =

∞ dt

t0

yft (u, y)dy.

(2.1)

0

, ft (x, y) is the density of the joint distrix=u

bution of random values X(t) and Y (t), where Y (t) is a derivative in the mean-square of process X(t). Let us divide the interval (t0 , T ) into non-crossing intervals (t0 , t1 ), [t1 , t2 ), . . . , [tj−1 , tj ), . . . , [tn−1 , T ), where t0 < t1 < . . . < tj−1 < tj < . . . < tn−1 < tn = T. We denote Δtj = tj − tj−1 , j = 1, 2, . . . , n. To each realization x(t) assign n random values V1 , V2 , . . . , Vn , where by definition Vj = 1 if the realization x(t) crossed the level u from below to top on the interval [tj−1 , tj ) and Vj = 0 if the realization x(t) did not cross the level u from below to top on the interval [tj−1 , tj ). Then Nu+ (t0 , T )

=

n 

Vj ,

j=1 n   Nu+ (t0 , T ) = M Nu+ (t0 , T ) = M {Vj }. j=1

From the definition of Vj it follows that

71

Crossings Problems

M {Vj } = 0 · P {Vj = 0} + 1 · P {Vj = 1} = P {Vj = 1}. Considering that the interval [tj−1 , tj ) is sufficiently small and neglecting the opportunity for more than one crossing of level u on this interval, we get P {Vj = 1} = P {X(tj−1 ) < u < X(tj )} = P {u−ΔXj < X(tj−1 ) < u}, where ΔXj = X(tj ) − X(tj−1 ). Assuming ΔXj = Y (tj−1 )Δtj , we can consider P {u−ΔXj < X(tj−1 ) < u} = = P {u−ΔXj < X(tj−1 ) < u; Y (tj−1 ) > 0} =

(2.2)

= P {u−Y (tj−1 )Δtj < X(tj−1 ) < u; Y (tj−1 ) > 0}. The last probability can be expressed as ∞ ftj−1 (u, y)yΔtj dy.

(2.3)

0

Thus, n 

M {Vj } =

j=1

n 

∞ Δtj

j=1

ftj−1 (u, y)ydy. 0

From this, passing to the limit max Δtj → 0 (n → ∞), we obtain j=1,...,n

the declared result (2.1). The formula Nu− (t0 , T )

T

0

=−

dt

yft (u, y)dy

(2.4)

−∞

t0

is established by similar reasonings. By Nu (t0 , T ) we denote an average number of crossings of level u on interval (t0 , T ). Suppose Nu (t0 , T ) = Nu+ (t0 , T ) + Nu− (t0 , T ), then combining this equality with (2.1) and (2.4), we get T Nu (t0 , T ) =

∞ dt

t0

−∞

|y|ft (u, y)dy.

(2.5)

72

Chapter 2

Of course, the results (2.1), (2.4), (2.5) were not obtained strictly or, as it is sometimes said, on a physical level of stictness. We will now formulate the corresponding exact and strictly mathematically provable statements.1 Let us introduce the following definitions. D e f i n i t i o n 1. Gu (t1 , t2 ) is a set of scalar functions continuous on the interval from t1 to t2 that do not identically equal u on any subinterval inside this interval; any interval from intervals [t1 , t2 ], [t1 , t2 ), (t1 , t2 ], (t1 , t2 ) can be understood as an interval from t1 to t2 . D e f i n i t i o n 2. The function h(t) ∈ Gu (t1 , t2 ) has the crossing of level u at the point t∗ ∈ (t1 , t2 ) if for any ε > 0 there exist t˜ ∈ (t∗−ε, t∗+ε) and tˆ ∈ (t∗−ε, t∗+ε) such that (h(t˜)−u)(h(tˆ)−u) < 0. We will consider processes X(t) such that with probability 1 sample functions x(t) of X(t) belong to the set Gu (t1 , t2 ). This condition is satisfied if the sample functions are continuous with probability 1 on the corresponding interval and the equality P {X(tr ) = u} = 0 holds for every rational point tr from this interval. Indeed, since the set of rational numbers is countable, we can number them: tr1 , tr2 , . . . . If some realization x(t) is identically equal to the value u on some subinterval of the interval (t1 , t2 ), then at least at one rational point trn it holds that x(trn ) = u. Therefore, the probability of the event (that the process X(t) is identically equal to the value u on some subinterval of the interval (t1 , t2 )) does not exeed the probability  ∞ ∞   {X(tri ) = u} ≤ P {X(tri ) = u} = 0. P i=1

i=1

By gt,τ (x, y) we denote the joint distribution density of X(t) and Y (t) ≡

X(t+τ )−X(t) . τ

T h e o r e m 1.2 Suppose 1) with probability 1 sample functions of the process X(t) belong to the set Gu (t1 , t2 ); 2) the function gt,τ (x, y) is continuous on the set of variables (t, x) at every τ, y; 3) there is a function ft (x, y) so that gt,τ (x, y) → ft (x, y) as τ → 0 evenly for (t, x) at each y; 4) there is the function h(y) so that 1 It is possible to find proof of these statements in a well-known work by Cramer and Leadbetter: H. Cramer and M.R. Leadbetter, Stationary and Related Stochastic Processes, New York-London-Sydney: Wiley, 1967. 2 This result was proved for the first time in the work: M.R. Leadbetter, On Crossings of Levels and Curves by a Wide Class of Stochastic Processes, Ann. Math. Statist., 1966, vol. 37, no. 1, pp. 260-267.

73

Crossings Problems

gt,τ (x, y) ≤ h(y) for all t, τ, x, and ∞ |y|h(y)dy < ∞. −∞

Then

t2 Nu (t1 , t2 ) =

∞ dt

t1

|y|ft (u, y)dy < ∞.

(2.6)

−∞

The proof of this result is based on the following lemma: L e m m a 1. Suppose for each natural n a process Xn (t) coincides with the process X(t) at points tn,r = 2rn , r = 1, 2, . . . , 2n , and arcwise changes between these points. We denote by Cn the number of crossings of the level u by the process Xn (t) on the interval (t1 , t2 ); by C, the number of crossings of level u by the process X(t) on this interval. Then with probability 1 Cn → C as n → ∞. The average number of crossings Nu (t1 , t2 ) is the mathematical expectation of the random value C. Proof of the formula (2.6) is based on the study of the random value Cn and the mathematical expectation M {Cn } as n → ∞. By X  (t) we denote a derivative in the mean-square of the process X(t). By definition, put K(t, τ ) = M {(X(t) − M {X(t)})(X(τ ) − M {X(τ )})}, σ 2 (t) = K(t, t), k1 (t) =

σ12 (t) = M {(X  (t) − M {X  (t)})2 },

M {(X(t) − M {X(t)})(X  (t) − M {X  (t)})} . σ(t)σ1 (t)

If X(t) is a normal process, Theorem 1 is formulated as follows. T h e o r e m 1a. Suppose 1) X(t), t ∈ [t1 , t2 ], is a normal process so that the sample functions of X(t) are continuous with probability 1; 2) mathematical expectation M {X(t)} ≡ m(t) has the continuous derivative dm dt , t ∈ [t1 , t2 ], and the correlation function K(t, τ ) has 2

K(t,τ ) the mixed derivative ∂ ∂t∂τ , which is continuous at all points of type (t, t); 3) a joint normal distribution of X(t) and X  (t) is not degenerated at everyone t, σ(t) > 0 and |k1 (t)| < 1. Then the formula (2.6) holds for an average number of crossings Nu (t1 , t2 ). If X(t) is a stationary normal process, Theorem 1 is formulated as follows.

74

Chapter 2

T h e o r e m 1b. Suppose 1) X(t), t ∈ [t1 , t2 ] is a stationary normal process with the mathematical expectation m and the correlation function K(τ ) ≡ M {(X(t + τ ) − m)(X(t) − m)}, 0 < K(0) < ∞; sample functions of X(t) are continuous with probability 1; 2) K(τ ) has the finite second derivative K  (0). Then the formula (2.6) holds for an average number of crossings Nu (t1 , t2 ). In this special case, the density ft (x, y) does not depend on t and is equal to (see Chapter 1)    1 1 (x − m)2 y2 exp − + , ft (x, y) = 2πσσ1 2 σ2 σ12 where σ 2 = K(0), σ12 = −K  (0); the integral in the right side of (2.6) is easily calculated: t2 Nu (t1 , t2 ) = t1

=

∞ |y|ft (u, y)dy =

dt −∞

   ∞  t2 − t1 (u − m)2 y2 exp − |y| exp − dy = 2πσσ1 2σ 2 2σ12 −∞



=

(u − m)2 (t2 − t1 ) σ1 exp − π σ 2σ 2

 .

This simple expression yields an average number of crossings of a fixed level by a stationary normal process on a given interval.

2.3 Finiteness of a Number of Crossings and Absence of Touchings In the previous section we defined the concept of a level crossing. We also considered crossings of a level from bottom to top and from top to bottom. Such crossings are called upcrossings and downcrossings. The exact definitions of these concepts are formulated below.1 1 The below definitions have been introduced in the work: H. Cramer and M.R. Leadbetter, Stationary and Related Stochastic Processes, New York-LondonSydney: Wiley, 1967.

Crossings Problems

75

D e f i n i t i o n 3. The function h(t) ∈ Gu (t1 , t2 ) at point t∗ ∈ (t1 , t2 ) has an upcrossing of level u (crossing of level u from bottom to top) if there is ε > 0 such that h(t) ≤ u for any t ∈ (t∗ − ε, t∗ ) and h(t) ≥ u for any t ∈ (t∗ , t∗ + ε). D e f i n i t i o n 4. The function h(t) ∈ Gu (t1 , t2 ) at point t∗ ∈ (t1 , t2 ) has a downcrossing of level u (crossing of level u from top to bottom) if there is ε > 0 such that h(t) ≥ u for any t ∈ (t∗ − ε, t∗ ) and h(t) ≤ u for any t ∈ (t∗ , t∗ + ε). The concept a touching of a level is defined as follows. D e f i n i t i o n 5. The function h(t) ∈ Gu (t1 , t2 ) at point t∗ ∈ (t1 , t2 ) has a touching of level u if h(t∗ ) = u and there exists ε > 0 such that h(t)−u does not change a sign on an interval (t∗ − ε, t∗ + ε). Definitions 1-5 imply that if h(t∗ ) = u then t∗ will be either a crossing point or a touching point for level u. Upcrossings and downcrossings are both crossings, but there may be crossings that are neither upcrossings nor downcrossings. For instance, function  t sin 1t at t = 0, h(t) = 0 at t = 0, has the crossing of level u = 0 at point t = 0, which is neither an upcrossing nor a downcrossing. It is also obvious that if the number of crossings is finite then any crossing is either an upcrossing or a downcrossing. In studies of crossing problems, it is often necessary to solve the question of the finiteness of a number of crossings and the absence of touchings. In one of the earliest works1 on this subject, it was shown that in order to ensure that there are no touchings of level u with probability 1 and the average number of crossings Nu (t1 , t2 ) is finite, it suffices to require that for every fixed t the one-dimensional distribution density of the process X(t) is bounded and that the sample functions x(t) are continuously differentiable with probability 1 on the corresponding interval. This question has also been studied by many mathematicians with regard to the general form of processes and the particular case of normal processes. The results of this research are formulated below in Theorems 2, 2a, and 2b.2 1 E.V. Bulinskaya, On the Average Number of Times a Stationary Gaussian Process Crosses a Certain Level, Theor. Veroyat. Primen., 1961, vol. 6, no. 4, pp. 474-478. 2 See: H. Cramer and M.R. Leadbetter, Stationary and Related Stochastic Processes, New York-London-Sydney: Wiley, 1967.

76

Chapter 2

T h e o r e m 2. Suppose the conditions of Theorem 1 hold; then the probability of touching the level u by the process X(t) on the interval (t1 , t2 ) is equal to zero, Nu+ (t1 , t2 ) + Nu− (t1 , t2 ) = Nu (t1 , t2 ), and Nu+ (t1 , t2 )

Nu− (t1 , t2 )

t2 =

∞ dt

t1

0

t2

0

=

dt t1

(3.1)

yft (u, y)dy,

(3.2)

|y|ft (u, y)dy,

(3.3)

−∞

Nu+ (t1 , t2 )

where is the average number of upcrossings of level u by the process X(t) on the interval (t1 , t2 ) and Nu− (t1 , t2 ) is the average number of downcrossings of level u by the process X(t) on the interval (t1 , t2 ). T h e o r e m 2a. Under the conditions of Theorem 1a, an average number of upcrossings of the level u by the process X(t) on the interval (t1 , t2 ) is given by formula (3.2), an average number of downcrossings of the level u by the process X(t) on the interval (t1 , t2 ) is given by formula (3.3), the probability of touching is equal to zero, and equality (3.1) holds. T h e o r e m 2b. Under the conditions of Theorem 1b, the probability of touching the level u by the process X(t) on the interval (t1 , t2 ) is equal to zero and   (u − m)2 Nu (t1 , t2 ) (t2 − t1 ) σ1 = exp − . Nu+ (t1 , t2 ) = Nu− (t1 , t2 ) = 2 2π σ 2σ 2

2.4 Characteristics of Extreme Values of a Random Process We will consider the random process X(t) that is twice differentiated on interval (t1 , t2 ). Let ft (x, y, z) be the density of the joint distribution of the process X(t) and the derivatives X  (t) and X  (t) at the same moment t. It is necessary to find the average number of local maxima Nmax (t1 , t2 ) of the process X(t) on the interval (t1 , t2 ).

77

Crossings Problems

Note that the maximum of the process X(t) corresponds to the downcrossing of the zero level by the process X  (t) at the same moment t. Therefore, Nmax (t1 , t2 ) is equal to an average number of downcrossings Nu− (t1 , t2 ) of the zero level by the process X  (t) on the interval (t1 , t2 ), i.e., t2

0

Nmax (t1 , t2 ) =

|z|f˜t (0, z)dz,

dt −∞

t1

where

∞ f˜t (y, z) =

ft (x, y, z)dx −∞

is the density of the joint distribution of random values X  (t) and X  (t). Similarly, for an average number of local minima Nmin (t1 , t2 ) we get t2 ∞ Nmin (t1 , t2 ) = dt z f˜t (0, z)dz. 0

t1

Let us consider the minima of the process X(t) on the interval (t1 , t2 ). By Nmin≤u (t1 , t2 ) denote the average number of minima that do not exceed level u. This number is equal to the average number of upcrossings of the zero level by the process X  (t) on the interval (t1 , t2 ) if only such upcrossinngs are considered when the condition X ≤ u holds at the moment of upcrossing. By almost repeating the nonrigorous reasoning from section 2.2 in the derivation of (2.1) (within this difference the process X  (t) will play a role in process X(t) and the condition X ≤ u will be added to the calculating probabilities (2.2)), we obtain t2 Nmin≤u (t1 , t2 ) =

u dt

t1

−∞

∞ dx

zft (x, 0, z)dz. 0

Similarly, for the average number of local maxima Nmax≤u (t1 , t2 ) we get t2 u 0 Nmax≤u (t1 , t2 ) = dt dx |z|ft (x, 0, z)dz. (4.1) t1

−∞

−∞

78

Chapter 2

Further, we will consider only maxima. Formulas for minima are obtained by means of similar reasonings. Let us consider the local maxima that range from u−Δu to u. From formula (4.1) it follows that at a small Δu an average number of these maxima on the interval (t1 , t2 ) is approximately equal to t2

0 |z|ft (u, 0, z)dz.

dt

Δu

−∞

t1

Let g(u) be a function of the distribution density for the height of a local maximum; then at the small Δu the product g(u)Δu is approximately equal to the probability that a height of a local maximum is in the range from u−Δu to u. We may assume that this probability A , where A is the average number of maxima that have is equal to B heights in the range from u−Δu to u, and B is the average number Nmax (t1 , t2 ) of all maxima. Then we get: "t2 g(u) =

"0

dt

t1

"t2 t1

−∞ "0

|z|ft (u, 0, z)dz

dt

−∞

.

(4.2)

|z|f˜t (0, z)dz

It can be proved that result (4.2) holds. We will formulate corresponding theorems for stationary normal processes. Let X(t) be a stationary process that has a continuous with probability 1 derivative in the mean square X  (t). We will suppose that X(t) and X  (t) have continuous one-dimensional functions of distribution and we suppose that an average number of crossings of level u by process X(t) and an average number of crossings of the zero level by process X  (t) are finite. The process X(t) has a local maximum at the point t = t0 if the process X  (t) has a downcrossing of the zero level at the point t = t0 . We denote by D = D (t0 − τ, t0 ) a number of downcrossings of the zero level by the process X  (t) on the interval (t0 − τ, t0 ); by Du = Du (t0 − τ, t0 ), a number of downcrossings such that the condition X(t) ≤ u holds at the moments of these downcrossings. Then, Nmax (t0 − τ, t0 ) = M {D (t0 − τ, t0 )}, Nmax≤u (t0 − τ, t0 ) = M {Du (t0 − τ, t0 )}, where M is a sign of a mathematical expectation.

79

Crossings Problems

By definition, put G(u) = lim P {Du (t0 − τ, t0 ) ≥ 1|D (t0 − τ, t0 ) ≥ 1} τ →0

(4.3)

if a limit of a given conditional probability exists. L e m m a 2.1 The limit in (4.3) exists and νu , G(u) = ν where ν = M {D (0, 1)} = Nmax (0, 1) is the average number of local maxima of the stationary process X(t) on a unit interval; νu = M {Du (0, 1)} = Nmax≤u (0, 1) is the average number of local maxima that do not exceed a level u. The above function G is a function of distribution. Since the process X(t) is stationary, we have Nmax (t1 , t2 ) = (t2 − t1 ) · Nmax (0, 1) = (t2 − t1 )ν, Nmax≤u (t1 , t2 ) = (t2 − t1 ) · Nmax≤u (0, 1) = (t2 − t1 )νu . T h e o r e m 3. Let X(t) be a stationary normal process and a correlation function of this process have the finite fourth derivative at zero: K (IV ) (0) < ∞. Then u 0 νu = dx |z|ft (x, 0, z)dz, (4.4) −∞

−∞

where ft (x, y, z) is the joint distribution density of values X(t), X  (t), and X  (t).2 The formula for ν follows from Theorem 2b. Indeed, since ν is the average number of downcrossings of the zero level by the process X  (t) on the unit interval and process X  (t) satisfies all conditions of Theorem 2b, we have 1 Lemma 2 and next Theorem 3 are proved in the work: H. Cramer and M.R. Leadbetter, Stationary and Related Stochastic Processes, New York-LondonSydney: Wiley, 1967. 2 Under the conditions of Theorem 3, this density does not depend on t because the process is stationary.

80

Chapter 2

0 |z|f˜t (0, z)dz,

ν=

(4.5)

−∞

where f˜t (y, z) is the joint distribution density of values X  (t) and X  (t).1 In the case of a stationary normal process, this integral for ν is easily calculated. Indeed, if X(t) is the stationary normal process with the correlation function K(τ ), then (see Chapter 1) X  (t) is a stationary normal process with zero mathematical expectation and the correlation function −K  (τ ) and    1 1 y2 z2 ˜ ft (y, z) = exp − + 2 , 2πσ1 σ2 2 σ12 σ2 where σ12 = −K  (0), σ22 = K (IV ) (0). We obtain 0 |z|f˜t (0, z)dz =

ν= −∞

1 = 2πσ1 σ2

∞ 0



1 2πσ1 σ2

z2 z exp − 2 2σ2





0 −∞

z2 |z| exp − 2 2σ2

1 1 σ2 = dz = 2π σ1 2π

' −

 dz =

K (IV ) (0) . K  (0)

Note that formula (4.5) can be considered as a special case of formula (4.4) when u → ∞. From Lemma 2 and Theorem 3 it follows that for a stationary normal process X(t) that satisfies the conditions of Theorem 1 the distribution function G(u) for the height of a local maximum can be written out as follows: ' u 0 K  (0) G(u) = 2π − (IV ) dx |z|ft (x, 0, z)dz. K (0) −∞

−∞

Therefore, the distribution density g(u) for the height of a local maximum can be calculated by the formula ' 0 K  (0) |z|ft (u, 0, z)dz. g(u) = 2π − (IV ) K (0) −∞

1

Note that as well as ft (x, y, z) in formula (4.4) of Theorem 3 this density does not depend on t owing to the stationarity of the process.

Crossings Problems

81

This result agrees with formula (4.2), which was previously obtained by non-strict arguments.

2.5 Problem Formulation about the First Achievement of Boundaries Consider the point M (t) moving in the n-dimensional Euclidean space Rn whose change of position is governed by an n-dimensional random process X(t) = {X1 (t), . . . , Xn (t)}. At the initial moment t0 , the point is located at position M (t0 ) = x0 = (x01 , . . . , x0n ) which, in general, may be unknown. The independent variable t may be any continuously and monotonely changing variable: for instance, time or one of the components of process X(t) that satisfies this condition. In addition, we suppose that x0 ∈ G, where G is a given region in Rn . We denote by ∂G the boundary of region G and by ∂1 G, some part of the boundary ∂G. We are to find the probability ψ(t , t , x0 ) of the event that point M will reach the boundary of region G for the first time at some moment from a given interval (t , t ) of the independent variable, and will specifically reach the part ∂1 G of the boundary ∂G.1 In applications, this problem for the various configurations of region G and boundary ∂1 G appears. For instance, when studying stochastic systems in situations when a normal system operation corresponds to the position of a point that depicts the system in a certain region G of the system’s phase space, and the consequences of the point leaving this region depend on which part of the region’s boundary it will exit from. In section 2.1 we have already given examples of problems where the decision is connected with a definition of probability ψ; for example, problems about the ruin of an economic agent, a failure of tracking, and the life expectancy of a biological population. However, we note that all these problems correspond to the simplest one-dimensional case n = 1 when a region G is some interval of a real axis and a boundary ∂1 G consists of one of the ends of this interval. We will now give a more substantial and multidimensional example. We will consider the problem of finding the probability of success1 This problem was first posed in the work: S.L. Semakov, First Arrival of a Stochastic Process at the Boundary, Autom. Remote Control, 1988, vol. 49, no. 6, part 1, pp. 757-764. However, in the special case when t = t0 this problem setting appears in a much earlier work: L.S. Pontryagin, A.A. Andronov, and A.A. Vitt, On Statistical Analysis of Dynamical Systems, Zh. Eksp. Teor. Fiz., 1933, vol. 3, no. 3, pp. 165-180.

82

Chapter 2

ful landing for an airplane. This problem was formulated in section 2.1 in the one-dimensional option and it is required to estimate probability that a landing, i.e., an initial contact between an airplane and a landing surface will occur on the given interval (l , l ), where l is the distance is counted from the forward edge of the landing strip. However, for an actual successful landing, the formulated condition is not sufficient. In order to avoid an aviation incident it is still necessary that the vertical velicity, as well as the angles of pitch, roll, and some other characteristics of flight do not reach a certain maximum permissible values at the moment of initial contact between the airplane and the landing surface.1 If we suppose X(l) = {X1 (l), X2 (l), . . . , Xn (l)} is an n-dimensional random process describing the movement of an airplane, the independent variable l is the flight length, component X1 (l) is the flight altitude, component X2 (l) is the vertical velocity of flight, and components X3 (l), . . . , Xn (l) are other scalar characteristics of flight, then the successful landing means that the first achievement of level x∗1 = 0 by the component X1 (l) occurs on the given interval (l , l ) and the conditions xi ≤ Xi ≤ xi ,

i = 2, . . . , n,

are satisfied at the moment of this achievement. Here, xi , xi , i = = 2, . . . , n, are fixed numbers. We denote by y1 , . . . , yn the coordinates of point y ∈ Rn . The considered task represents a special case of the general task of the definition of probability ψ, if the region G is the half-space G = {y : y1 > x∗1 } and part ∂1 G of the boundary ∂G = {y : y1 = x∗1 } is ∂1 G = {y : y1 = x∗1 , (y2 , . . . , yn ) ∈ D}, where D ⊂ Rn−1 is determined by inequalities xi ≤ yi ≤ xi ,

i = 2, . . . , n.

(5.1)

In this important special case of region G, the exact mathematical problem definition is formulated as follows. Let X(t) = 1 A detailed description of the safety problems of airplane landings and the method of its decision are stated in Chapter 3; the reduced statement of Chapter 3 is given in the work: S.L. Semakov, The Application of the Known Solution of a Problem of Attaining the Boundaries by Non-Markovian Process to Estimation of Probability of Safe Airplane Landing, J. Comput. Syst. Sci. Ins., 1996, vol. 35, no. 2, pp. 302-308.

83

Crossings Problems

= (X1 (t), . . . , Xn (t)) be an n-dimensional random process, and x∗1 be a fixed number. We will suppose that an independent variable t changes on the segment [t0 , t ] and the condition P {X1 (t0 ) > x∗1 } = 1 holds. Let t ∈ (t0 , t ) and D be some subset of the (n−1)-dimensional Euclidean space Rn−1 ; in particular, this subset can be determined by system of inequalities (5.1). We are to find the probability of event 

there exists t∗ ∈ (t , t ) such that for every t < t∗ X1 (t) > x∗1 , X1 (t∗ ) = x∗1 , (X2 (t∗ ), . . . , Xn (t∗ )) ∈ D

 ,

i.e., the probability that the first achievement of level x∗1 by the component X1 (t) of the process X(t) = {X1 (t), X2 (t), . . . , Xn (t)}, occurs at moment t∗ from the interval (t , t ) and, at this moment t∗ , the condition (X2 (t∗ ), . . . , Xn (t∗ )) ∈ D holds.

2.6 The First Achievement of Boundaries by the Markov Process We will show that for a certain class of Markov processes the problem of finding out the probability ψ can be reduced to solving some task for an equation in partial derivatives relative to the function ψ. In order to illustrate the idea of obtaining this result, it is enough to consider the one-dimensional case. Let X(t) be a random (and still any) process describing the movement of a point along a real axis and (q, r) be some interval of this axis. We denote by φ(t, x) the probability that during time t this point will reach one of the ends, q or r, if, at the initial moment, it is located at position x ∈ (q, r). In other words, φ(t, x) is the conditional probability     ∃ t∗ ∈ (0, t] such that  P  X(0) = x = φ(t, x). (X(t∗ ) − q)(X(t∗ ) − r) = 0  Let Z be the event that during the time from 0 to τ > 0, inclusively, the process X(t) will not go beyond interval (q, r). Let Z be the event opposite to Z, i.e., the event that during a given period [0, τ ] the process X(t) will go beyond interval (q, r). Then φ(t + τ, x) = P {Z | X(0) = x} +     ∃ t∗ ∈ (τ, t + τ ] such that  + P Z·  X(0) = x . (X(t∗ ) − q)(X(t∗ ) − r) = 0  

(6.1)

84

Chapter 2

Indeed, consider the event that an exit has occurred from the interval (q, r) during the time t + τ under the condition X(0) = x. This event means that either 1) the exit occurred during the time τ or 2) the exit did not occur during the first τ units of time and instead occurred during the following t units of time. The first summand in the right part of (6.1) is the probability of event 1) and the second summand in the right part of (6.1) is the probability of event 2). Note also that P {Z|X(0) = x} = φ(τ, x). Since the product of events Z and {X(τ ) ∈ (q, r)} coincides with the event Z, we transform the second summand in (6.1) as follows:      ∃ t∗ ∈ (τ, t + τ ] such that  P Z·  X(0) = x = (X(t∗ ) − q)(X(t∗ ) − r) = 0  



= P Z · {X(τ ) ∈ (q, r)} · 

   ∃ t∗ ∈ (τ, t + τ ] such that   X(0) = x (X(t∗ ) − q)(X(t∗ ) − r) = 0 



   ∃ t∗ ∈ (τ, t + τ ] such that  = P {X(τ ) ∈ (q, r)} ·  X(0) = x − (X(t∗ ) − q)(X(t∗ ) − r) = 0       ∃ t∗ ∈ (τ, t + τ ] such that  − P Z · {X(τ ) ∈ (q, r)} ·  X(0) = x . (X(t∗ ) − q)(X(t∗ ) − r) = 0  By definition, put ε(τ, x) = P {Z | X(0) = x} −     ∃ t∗ ∈ (τ, t + τ ] such that  − P Z · {X(τ ) ∈ (q, r)} · X(0) = x . (X(t∗ ) − q)(X(t∗ ) − r) = 0  

Obviously, 0 ≤ ε(τ, x) ≤ φ(τ, x) and for the probability φ(t + τ, x) we obtain φ(t + τ, x) = ε(τ, x) +     ∃ t∗ ∈ (τ, t + τ ] such that  + P {X(τ ) ∈ (q, r)}·  X(0) = x . (X(t∗ ) − q)(X(t∗ ) − r) = 0  

85

Crossings Problems

We will now specify the process X(t). At first, for simplicity, we will consider that X(t) can accept only a finite number of values ξ1 , . . . , ξn , where ξi ∈ (q, r) ∀i = 1, . . . , n, and each value ξi is accepted with a non-zero probability. Then n  {X(τ ) ∈ (q, r)} = {X(τ ) = ξi } i=1

and for probability φ(t + τ, x) we obtain φ(t + τ, x) = ε(τ, x) +    n   ∃ t∗ ∈ (τ, t + τ ] such that  + P {X(τ ) = ξi }· X(0) = x . (X(t∗ ) − q)(X(t∗ ) − r) = 0  

i=1

For events A, B, C such that P {C} =  0, P {AC} = 0 we have P {AB|C} =

P {ABC} P {ABC} P {AC} = · = P {B|AC} · P {A|C}. P {C} P {AC} P {C}

Therefore, φ(t + τ, x) = ε(τ, x) +

n 

pi ·P {X(τ ) = ξi |X(0) = x},

i=1

where

   ∃ t∗ ∈ (τ, t + τ ] such that  X(0) = x, pi = P .  (X(t∗ )−q)(X(t∗ )−r) = 0  X(τ ) = ξi 

If X(t) is a Markov process, then     ∃ t∗ ∈ (τ, t + τ ] such that  X(0) = x, P =  (X(t∗ )−q)(X(t∗ )−r) = 0  X(τ ) = ξi     ∃ t∗ ∈ (τ, t + τ ] such that  =P  X(τ ) = ξi . (X(t∗ ) − q)(X(t∗ ) − r) = 0 The last probability is φ(t, ξi ). Therefore, φ(t + τ, x) = ε(τ, x) +

n  i=1

φ(t, ξi )P {X(τ ) = ξ1 |X(0) = x}.

(6.2)

86

Chapter 2

Let X(t) now be a continuous Markov process. We denote by f (ξ, τ ; x) the conditional density of the probability that the point will be at position ξ at moment τ (i.e., X(τ ) = ξ) under the condition that the point is at position x at moment 0 (i.e., X(0) = x). The sum in (6.2) will pass into the corresponding integral: r (6.3) φ(t + τ, x) = ε(τ, x) + φ(t, ξ)f (ξ, τ ; x)dξ. q

Furthermore, we assume that derivatives φξ , φξξ , and φ ξξξ exist and the function φ(t, ξ) allows Taylor’s decomposition in the neighborhood of x: 1 φ(t, ξ) = φ(t, x) + φξ (t, x)(ξ − x) + φξξ (t, x)(ξ − x)2 + 2 1  3 + φξξξ (t, x + ϑ(ξ − x))(ξ − x) , where 0 ≤ ϑ = ϑ(ξ) ≤ 1. 6 Substituting this decomposition for φ(t, ξ) in (6.3) and dividing both sides by τ , we get φ(t + τ, x) φ(t, x) − τ τ

r f (ξ, τ ; x)dξ =

ε(τ, x) + τ

q

+

1 φξ (t, x)

r

τ

1 1 f (ξ, τ ; x)(ξ −x)dξ + φξ (t, x) 2 τ

q

1 + 6τ

r

f (ξ, τ ; x)(ξ −x)2 dξ +

q

r

φ ξξξ (t, x

(6.4) 3

+ ϑ(ξ − x))f (ξ, τ ; x)(ξ − x) dξ.

q

We pass to a limit as τ → 0. We consider that at every x ∈ (q, r) lim

τ →0

φ(τ, x) = 0. τ

(6.5)

This assumption can be justified as follows. The value φ(τ, x) is the conditional probability that the point will reach one of the ends of interval (q, r) during the time τ if it is at position x ∈ (q, r) at an initial moment. If we suppose that with probability 1− o(τ ), where ) lim o(τ τ = 0, the point does not pass the finite distance min{x−q, r−x} τ →0

87

Crossings Problems

quicker than some small time τ , then φ(τ, x) ≤ o(τ ) at small τ and the condition (6.5) is satisfied. Together with (6.5) the condition ε(τ, x) =0 τ

lim

τ →0

is also satisfied. Further, r f (ξ, τ ; x)dξ = P {X(τ ) ∈ (q, r)|X(0) = x}. q

Obviously, P {X(τ ) ∈ (q, r)|X(0) = x} ≥ ≥ P {Z|X(0) = x} = 1 − P {Z|X(0) = x} = 1 − φ(τ, x). Therefore,

r 1≥

f (ξ, τ ; x)dξ ≥ 1 − φ(τ, x).

(6.6)

q

It follows from (6.6) and (6.5) that the limit of the left side in (6.4) is equal to   r ∂φ(t, x) φ(t + τ, x) φ(t, x) − . lim f (ξ, τ ; x)dξ = τ →0 τ τ ∂t q

By definition, put 1 a ˜(x) = lim τ →0 τ

r f (ξ, τ ; x)(ξ − x)dξ, q

˜b(x) = lim 1 τ →0 τ

r

f (ξ, τ ; x)(ξ − x)2 dξ;

q

then from (6.4) we obtain ∂φ(t, x) ∂φ(t, x) 1 ˜ ∂ 2 φ(t, x) =a ˜(x) + b(x) + ∂t ∂x 2 ∂x2 1 1 + lim 6 τ →0 τ

r q

3 φ ξξξ (t, x + ϑ(ξ − x))f (ξ, τ ; x)(ξ − x) dξ.

88

Chapter 2

We calculate the limit in the right side of this equality. We assume that at every x 1 lim τ →0 τ

∞ f (ξ, τ ; x)|ξ − x|n dξ = 0 for every n = 3, 4, . . . ,

(6.7)

−∞

i.e., the probability of big deviations |X(τ ) − X(0)| decreases with the decreasing τ , and also the third and all following moments of random value |X(τ ) − X(0)| have a higher order of smallness in comparison with τ . Such Markov processes are called diffusive processes. If we suppose function φ ξξξ (t, ξ) is limited on interval (q, r) at every t, then 1 lim τ →0 τ

r

3 φ ξξξ (t, x + ϑ(ξ − x))f (ξ, τ ; x)(ξ − x) dξ = 0,

q

and

∂φ(t, x) ∂φ(t, x) 1 ˜ ∂ 2 φ(t, x) =a ˜(x) + b(x) . ∂t ∂x 2 ∂x2

By definition, put 1 τ →0 τ

∞ f (ξ, τ ; x)(ξ − x)dξ,

a(x) = lim

1 b(x) = lim τ →0 τ

−∞

∞

f (ξ, τ ; x)(ξ − x)2 dξ.

−∞

The numbers a(x) and b(x) are called coefficients of the drift and diffusion of the Markov process X(t); these numbers were already considered earlier (see Chapter 1) with corresponding comments. We claim that a ˜(x) ≡ a(x), ˜b(x) ≡ b(x) (6.8) if equalities (6.5) and (6.7) hold. Indeed, without loss of generality it can be assumed that x − 1 < q < x < r < x + 1.

89

Crossings Problems

We have 1 lim τ →0 τ

+ lim

τ →0

∞

1 f (ξ, τ ; x)(ξ − x)dξ = lim τ →0 τ

−∞

x−1 

f (ξ, τ ; x)(ξ − x)dξ + −∞

q

1 τ

f (ξ, τ ; x)(ξ − x)dξ + lim

τ →0

1 τ

f (ξ, τ ; x)(ξ − x)dξ + q

x−1

1 + lim τ →0 τ

(6.9)

r

x+1 

∞

1 f (ξ, τ ; x)(ξ − x)dξ + lim τ →0 τ

r

f (ξ, τ ; x)(ξ − x)dξ. x+1

Due to condition (6.7), we get 1 lim τ →0 τ

x−1 

f (ξ, τ ; x)(ξ − x)dξ = 0, −∞

1 lim τ →0 τ

∞ f (ξ, τ ; x)(ξ − x)dξ = 0. x+1

Consider 1 lim τ →0 τ

x+1 

f (ξ, τ ; x)(ξ − x)dξ. r

We estimate the expression under a sign of limit: 1 τ

x+1 

f (ξ, τ ; x)(ξ − x)dξ < r

1 τ

x+1 

f (ξ, τ ; x)dξ. r

Taking into account the equality ∞ f (ξ, τ ; x)dξ = 1, −∞

from (6.6) we obtain q

∞ f (ξ, τ ; x)dξ ≤ φ(τ, x),

f (ξ, τ ; x)dξ + −∞

r

90

Chapter 2

and 1 τ

x+1 

f (ξ, τ ; x)(ξ − x)dξ
0. The second and third conditions from (6.11) mean that at x = q or x = r the probability of achieving the boundaries during any time t ≥ 0 is equal to 1: the point is already on the boundary at the initial moment. Consequently, the probability φ(t, x) satisfies the differential equation in partial derivatives (6.10) with boundary conditions (6.11).

91

Crossings Problems

If we are interested in the exit of process X(t) from interval (q, r) only through the left end q (similarly, only through the right end r), then φ(t, x) denotes the probability that the first exit of process X(t) from interval (q, r) will occur until moment t through the left end q. The last circumstance does not affect a derivation of the equation for probability φ(t, x), function φ(t, x) will continue to satisfy the equation (6.10) and only boundary conditions will change: φ(0, x) = 0 ∀x ∈ (q, r), φ(t, q) = 1 ∀t ≥ 0, φ(t, r) = 0 ∀t ≥ 0. (6.12) Now consider the general n-dimensional problem about the calculation of probability ψ(t , t , x) at t = 0, t = t. This problem was formulated in section 2.5. By definition, put     φ(t, x) = ψ(t , t , x) . t =0, t =t

Then (see section 2.5) φ(t, x) is the probability that point M will, for the first time, come to the boundary of region G at some moment t∗ from interval (0, t) (i.e., during t time) and this will occur on part ∂1 G of boundary ∂G. In the one-dimensional case when G = (q, r) and ∂G = {q, r}, for φ(t, x) we obtain problem (6.10) with conditions (6.11) if ∂1 G = ∂G = {q, r} and we obtain problem (6.10) with conditions (6.12) if ∂1 G = {q}. In the multidimensional case, it is similarly possible to repeat all reasonings which are carried out at the derivation of equation (6.10) and to show that if X(t) is a diffusive Markov process, then φ(t, x) satisfies the equation1 n n ∂φ  ∂φ 1  ∂2φ = ak (x) + bkm (x) ∂t ∂xk 2 ∂xk ∂xm k=1

k,m=1

with boundary conditions: φ(0, x) = 0 if x ∈ G, φ(t, x) = 1 if x ∈ ∂1 G, φ(t, x) = 0 if x ∈ / ∂1 G but x ∈ ∂G; and coefficients ak (x) and bkm (x) are defined as follows: ∞ ∞ 1 ak (x) = lim ... f (ξξ , τ ; x)(ξk − xk )dξ1 . . . dξn , τ →0 τ −∞

−∞

1 This equation was published in a work that has already been mentioned above: L.S. Pontryagin, A.A. Andronov, and A.A. Vitt, On Statistical Analysis of Dynamical Systems, Zh. Eksp. Teor. Fiz., 1933, vol. 3, no. 3, pp. 165-180.

92

Chapter 2

1 bkm (x) = lim τ →0 τ

∞

∞ ...

−∞

f (ξξ , τ ; x)(ξk − xk )(ξm − xm )dξ1 . . . dξn ,

−∞

where f (ξξ , τ ; x) is the conditional density of probability that point M will be at position ξ at moment τ given that this point is at position x at the moment 0. The probability ψ(t , t , x) is connected with the probability φ(t, x) by equality ψ(t , t , x) = φ(t , x) − φ(t , x). Indeed, let At∂1 G be the event that the point will reach the boundary of region G for the first time at some moment from interval (0, t) and, specifically, it will reach the part ∂1 G of boundary ∂G; this is, by definition,  P At∂1 G = φ(t, x). If At denotes the event that the point will reach the boundary of region G for the first time at some moment from interval (0, t), then (  ) ψ(t , t , x) = P At∂1 G At ,

(6.13)

where the line over an event denotes its complement. Let ∂G∂1 G be the part of boundary ∂G such that {∂G  ∂1 G} ∩ ∂1 G = ∅ and   {∂G∂1 G} ∪ ∂1 G = ∂G. Since the events At∂1 G + At and At∂G∂1 G are not crossed and the sum of these events is a persistent event, we have (  ) (  ) P At∂1 G + At + P At∂G∂1 G = 1. (6.14) From the equality ) (  ) ( ) (  ) (  P At∂1 G At = P At∂1 G + P At − P At∂1 G + At , and equalities (6.13) and (6.14), we get (  ) ( ) (  ) ψ(t , t , x) = P At∂1 G + P At − 1 + P At∂G∂1 G .

93

Crossings Problems 



Since the events At , At∂G∂1 G , and At∂1 G are not crossed and the sum of these events is a persistent event, we obtain (  )  ( ) (  ) ψ(t , t , x) = P At∂1 G − 1 − P At − P At∂G∂1 G = (  ) (  ) = P At∂1 G − P At∂1 G = φ(t , x) − φ(t , x). The statement is proved.

2.7 Examples Let W (t), t ≥ 0, W (0) = 0, be the Wiener process and X(t) be a random process that satisfies the stochastic differential equation (see Chapter 1) (7.1) dXt = h(Xt , t)dt + g(Xt , t)dWt , where h and g are continuously differentiable functions of their arguments. Then the process Xt is a diffusion Markov process. This statement is known as Doob’s theorem.1 The coefficient of drift a and the coefficient of diffusion b from process Xt can be determined by the following formulas: a(x, t) = h(x, t),

b(x, t) = g 2 (x, t).

(7.2)

We will prove equalities (7.2) when h(x, t) ≡ c and g(x, t) ≡ σ, σ > 0, where c and σ are given numbers. In this case, dXt = cdt + σdWt ; from (7.3) it follows that X(t) = X(0) + ct + σW (t). Let, for simplicity, X(0) = 0. Then X(t) = ct + σW (t). 1

J.L. Doob, Stochastic Processes, New York: Wiley, 1953.

(7.3)

94

Chapter 2

Earlier it was noted (see Chapter 1) that all finite-dimensional distributions of process W (t) are normal with a zero vector of mathematical expectations and the correlation function K(t1 , t2 ) = min(t1 , t2 ). Therefore, the distribution density f (x, t) of random value X(t) is   (x − ct)2 1 √ exp − f (x, t) = √ ; 2σ 2 t 2π · σ t the joint distribution density f (z, t+Δ; x, t) of random values X(t+Δ) and X(t) is 1 1 √ √ · 2π · σ t · σ t + Δ 1−

f (z, t+Δ; x, t) =  · exp −

1 t 2(1 − t+Δ ) *

−2



t t+Δ

·

(x − ct)2 − σ2 t

t (x − ct)(z − c(t + Δ)) (z − c(t + Δ))2 √ √ + t+Δ σ 2 (t + Δ) σ tσ t + Δ

 ;

the conditional distribution density f (z, t+Δ|x, t) is f (z, t+Δ|x, t) =

=√

1 √

2πσ t + Δ



1

· 1−

f (z, t+Δ; x, t) = f (x, t)

t t+Δ

· exp

(x − ct)2 2σ 2 t

 1−

1 1−

t t+Δ

 ·

  * t (x−ct)(z−c(t+Δ)) (z−c(t+Δ))2 1 √ √ − = · exp − t 2σ 2 (t+Δ) t+Δ 1− t+Δ σ tσ t+Δ  (x − ct)2 1 √ exp − − =√ 2σ 2 Δ 2πσ Δ  (x − ct)(z − c(t + Δ)) (z − c(t + Δ))2 + − = 2σ 2 Δ σ2 Δ   2  1 1 √ exp − 2 = (z − c(t + Δ)) − (x − ct) =√ 2σ Δ 2πσ Δ 

95

Crossings Problems

=√

  1 1 √ exp − 2 (z − x − cΔ)2 . 2σ Δ 2πσ Δ

We obtain for the coefficient of drift ∞ 1 a(t, x) = lim (z − x)f (z, t + Δ|x, t)dz = Δ→0+ Δ −∞

1 = lim Δ→0+ Δ

∞ −∞

  1 (z − x) √ exp − 2 (z − x − cΔ)2 dz. √ 2σ Δ 2πσ Δ

To calculate the last integral, we will introduce a new variable of integration using the formula u=

z − x − cΔ √ ; σ Δ

then 1 a(t, x) = lim Δ→0+ Δ

 2 ∞ √ √ u σ Δu + cΔ √ √ exp − σ Δdu = 2 2πσ Δ

−∞

⎧ ∞ ⎫ √  2  2 ⎬  ∞ 1⎨ σ Δ cΔ u u √ u exp − √ exp − = lim du + du = Δ→0+ Δ ⎩ ⎭ 2 2 2π 2π −∞



−∞

1 0 + cΔ = lim Δ→0+ Δ



= c.

Similarly, we get for the diffusion coefficient: 1 b(t, x) = lim Δ→0+ Δ

1 = lim Δ→0+ Δ

∞ −∞

∞

(z − x)2 f (z, t + Δ|x, t)dz =

−∞

  1 (z − x)2 √ exp − 2 (z − x − cΔ)2 dz = √ 2σ Δ 2πσ Δ

96

Chapter 2

1 = lim Δ→0+ Δ

∞ −∞

√  2 (σ Δu + cΔ)2 u √ exp − du = 2 2π

⎧ ∞  2  1 ⎨ σ2 Δ u √ u2 exp − = lim du + Δ→0+ Δ ⎩ 2 2π −∞

∞ + −∞

√



2



∞

2

2



2

2σcΔ Δ c Δ u u √ √ u exp − exp − du + 2 2 2π 2π −∞   1 σ2 Δ √ 2 2 √ · 2π + 0 + c Δ = σ 2 . = lim Δ→0+ Δ 2π

⎫ ⎬

 du

⎭

=

We will give two calculation examples for the probability φ of the first achievement of boundaries in the one-dimensional case when a random process is described by equation (7.3). 1. We will consider the differential equation dN dN = ε(t)N, or = ε(t)dt, dt N which is met in many application domains. We assume that the right side ε(t)dt can be represented as μdt + σdWt , where μ and σ are positive constants; then dN = μdt + σdWt . N

(7.4)

The first summand in the right side of (7.4) is interpreted as a nonrandom tendency in change of Nt , and the second summand in the right side of (7.4) is interpreted as a random component. The equation (7.4) has already been discussed above (see Chapter 1) in the study shares prices. It is possible to also give other examples when this equation is used. For example, (7.4) describes the number of individuals N (t) in an isolated biological population. Suppose that N (0) = N0 > 0. Consider the random process N (t) , t ≥ 0. N0 Using Ito’s formula (see Chapter 1), we obtain   1 1 1 1 dt + σNt dWt + dXt = μNt · σ 2 (Nt )2 dt, − Nt Nt 2 (Nt )2 X(t) = ln

97

Crossings Problems

 dXt =

 1 μ − σ 2 dt + σdWt . 2

(7.5)

ˇ, N ˆ ), We find the probability that N (t) leaves the interval (N ˇ ˆ N < N0 < N , before the moment T under the condition N (0) = N0 . Obviously, this probability coincides with the probability that X(t) ˇ ˆ ˆ = ln NN0 , before the moment T leaves the interval (ˇ x, x ˆ), x ˇ = ln NN0 , x

under the condition X(0) = ln NN(0) = ln 1 = 0. If φ(t, x) denotes the 0 probability that the random process X(t) leaves the interval (ˇ x, x ˆ) before the moment t under the condition X(0) = x, x ˇ x φ(t, x ˇ) = 1, t ≥ 0; ⎩ φ(0, x) = 0, x > x ˇ. ∂φ ∂t

(7.7)

We have introduced a new independent variable y = x−ˇ x, the function ˜ x) = φ(t, y + x φ(t, ˇ), and the function   c c2 ˜ u(t, y) = (1 − φ(t, y)) exp 2 y + 2 t , σ 2σ where c = μ − 12 σ 2 . Then for u(t, y) we get ⎧ ⎨

2

= 12 σ 2 ∂∂yu2 , t > 0, y > 0; u(t, 0) = 0, t ≥ 0; ⎩ u(0, y) = exp σc2 y , y > 0. ∂u ∂t

(7.8)

Problem (7.8) is a well known mixed problem for the heat conductivity equation on a semi-infinite straight line with a zero boundary condition. The solution of (7.8) is given by the integral 1 u(t, y) = √ 2π

∞ 0

  2 2 − exp − (y+v) exp − (y−v) 2σ 2 t 2σ 2 t √ σ t

 exp

 c v dv. σ2

Using the special function 1 Φ(y) = √ 2π we obtain



c2 t u(t, y) = exp 2σ 2

y −∞

 2 v exp − dv, 2

-    .     ct+y ct−y c c √ exp 2 y −Φ √ exp − 2 y . Φ σ σ σ t σ t

Coming back to the function φ(t, x), we get .     -  ct − x + x ˇ ct + x − x ˇ 2c √ √ ˇ) . −Φ exp − 2 (x − x φ(t, x) = 1− Φ σ σ t σ t Direct checking has convinced us that this function φ(t, x) satisfies conditions (7.7), as well as the condition 0 ≤ φ(t, x) ≤ 1; the condition

100

Chapter 2

φ(0, x) = 0, x > x ˇ, from (7.7) means that lim φ(t, x) = 0, where x > x ˇ. t→0

Therefore, the probability of interest to us is .     -  cT + x ˇ 2c cT − x ˇ √ √ ˇ . −Φ exp 2 x φ(T, 0) = 1 − Φ σ σ T σ T

2.8 The First Achievement of Boundaries by Non-Markovian Process The problem from section 2.5 about finding the probability ψ is not solved for non-Markovian processes. This is explained by the fact that the solution to this problem is related to overcoming a fundamental difficulty. The difficulty lies in the fact that an event of interest to us is determined by the behavior of a random process at a non-enumerable number of points t. For such events, even if a family of finite-dimensional distributions of the process is known, there is no general method for finding probabilities. Hence, in each case, 1) it is necessary to use the specific features of the given random process as much as possible, and 2) to use the specific features of the given event as much as possible. An exception only exists for Markov processes where similar problems are solved thanks to the Markov property that the future is independent from the past provided that we know the present. In particular, in the case of diffusion Markov processes, these problems are reduced to the known equations of mathematical physics. Examples of this were considered in sections 2.6 and 2.7. In the present section, we will state the results1 that are concerned with estimating the probability ψ for a wide class of continuous processes X(t) in the case when the region G is a half-space. This form of region G often occurs in application domains. A certain reasoning scheme is proposed for estimating probabilities. This scheme can be used for finding the probabilities of any events related to crossings of a level by a random process. This scheme involves the following. 1) By means of a direct search we consider possible options of behavior for the sample functions of a random process in relation to crossing a given level. 1 These results were published for the first time in the work: S.L. Semakov, The Probability of the First Hiting of a Level by a Component of a Multidimensional Process on a Prescribed Interval under Restrictions of the Remaining Components, Theor. Prob. App., 1989, vol. 34, no. 2, pp. 357-361.

101

Crossings Problems

2) By any criterion (such as, a number of upcrossings and downcrossings) we have conveniently partitioned sample functions into bunches. 3) We will express the probabilities of interest to us through the introduction of the probabilities of such bunches, i.e., probabilities that a sample function belongs to a particular bunch. 4) The probabilities of bunches are estimated by means of any characteristics from a random process (for example, by the average number of upcrossings and downcrossings) and the joint distribution density of values of the process at fixed points. This scheme is realized in the example of estimating the probability ψ when region G is a half-space. We will solve the following problem. Let X(t) be an ndimensional random process, and let u be a given number. We will consider the processes X(t) = {X1 (t), . . . , Xn (t)} of two types: a) t ∈ [t0 , t ], and P {X1 (t0 ) > u} = 1;

(8.1)

b) t ∈ (t0 , t ], t0 ≥ −∞, and lim P {X1 (t) > u} = 1.

t→t0

(8.2)

Let t ∈ (t0 , t ), and let D be a subset of the (n − 1)-dimensional Euclidean space Rn−1 . We introduce the event  ∗  ∃ t ∈ (t , t ) such that for every t < t∗ X1 (t) > u, ZD = . X1 (t∗ ) = u, (X2 (t∗ ), . . . , Xn (t∗ )) ∈ D When D = Rn−1 we will also use symbol Z for a designation of event ZD . It is necessary to find the conditional probability P {ZD |L} of event ZD provided that the event L occurred, where L = {X1 (t0 ) > u} if processes of type a) are considered, and L = {∃ t˜ > t0 such that for every t ∈ (t0 , t˜) X1 (t) > u} if processes of type b) are considered. We have already mentioned above that problems related to finding the probability P {ZD |L} appear when studying stochastic systems

102

Chapter 2

in various application domains and these problems are of great practical importance. The physical meaning of conditions of type (8.1), (8.2) is that at the initial time moment the stochastic system, whose behavior is described by process X(t), is located inside a known region of its phase space. In the case under consideration, this region is a half-space. We note that P {ZD |L}=P {ZD } if processes of type a) are considered. We will assume that with probability 1 the sample functions x1 (t) from process X1 (t) belong to the set Gu (t0 , t ) and do not touch the level u; additionally, the average number of crossings of level u by process X1 (t) on the interval (t0 , t ) is finite. Then it is easy to see that for processes of type b) the equality P {ZD |L} = P {ZD } also holds. Therefore, instead of estimating the conditional probability P {ZD |L} we can estimate the unconditional probability P {ZD }. We will speak about crossings, upcrossings, downcrossings, and touchings of the level u by the process X1 (t). We denote by N (t1 , t2 ), N + (t1 , t2 ), and N − (t1 , t2 ) the average numbers of crossings, upcrossings, and downcrossings on interval (t1 , t2 ). To avoid extra clutter, we will usually proceed with our reasonings in such a way as if properties of the sample functions x1 (t) that are supposed to hold with probability 1 actually hold for all x1 (t). This does not violate the correctness of the final results. We will begin with a one-dimensional case, i.e., at first we assume D = Rn−1 . 2.8.1 Probability Estimates in a One-Dimensional Case We denote by A− j (t1 , t2 ), j = 1, 2, . . . the event that the number of crossings on interval (t1 , t2 ) is equal to j and the first crossing is a downcrossing. T h e o r e m 4. Suppose that 1) with probability 1 sample functions x1 (t) belong to the set Gu (t0 , t ) and do not touch the level u on the interval (t0 , t ), N (t0 , t ) < ∞; 2) P {X1 (t ) = u} = 0; 3) the condition (8.1) is satisfied if the variable t changes on the segment [t0 , t ], t0 > −∞, t < ∞, and the condition (8.2) is satisfied if t changes on the interval (t0 , t ], t0 ≥ −∞, t < ∞. Then   ∞ −  −   +  N (t , t ) − N (t0 , t ) + P A2k (t0 , t ) ≤ P {Z} ≤ N − (t , t ). k=1 (8.3)

103

Crossings Problems

P r o o f. We denote by Ak (t0 , t), k = 0, 1, 2, . . . , the event that exactly k crossings occurred on interval (t0 , t). Then   ∞ P Ak (t0 , t) = 1, Ai (t0 , t) ∩ Aj (t0 , t) = ∅ ∀i = j, k=0

and

∞ 

P {Ak (t0 , t)} = 1.

(8.4)

k=0

We will now introduce the event  ∗  ∃ t ∈ (t , t ) such that for every t < t∗ X1 (t) < u ∗ Z = and X1 (t∗ ) = u and the union of events  ∞





Ak (t0 , t ) ∪ A0 (t0 , t ).

(8.5)

k=1

Event (8.5) implies that at least one crossing occurred on the interval (t0 , t ) or did not occur on the interval (t0 , t ). The event Z ∪ Z ∗ that Z ∪ Z ∗ did not occur and the event (8.5) can differ only by such sample functions x1 (t), where x1 (t ) = u or x1 (t) touches the level u at least once until moment t . Under the conditions of the theorem, we have     ∞   ∗ P {Z ∪ Z } = P Ak (t0 , t ) ∪ A0 (t0 , t ) . k=1

From this, using (8.4) and the incompatibility of events A0 (t0 , t ), Ak (t0 , t ), k = 1, 2, . . . , we get P {Z ∪ Z ∗ } =

∞  k=1

P {Ak (t0 , t )} −

∞ 

P {Ak (t0 , t )}.

k=1

Due to the conditions of the theorem, P {Z ∗} = 0. Therefore, P {Z ∪ Z ∗} = P {Z}. We denote by A+ k (t0 , t), k = 1, 2, . . . , the event that the number of crossings on interval (t0 , t) are equal to k and the first crossing is an upcrossing. Under the conditions of the theorem, we have P {A+ k (t0 , t)} = 0 ∀ k = 1, 2, . . . .

(8.6)

104

Chapter 2

Therefore, P {Ak (t0 , t)} = P {A− k (t0 , t)} ∀ k = 1, 2, . . . , and P {Z} =

∞ 

 P {A− k (t0 , t )} −

k=1

∞ 

 P {A− k (t0 , t )}.

(8.7)

k=1

Let t˜ ∈ (t0 , t ], k = 1, 2, . . . . By definition, put ⎧   ⎪ − ⎪ ˜ ˜ 0 when x (t) ∈ A (t , t ) ∪ A (t , t ) , ⎪ 1 0 0 1 0 ⎪ ⎪ ⎪  ⎨ − + ˜ ˜ ˜ k when x1 (t) ∈ A− N + (t0 , t˜) = 2k (t0 , t) ∪ A2k+1 (t0 , t) ∪ A2k (t0 , t)∪ ⎪ ⎪  ⎪ ⎪ ⎪ + ⎪ ˜ (t , t ) . ∪A ⎩ 2k−1 0 It is easy to see that the random value N + (t0 , t˜) is equal to the number of upcrossings on interval (t0 , t˜). Taking into account (8.6), we obtain the mathematical expectation N + from this random value: N + (t0 , t) =

∞ 

− k[P {A− 2k (t0 , t)} + P {A2k+1 (t0 , t)}]

(8.8)

k=1

at every t ∈ (t0 , t ]. Similarly, N − (t0 , t) =

∞ 

− k[P {A− 2k (t0 , t)} + P {A2k−1 (t0 , t)}].

(8.9)

k=1

Consider the series ∞ 

− k[P {A− 2k+1 (t0 , t)} + P {A2k+2 (t0 , t)}].

k=1

This series converges because it is majorized by the converging series ∞ 

− (k + 1)[P {A− 2k+1 (t0 , t)} + P {A2k+2 (t0 , t)}] =

k=1

=

∞  m=2

− m[P {A− 2m−1 (t0 , t)} + P {A2m (t0 , t)}] =

105

Crossings Problems

=

∞ 

− m[P {A− 2m−1 (t0 , t)} + P {A2m (t0 , t)}] −

m=1

− − P {A− 1 (t0 , t)} − P {A2 (t0 , t)} = − = N − (t0 , t) − P {A− 1 (t0 , t)} − P {A2 (t0 , t)}.

By definition, put S(t0 , t) =

∞ 

− k[P {A− 2k+1 (t0 , t)} + P {A2k+2 (t0 , t)}].

k=1

It follows from this definition and from equality (8.9) that ∞ 

− P {A− k (t0 , t)} = N (t0 , t) − S(t0 , t).

k=1

Due to this, we get from (8.7) the following equality:   −   −   P {Z} = N (t0 , t ) − S(t0 , t ) − N (t0 , t ) − S(t0 , t ) = = N − (t0 , t ) − N − (t0 , t ) − S(t0 , t ) + S(t0 , t ) =

(8.10)

= N − (t , t ) − S(t0 , t ) + S(t0 , t ). Combining (8.8) and the definition of S(t0 , t), we obtain   ∞ A− (t , t) . S(t0 , t) = N + (t0 , t) − P 2k 0

(8.11)

k=1

If we combine this with (8.10), we get P {Z} ≥ N − (t , t ) − S(t0 , t ) =   ∞ − −   +   = N (t , t ) − N (t0 , t ) + P A2k (t0 , t ) , k=1

i.e., the left inequality from (8.3) is proved. Now we will prove the right inequality from (8.3). Let us turn to the definition of value S(t0 , t). Since a converging series from nonnegative terms converges to one and the same value at any order of summation, we have    ∞ ∞ ∞ ∞    − − P {Ak (t0 , t)} = P Ak (t0 , t) . S(t0 , t) = m=1 k=2m+1

m=1

k=2m+1

106

Chapter 2

Since for every m = 1, 2, . . .     ∞  A− (t , t ) ⊂ k 0 k=2m+1

∞ 

  A− (t , t ) , k 0

k=2m+1

we obtain S(t0 , t ) ≤ S(t0 , t ). From this inequality and from (8.10) it follows that P {Z} ≤ N − (t , t ). Theorem 4 is proved. We state the simple consequence from inequalities (8.3). C o r o l l a r y 1. Under the conditions of Theorem 4, N − (t , t ) − N + (t0 , t ) ≤ P {Z} ≤ N − (t , t ).

(8.12)

Result (8.12) is used to solve some problems in aviation.1 Estimates (8.12) are constructive: i.e., these estimates can be calculated. However, the lower estimate from (8.3) is unconstructive because the method of calculating the probability   ∞  A− (t , t ) P 2k 0 k=1

is not proposed. The following theorem gives the non-decreasing sequence (i.e., calculated) lower estimates for  ∞ of constructive  / −  P A2k (t0 , t ) . This allows to improve the calculated lower estik=1

mate for the probability P {Z} of interest to us. T h e o r e m 5. Suppose that the conditions of Theorem 4 are satisfied. Then at any partition of the interval (t0 , t ) t0 < t1 < . . . < tn−1 < tn = t , where for every i = 1, 2, . . . , n−1 P {X1 (ti ) = u} = 0, the inequalities N − (t , t ) − N + (t0 , t ) + Δ ≤ P {Z} ≤ N − (t , t ) hold, where Δ = Δ(t1 , . . . , tn ) =

n−1  i=1

 P

 X1 (ti ) < u ∩

- ! n 

(8.13)

. . X1 (tj ) > u

j=i+1

(8.14) 1

See chapter 3; see also the work: S.L. Semakov, The Application of the Known Solution of a Problem of Attaining the Boundaries by Non-Markovian Process to Estimation of Probability of Safe Airplane Landing, J. Comput. Syst. Sci. Ins., 1996, vol. 35, no. 2, pp. 302-308.

107

Crossings Problems

If new points tk such that P {X1 (tk ) = u} = 0 are added to the available points of division ti , i = 1, 2, . . . , n−1, the value Δ can only increase. ∞ /  A− P r o o f. Take point tn−1 ∈ (t0 , t ). The event 2k (t0 , t ) k=1

can be represented as:     ∞ ∞  − −   A2k (t0 , t ) = X1 (t ) > u ∩ A2k (t0 , tn−1 ) ∪ k=1

 ∪





k=1



X1 (tn−1 ) < u ∩ X1 (t ) > u ∩  ∪

A0 (t0 , tn−1 ) ∩

 ∞

 ∞

A− 2k−1 (t0 , tn−1 )

k=1  A− 2k (tn−1 , t )

 ∪

 .

k=1

Consider three events written here in brackets {}. These events are not crossed. Under the conditions of the theorem, the probability of the second events coincides with   from these P P

X1 (tn−1 )u . Therefore,  ∞

 A− 2k (t0 , t )

k=1



 = P A0 (t0 , tn−1 ) ∩







 ∞



 A− 2k (tn−1 , t )

k=1







X1 (t ) > u ∩

+P

 ∞

+



X1 (t ) > u ∩ X1 (tn−1 ) < u

+P



A− 2k (t0 , tn−1 )

+

(8.15)

 .

k=1

Now take point tn−2 ∈ (t0 , tn−1 ). Similarly, the third summand in the right side of (8.15) can be represented as     ∞ A− (t , t ) = P X1 (t ) > u ∩ 2k 0 n−1 k=1





X1 (t ) > u ∩A0 (t0 , tn−2 )∩

=P  +P  +P



 ∞

A− 2k (tn−2 , tn−1 )

 +

k=1

     X1 (t ) > u ∩ X1 (tn−1 ) > u ∩ X1 (tn−2 ) < u +

     ∞ A− (t , t ) . X1 (t ) > u ∩ X1 (tn−1 ) > u ∩ 0 n−2 2k k=1

108

Chapter 2

In the same way, take point tn−3 ∈ (t0 , tn−2 ) and decompose the third summand in the right side of the last equality and so on for remaining points tn−4 , tn−5 , . . . , t2 , t1 . Uniting all decompositions in one, we obtain   ∞  −  P A2k (t0 , t ) = Δ(t1 , . . . , tn ) + δ(t1 , . . . , tn ), (8.16) k=1

where Δ(t1 , . . . , tn ) is calculated by formula (8.14), and δ(t1 , . . . , tn ) is calculated by the formula    ∞ − δ(t1 , . . . , tn ) = P A0 (t0 , tn−1 ) ∩ A2k (tn−1 , tn ) + +P

- ! n  i=2

+

. X1 (ti ) > u

∩

 ∞

k=1

A− 2k (t0 , t1 )

 +

(8.17)

k=1

- ! .   n n  ∞  P A− (t , t ) . X1 (ti ) > u ∩A0 (t0 , tj−2 )∩ 2k j−2 j−1 j=3

i=j

k=1

Since δ(t1 , . . . , tn ) ≥ 0, we see that inequalities (8.13) follow from (8.3), (8.16). Now suppose that point tˆ is located between points ti and ti+1 , where i is equal to any of numbers 0, 1, 2, . . . , n − 1. Let us turn to formula (8.14). The events whose probabilities are summarized in the right side of (8.14) are not crossed. We denote by H(t1 , . . . , ti , ti+1 , . . . , tn ) the union of these events. Then Δ(t1 , . . . , ti , ti+1 , . . . , tn ) = P {H(t1 , . . . , ti , ti+1 , . . . , tn )}, Δ(t1 , . . . , ti , tˆ, ti+1 , . . . , tn ) = P {H(t1 , . . . , ti , tˆ, ti+1 , . . . , tn )}. It is easy to see that any sample function x1 (t) leading to the event H(t1 ,..., ti , ti+1 ,..., tn ) then leads to the event H(t1 ,..., ti , tˆ, ti+1 ,..., tn ). Therefore, Δ(t1 , . . . , ti , ti+1 , . . . , tn ) ≤ Δ(t1 , . . . , ti , tˆ, ti+1 , . . . , tn ). Theorem 5 is proved. We note the obvious fact. R e m a r k 1. At the fixed number n − 1 of points of division t1 , . . . , tn−1 , n ≥ 2, the value Δ(t1 , . . . , tn ) determined by formula (8.14) is not fixed and the value Δ(t1 , . . . , tn ) can be varied depending on locations of points t1 , . . . , tn−1 . The best choice of points t1 , . . . , tn−1 corresponds to the largest size of Δ.

109

Crossings Problems

In other words, the best choice of a fixed number n−1 of points of the division is reduced to finding the maximum function of the n−1 variables. 2.8.2 Limits of the Sequence of Estimates The following theorem concerns a question about the existence of lim Δ(t1 , . . . , tn ) as a cross-partition of the interval (t0 , t ). We have formulated the corresponding statement for processes of type a). For processes of type b) this statement is similarly formulated and proved. T h e o r e m 6. Suppose that conditions 1) and 2) of Theorem 4 are satisfied and the process of type a) is considered, i.e., the variable t changes on segment [t0 , t ], t0 > −∞, t < ∞, and also the condition (8.1) is satisfied. Let t0 < t1 < . . . < ti . . . < tn = t and t0 = t˜0 < t˜1 < . . . < t˜j < . . . < t˜m = t be any partitions of segments [t0 , t ] and [t0 , t ]; P {X1 (ti ) = u} = 0 for every i = 1, . . . , n − 1 and P {X1 (t˜j ) = u} = 0 for every j = 1, . . . , m. In addition, suppose that lim

max (ti −ti−1 )→0

i=1,...,n

  n ∞  P A− (t , t ) = 0. 2k i−1 i i=1

(8.18)

k=1

Then limits exist: lim

Δ(t1 , . . . , tn ) = Δlim (t0 , t ),

lim

Δ(t˜1 , . . . , t˜m ) = Δlim (t0 , t ),

max (ti −ti−1 )→0

i=1,...,n

max

j=1,...,m

(t˜j −t˜j−1 )→0

and N − (t , t ) − P {Z} = N + (t , t ) − Δlim (t0 , t ) + Δlim (t0 , t ). The numbers Δlim (t0 , t ) and Δlim (t0 , t ) have the following sense: 

Δlim (t0 , t ) = P

 ∞ k=1



 A− 2k (t0 , t )



, Δlim (t0 , t ) = P

 ∞ k=1

P r o o f. By equalities (8.16) and (8.17), it follows that 0≤P

 ∞ k=1

 .

 A− 2k (t0 , t )

  A− (t , t ) − Δ(t1 , . . . , tn ) = δ(t1 , . . . , tn ) ≤ 2k 0

110

Chapter 2

≤P

 ∞

A− 2k (tn−1 , tn )

 +P

 ∞

k=1

A− 2k (t0 , t1 )

 +

k=1

     n ∞ n ∞  − − + P A2k (tj−2 , tj−1 ) = P A2k (ti−1 , ti ) . j=3

i=1

k=1

Passing to the limit as

max (ti − ti−1 ) → 0 and using condition

i=1,...,n

(8.18), we get lim

k=1

max (ti −ti−1 )→0

Δ(t1 , . . . , tn ) = P

 ∞ k=1

i=1,...,n

Similarly, max

j=1,...,m

  A− (t , t ) . 2k 0

Δ(t˜1 , . . . , t˜m ) = P

lim

(t˜j −t˜j−1 )→0

 ∞

 A− 2k (t0 , t )

 .

k=1

It follows from equalities (8.10) and (8.11) that P {Z} = N − (t , t ) − N + (t , t ) +     ∞ ∞ − −   A2k (t0 , t ) − P +P A2k (t0 , t ) . k=1

k=1

Theorem 6 is proved. Theorem 7 gives sufficient conditions for holding assumption (8.18). T h e o r e m 7. Let with probability 1 sample functions x1 (t) be continuous on the segment [tI , tII ] and belong to the set Gu (tI , tII ). Let P {X1 (t) = u} = 0 for every point t ∈ [tI , tII ] except, perhaps, a finite number of points and there exist a constant C > 0 such that the inequality       ti+1 − ti ) < u ∩ X1 (ti+1 ) > u ≤ P X1 (ti ) > u ∩ X1 (ti + 2 (8.19) ≤ Cε(ti+1 − ti ), i = 1, . . . , n−1, holds for every rather small partition tI = t1 < t2 < . . . < tn−1 < tn = = tII , where the function ε(τ ) satisfies the condition

max

lim

i=1,...,n−1

(ti+1 −ti )→0

n−1 ∞  i=1 m=0

 2m ε

ti+1 − ti 2m

 = 0.

(8.20)

111

Crossings Problems

Then it holds that max

lim

i=1,...,n−1

n−1 

(ti+1 −ti )→0

P

 ∞

i=1

A− 2k (ti , ti+1 )

 = 0.

(8.21)

k=1

P r o o f. Consider the sample functions x1 (t) so that at every m = 0, 1, 2, . . . , every l = 1, 2, 3, . . . , 2m , and every i = 1, 2, . . . , n−1   ti+1 − ti l = u. (8.22) x 1 ti + 2m Under the conditions of the theorem, the probability of such sample functions is equal to 1. We will introduce the event     k−1 Bm,k (ti , ti+1 ) = X1 ti + m (ti+1 − ti ) > u ∩ 2         1 k− 2 k ∩ X1 ti + m (ti+1 −ti ) < u ∩ X1 ti + m (ti+1 −ti ) > u . 2 2 If the event

∞ / k=1

A− 2k (ti , ti+1 ) occurs, then either at least one of a de-

numerable number of events B0,1 (ti , ti+1 ); B1,1 (ti , ti+1 ), B1,2 (ti , ti+1 ); B2,1 (ti , ti+1 ), B2,2 (ti , ti+1 ), B2,3 (ti , ti+1 ), B2,4 (ti , ti+1 ); ... Bm,1 (ti , ti+1 ), Bm,2 (ti , ti+1 ), Bm,3 (ti , ti+1 ), . . . Bm,2m (ti , ti+1 ); ... occurred or at some m (from the set of0 0, 1, 2, . . .) and 1 some l (from the −ti set of 1, 2, 3, . . . , 2m ) the equality x1 ti + ti+1 l = u was satisfied. 2m ( 0 1 ) −ti Since P X1 ti + ti+1 2m l = u = 0, we have P

 ∞ j=1

A− 2j (ti , ti+1 )



m

≤

∞  2 

P {Bm,k (ti , ti+1 )}.

m=0 k=1

Under the conditions of the theorem, at every m = 0, 1, 2, . . . and every k = 1, 2, 3, . . . , 2m

112

Chapter 2

  ti+1 − ti P {Bm,k (ti , ti+1 )} ≤ Cε . 2m Therefore, n−1 

P

 ∞

i=1

A− 2j (ti , ti+1 )



  ti+1 − ti 2m Cε . 2m i=1 m=0

∞ n−1 

≤

j=1

Passing this inequality to a limit as

max

i=1,...,n−1

(ti+1 − ti ) → 0 and using

condition (8.20), we obtain result (8.21). Theorem 7 is proved. We note one important case of function ε(τ ) when condition (8.20) is satisfied. R e m a r k 2. Condition (8.20) is satisfied if ε(τ ) = τ 1+α , where α > 0. P r o o f:   n−1 ∞ ∞ n−1   ti+1 − ti 2m (ti+1 − ti )1+α m 2 ε = = m 2 2m 2mα i=1 m=0 i=1 m=0 =

n−1  i=1

 max

≤

(ti+1 − ti )1+α

i=1,...,n−1

1−

α (ti+1 −ti ) n−1 

(1/2)α

∞  m=0

1 2mα

n−1 

(ti+1 − ti )1+α ≤ 1 − (1/2)α i=1 α  max (ti+1 −ti )

=

(ti+1−ti ) =

i=1,...,n−1

1 − (1/2)α

i=1

The last expression tends to zero as

max

i=1,...,n−1

(tII−tI ).

(ti+1 − ti ) → 0. The

remark is proved. 2.8.3 The Application of Gaussian Processes Suppose that X1 (t) is a normal random process. We represent it as

◦

X1 (t) = m(t)+ X1(t), where m(t) = M {X1 (t)} is the mathematical expectation of process ◦

X1 (t), and X1(t) is a centered normal process. We find conditions for ◦

m(t) and X1 (t) when Theorem 7 can be used. For this purpose we consider the probability p(t, τ ) = P {(X1 (t) > u) ∩ (X1 (t + τ ) < u) ∩ (X1 (t + 2τ ) > u)}.

113

Crossings Problems

Let us introduce the following notation:   ◦ ◦ σ1 = σ(t) = M {X1(t)}2 , σ2 = σ(t + τ ) = M {X1(t + τ )}2 , 

r12

r13

◦

M {X1(t + 2τ )}2 ,

σ3 = σ(t + 2τ ) =

◦

◦

◦

◦

M {X1(t) X1(t + τ )} = r(t, t + τ ) = , σ1 σ2

M {X1(t) X1(t + 2τ )} = r(t, t + 2τ ) = , σ1 σ3 ◦

r23

r11 = r22 = r33 = 1,

◦

M {X1(t + τ ) X1(t + 2τ )} = r(t + τ, t + 2τ ) = , σ1 σ3 r21 = r12 ,

r31 = r13 ,

r32 = r23 ;

m1 = m(t), m2 = m(t + τ ), m3 = m(t + 2τ );    r11 r12 r13    2 2 2 R =  r21 r22 r23  = 1 − 2r12 r13 r23 − r12 − r13 − r23 ;  r31 r32 r33  Rij is the algebraic addition of the element rij of the determinant R. Without loss of generality it can be assumed that u = 0. Then p(t, τ ) =

1 (2π)3/2 σ

∞ ·

0 dz1

0

1 σ2 σ3

∞ dz2

−∞

√

0

R

·

  3 (zi − mi )(zj − mj ) 1  exp − Rij dz3 . 2R i,j=1 σi σj

By means of replacement of variables z˜1 =

z1 − m 1 , σ1

z˜2 =

z2 − m2 , σ2

z˜3 =

z3 − m3 σ3

and by means of subsequent replacement  2 1 − r23 r13 r23 − r12 r12 r23 − r13 √ z˜1 +  v1 = z˜2 +  z˜ , 2 2 ) 3 R R(1 − r23 ) R(1 − r23

114

Chapter 2

v2 = 

1 1−

z˜ 2 2 r23

(−r23 ) + z˜ , 2 3 1 − r23

v3 =

√

R˜ z3

this integral is reduced to the form ∞

1 p(t, τ ) = √ √ 2π R

e −

−

v2 3 2

 μ(v  3)

√ m3 R σ3

−∞

   v2 1 2 √ e− 2 Φ η(v2 , v3 ) dv2 dv3 , 2π (8.23)

where

m r23 v3  2 − , 2 2 ) σ2 1 − r23 R(1 − r23   2 2 m1 1 − r23 r13 1 − r23 r13 r23 − r12 √ √ − v2 + v3 , η(v2 , v3 ) = R σ1 R R μ(v3 ) = −

1 Φ(y) = √ 2π

y −∞



 u2 exp − du. 2 ◦

Assuming the stationarity of the process X1 (t), the following ◦

theorem establishes conditions for m(t) and X1 (t) when Theorem 7 can be used and equalities of type (8.21) take place. ◦

T h e o r e m 8. Let X1 (t) = m(t)+X1(t), where m(t) = M {X1 (t)} ◦

is a mathematical expectation of process X1 (t) and X1(t) is a station◦

ary normal process with variance σ 2 = M {X1 (t)}2 and normalized ◦

◦

correlation function r(τ ) = M {X1(t)σX2 1(t+τ )} . Let I be a finite open or closed interval from a domain of definition of process X1 (t), and there exist constants τ0 > 0, A > 0, α > 0, B > 0, and β > 0 such that 1) r(τ ) is twice differentiated on the segment [0, τ0 ], r (0)=0, r (0) < 0, and |r (τ1 ) − r (τ2 )| ≤ A|τ1 − τ2 |α if τ1 , τ2 ∈ (0, τ0 );

(8.24)

2) m(t) is continuous on I and differentiated at every internal point t ∈ I, and |m (t1 ) − m (t2 )| ≤ B|t1 − t2 |β if |t1 − t2 | < τ0 .

(8.25)

115

Crossings Problems

Then positive constants τ ∗ , C, and γ exist so that the inequality p(t, τ ) ≤ Cτ 1+γ holds for every τ ∈ (0, τ ∗ ) and every t ∈ I, where t + 2τ ∈ I. ◦

P r o o f. Since the process X1 (t) is stationary, we have r12 = = r23 = r = r(τ ), r13 = r(2τ ). Without loss of generality it can be assumed that σ = 1. Formula (8.23) for the probability p(t, τ ) takes the form 1 p(t, τ ) = √ √ 2π R

∞ e

−

v2 3 2

√

−m(t+2τ ) R

    μ(v  3) v2 1 2 √ e− 2 Φ η(v2 , v3 ) dv2 dv3 , 2π −∞

(8.26) where

m(t + τ ) rv3 μ(v3 ) = − √ , − 2 1−r R(1 − r2 ) *   1 − r2 r13 r(1 − r13 ) v2 + √ v3 . η(v2 , v3 ) = m(t) + √ R 1 − r2 R

(8.27)

If for every fixed v3 the variable v2 increases from −∞ to μ(v3 ), then the expression (8.27) increases monotonically from −∞ to the value *     1 − r2 r13 rv3 r(1 − r13 ) m(t + τ ) √ + √ v3 = + m(t) − √ R 1 − r2 1 − r2 R R(1 − r2 ) *   1 − r2 r13 − r2 r(1 − r13 ) v3 . m(t + τ ) − √ = m(t) − R 1 − r2 R(1 − r2 ) We denote by h(t, τ, v3 ) this quality. Since Φ(·) is a monotonically increasing function, the following estimate for the inner integral is obtained from (8.26): μ(v  3)

−∞

v2 1 2 √ e− 2 Φ 2π

*

  1 − r2 r13 r(1 − r13 ) v2 + √ v3 dv2 ≤ m(t) + √ R 1 − r2 R



 μ(v  3)

≤ Φ h(t, τ, v3 ) −∞

  v2 1 2 √ e− 2 dv2 ≤ Φ h(t, τ, v3 ) . 2π

116

Chapter 2

Therefore, ∞

1 p(t, τ ) ≤ √ R

√

  v2 1 3 √ e− 2 Φ h(t, τ, v3 ) dv3 . 2π

(8.28)

−m(t+2τ ) R

We will show that the inequality r13 − r2 < 0 is satisfied for a sufficiently small τ . Using Taylor’s formula, we get 1 r = r(τ ) = r(0) + r (0)τ + r (θτ )τ 2 = 1 − aτ 2 , 2 1 r13 = r(2τ ) = r(0) + r (0)2τ + r (θ1 2τ )(2τ )2 = 1 − 4bτ 2 , 2 where 1 a = a(τ ) = − r (θτ ), 0 < θ = θ(τ ) < 1; 2 1  b = b(τ ) = − r (θ1 2τ ), 0 < θ1 = θ1 (τ ) < 1. 2 Hence, r13 − r2 = 1 − 4bτ 2 − (1 − aτ 2 )2 = 2τ 2 (a − 2b) − a2 τ 4 .

(8.29)

We prove that lim r (τ ) = r (0). For this purpose we will consider τ →0+





)−r (τ ) −r (0), Δτ = 0, the function of two variables f (τ, Δτ ) = r (τ +Δτ Δτ

and we find a limit of this function as (τ, Δτ ) → (0+, 0). Representing r (0) as r (0) =

r (Δτ ) − r (0) + ε˜(Δτ ), Δτ

where

lim ε˜(Δτ ) = 0,

Δτ →0

we obtain lim



=

(τ,Δτ )→(0+,0)

f (τ, Δτ ) =

r (τ + Δτ ) − r (τ ) r (Δτ ) − r (0) − − ε˜(Δτ ) Δτ Δτ (τ,Δτ )→(0+,0)   = lim r (τ + θ2 Δτ ) − r (θ3 Δτ ) − ε˜(Δτ ) , lim

(τ,Δτ )→(0+,0)

where 0 < θ2 = θ2 (τ, Δτ ) < 1,

0 < θ3 = θ3 (Δτ ) < 1.

 =

117

Crossings Problems

From (8.24) it follows that lim

(τ,Δτ )→(0+,0)

f (τ, Δτ ) = 0.

Since there exists lim f (τ, Δτ ) = r (τ ) − r (0), we have Δτ →0

lim

lim f (τ, Δτ ) =

τ →0+ Δτ →0

and

lim (r (τ ) − r (0)) = 0,

τ →0+

Therefore,

lim

(τ,Δτ )→(0+,0)

or

f (τ, Δτ ),

lim r (τ ) = r (0).

τ →0+

1 lim a(τ ) = lim b(τ ) = − r (0) > 0, τ →0+ 2

τ →0+

(8.30)

and from (8.29) it follows that there exists τ  > 0 such that for any τ ∈ (0, τ  ) r13 − r2 < 0. We denote by v˜3 = v˜3 (t, τ ) the root of equation h(t, τ, v3 ) = 0 for fixed t and τ , i.e.,   √ r(1 − r13 ) 1 − r2 m(t + τ ) − m(t) . v˜3 = R 1 − r2 r13 − r2 We represent the right side of inequality (8.28) as 1 √ R

√ v ˜3 +ε  R √ −m(t+2τ ) R

  1 − v32 √ e 2 Φ h(t, τ, v3 ) dv3 + 2π 1 +√ R

∞ √ v ˜3 +ε R

  1 − v32 2 √ e Φ h(t, τ, v3 ) dv3 , 2π

where ε = ε(τ ) is a positive number depending on τ . For any τ ∈ ∈ (0, τ  ) and any t ∈ I (where t + τ ∈ I) the value h(t, τ, v√ 3 ) decreases monotonically from 0 to −∞ as v3 increases from v˜3 + ε R to +∞. Let τ ∈ (0, τ  ). Then     √ √  √ 1  1  p(t, τ ) ≤ √ v˜3 + ε R + m(t + 2τ ) R + √ Φ h(t, τ, v˜3 + ε R) , R R   *   1  1 1 − r2 r13 − r2   ε +ε. (8.31) p(t, τ ) ≤  √ v˜3 +m(t+2τ )+ √ Φ R 1 − r2 R R

118

Chapter 2

We estimate the first two terms of the sum on the right side of inequality (8.31). For the first term we have: 1 r(1 − r13 ) 1 − r2 √ v˜3 + m(t + 2τ ) = m(t + τ ) − m(t) + m(t + 2τ ) = r13 − r2 r13 − r2 R =

(1 − aτ 2 )4bτ 2 2aτ 2 − a2 τ 4 m(t+τ )− m(t)+m(t+2τ ) = 2 2 2 4 2aτ − 4bτ − a τ 2aτ 2 − 4bτ 2 − a2 τ 4 =

2a(m(t + 2τ ) − m(t)) − 4b(m(t + 2τ ) − m(t + τ )) + 2a − 4b − a2 τ 2 +

τ 2 [−4bam(t + τ ) + a2 (m(t) − m(t + 2τ ))] . 2a − 4b − a2 τ 2

This implies that 1 √ v˜3 + m(t + 2τ ) = ε1 (t, τ ) + ε2 (t, τ ), R

(8.32)

where −4(− 12 r (0))(− 12 r (0))m(t) ε2 (t, τ ) = −r (0)m(t), (8.33) = τ →0+ τ2 −r (0) + 2r (0) lim

2a(m(t + 2τ ) − m(t)) − 4b(m(t + 2τ ) − m(t + τ )) = 2a − 4b − a2 τ 2 2am (t + θ4 2τ )2τ − 4bm (t + τ + θ5 τ )τ = = (8.34) 2a − 4b − a2 τ 2   (4a − 4b)τ m (t + θ4 2τ ) + 4bτ m (t + θ4 2τ ) − m (t + τ + θ5 τ ) , = 2a − 4b − a2 τ 2 ε1 (t, τ ) =

0 < θ4 = θ4 (t, τ ) < 1, 0 < θ5 = θ5 (t, τ ) < 1. From condition 2) of the theorem it follows that m(t) and m (t) are bounded on I. Due to the boundedness of m(t) and due to (8.33), we see that there exists a constant D1 such that |ε2 (t, τ )| ≤ D1 τ 2 for any t ∈ I and any sufficiently small τ . If τ ∈ (0, τ20 ), then from (8.24) and (8.25) it follows that |2a − 2b| ≤ A(2τ )α , |m (t + θ4 2τ ) − m (t + τ + θ5 τ )| ≤ B(2τ )β . (8.35)

119

Crossings Problems

By definition, put δ˜ = min(α, β). Taking into account inequalities (8.35) and the boundedness of the function m (t), we get from (8.34) that there exists a constant D2 such that ˜

|ε1 (t, τ )| ≤ D2 τ 1+δ

for any t ∈ I and any sufficiently small τ . Using equality (8.32), we conclude that there exist positive constants D and τ  so that    1   √ v˜3 + m(t + 2τ ) ≤ Dτ 1+δ ∀τ ∈ (0, τ  ) and ∀t ∈ I, (8.36)  R  ˜ = min(1, α, β). where δ = min(1, δ) We will now estimate the second term on the right side of (8.31). For the determinant R we have R = 1 + 2r2 r13 − r13 − 2r2 = (1 − r13 )(1 + r13 − 2r2 ) =   a2 τ 2 = 4bτ 2 (2 − 4bτ 2 − 2 + 4aτ 2 − 2a2 τ 4 ) = τ 4 (a − b)16b 1 − . 2(a − b) Hence, 2 * 2τ 2 (a − 2b) − a2 τ 4 1 − r2 r13 − r2 3 2aτ 2 − a2 τ 4 3   =3 = 2 4 4 R 1−r 2aτ 2 − a2 τ 4 a2 τ 2 τ (a − b)16b 1 − 2(a−b) 2(a − 2b) − a2 τ 2 .   a2 τ 2 (a − b)16b 1 − 2(a−b) (2a − a2 τ 2 )

= ' τ Let

ε = τ 1+δ1 ,

where 0 < δ1
h0 } = 0.03.

wind force

0

capacity of wind wind velocity at altitude 6 m above sea level, m/s average when squall 00.0÷00.5 01.0

1 2

00.6÷01.7 01.8÷03.3

03.2 06.2

3 4 5 6 7 8

03.4÷05.2 05.3÷07.4 07.5÷09.8 09.9÷12.4 12.5÷15.2 15.3÷18.2

09.6 13.6 17.8 22.2 26.8 31.6

9 10 11

18.3÷21.5 21.6÷25.1 25.2÷29.0

36.7 42.0 47.5

Table 3–3 capacity of wave sea h0 , m force B

1

0.00÷0.25

2

0.25÷0.75

3 4 5 6

0.75÷1.25 1.25÷2.00 2.00÷3.50 3.50÷6.00

7

6.00÷8.50

8

8.50÷11.0

9 12

> 29

> 11

53.0

We will consider values B = 0, 1, 2, 3, 4, 5, 6 and we have assumed that b0 (B) = b0 (6) B6 . The coefficient b0 (6) is chosen so that the

147

Applications in Aviation

conditions

P {heave amplitude > 1.3 m} = 0.03 and P {pitch amplitude > 1.6◦ } = 0.03

are satisfied at sea force B = 6. In other words, we consider that the heave amplitude does not exceed 1.3 m with a probability of 0.97 at sea force B = 6, and we consider that the pitch amplitude does not exceed 1.6 ◦ with a probability of 0.97 at sea force B = 6. Assuming the random process yk (t) for heaving motions and the random process ϑk (t) for pitching motions as the solutions to equation (3.14), we have obtained the following results (see Fig. 3–5) for the standard deviations σyk (B) and σϑk (B): σyk , m; 6

σϑ k , ◦

0.50 σ  ϑk   σ    yk 0.25                      -B 0.00  0 2 4 6 Fig. 3–5 The intensity of heaving and pitging motions

3.3.5 Control Law The control law for tracking the nominal trajectory is chosen as follows: Δδ = δ − δ0 = kα Δα + kωz ωz + kh ε + kvy ε1 , (3.15) 1 1 dε 1 where Δα = Δαg + αw , dε dt = − T ε1 + T dt , T is a time constant. Three types of linear deviations are used as an error signal ε (see Fig. 3–6): 1) ε = Δy if the airplane keeps track of the glissade that is stabilized in space over an angle and responds to vertical and keel of ship oscillations;

148

Chapter 3

2) ε = Δyg − yk = Δy + xϑk if the airplane keeps track of the glissade that is stabilized in space over an angle and responds only to vertical ship oscillations; 3) ε = Δyg = Δy + yk + xϑk if the airplane keeps track of the glissade that is immovable for an observer of the system Og xg yg . In Fig. 3–6, the symbol I is used to denote the glissade in case 1), the symbol II is used to denote the glissade in case 2), and the symbol III is used to denote the glissade in case 3). rAirplane centre of mass y yg 6 Δyg BM.. . 6 BM .. ......B B III Q?B Δy QB B Q B I Q B Q II Q ....B.BN. QQ Q Q B x . ... Q...... Q Q B 1 . . ...  . Q ......Q Q   ...... Q Q .. .......  ...... Q.... B .... Q Q  .. .... . Q B  . . . . . . . . .Q ... Q . ..  .. .. ϑk Q QQ .....Q . .. . .B.. .. . .. . . . . O .. ... ..... Q  ..... . . . Og xg D . . . . D ....... D ... D .. .. .. Fig. 3–6 Types of followed glissades

In order to choose the coefficients kα , kωz , kh , kvy , we explore the solution of the system ⎛ ⎞ ⎞ ⎛ Δαg Δαg ⎜ ⎟ ⎟ d ⎜ ⎜ Δϑ ⎟ = B ⎜ Δϑ ⎟ , (3.16) ⎝ ωz ⎠ dt ⎝ ωz ⎠ Δyg Δyg where B = ||bij || is a 4×4-matrix: b11 = −Y α+Y δ (kvy v0−kα ),

b12 =

sin θ0 g−Y δ kvy v0 , v0

b13 = 1−Y δ kωz ,

149

Applications in Aviation

b14 = −Y δ kh , b31 =

Mzα

−

b21 = 0,

Mzδ (kvy v0

b34 = Mzω¯ z + Mzδ kωz ,

b22 = 0,

− kα ),

b32 =

b41 = −v0 ,

b23 = 1,

Mzδ kvy v0 ,

b42 = v0 ,

b24 = 0, b33 = Mzδ kh ,

b43 = 0,

b44 = 0.

System (3.16) is obtained from linearized equations (3.2) and control law (3.15) when αw = 0, wx = 0 and when ε1 = dε dt , ε = Δy = Δyg (in this case, deviations Δy and Δyg coincide because to choose the coefficients kα , kωz , kh , kvy the ship motion is not taken into consideration and the airplane is studied as an isolated dynamic system). The coefficients kα and kωz are chosen so that the damping coefficient ξ ≈ 0.7 and the response time trsp ≈ 1 s are in the short-periodic motion of the airplane. For this purpose, we will consider the characteristic equation of system (3.16) with kh = 0 and kvy = 0: (−λ)[λ2 − (˜b11 + b13 )λ + ˜b11 b13 − b13˜b31 ] + ˜b12 b31 = 0, where ˜b11 = −Y α − Y δ kα ,

˜b12 = − sin θ0 g, v0

˜b31 = M α + M δ kα . z z

We have neglected the free term ˜b12 b31 in this characteristic equation and we have found the roots of the quadratic equation λ2 + 2ξω0 λ + ω02 = 0, where

ω02 = ˜b11 b13 − b13˜b31 ,

2ξω0 = −˜b11 − b33 .

The response time1 trsp is expressed through ω0 and ξ: *   1 1 trsp =  −1 . π − arctan ξ2 ω0 1 − ξ 2

(3.17)

For calculations, we have selected the values kα and kωz , so that the damping coefficient is ξ = 0.73 and the response time is trsp = 1.06 s. 1 The characteristic equation λ2 +2ξω λ+ω 2 = 0 corresponds to the differential 0 0 equation z  + 2ξω0 z  + ω02 z = 0. The solution of this differential equation with initial conditions z(0) = z0 , z  (0) = 0 is 1 0   ξ sin(ω0 1 − ξ 2 t) . z(t) = z0 exp{−ξω0 t} cos(ω0 1 − ξ 2 t) +  1 − ξ2

The response time trsp is defined as the time when function z(t) vanishes for the first time: z(trsp ) = 0. From this we get result (3.17).

150

Chapter 3

Δyg , m 6

  k h , kv y 3 1.0 ..................... .................. ... .... .........   ... ..... ........ ... .... ......... k h , kv y 2 ... .... ...... ..... ... .... ..... ... ....   .... ... ... .... k h , kvy 1 . . ... .... ... 0.5 .... ... ... .... ... ... .... ... ... .... . . .... ... ... .... ... ... ..... . ... ... ...... ... ... ........ .... ......... ... . ........... . ... ..... ........... . . .... ....... ......... - t, s 0.0 . ..... . ............ ............................................................................... . . . . . . . . . . . . . . . . . . . . ............................. ...................... ...... 0 5 10 15 Fig. 3–7 Transient response plots

To select the coefficients kh , kvy , the root hodograph was built for the characteristic equation det(B − λE) = 0 with the selected values of the coefficients kα and kωz , where E is the identity 4×4-matrix. The calculations were performed   for all three  pairs of values from the coefficients kh and kvy : kh , kvy 1 , kh , kvy 2 ,   and kh , kvy 3 . The transient processes Δyg (t) in system (3.16) with the selected values of the coefficients kα , kωz , kh , kvy and the initial conditions Δα = 0, Δϑ = 0, ωz = 0, and Δyg = 1 m are shown in Fig. 3–7. 3.3.6 Calculation System of Equations and Calculation Formulas for Estimating the Probability In the coordinate system Oxy, which is connected with the ship, the airplane movement as a material point is described by the random processes x(t) and y(t). These processes are projections from the −−→ radius-vector OC of the airplane’s centre mass at the axis Ox and Oy. The mathematical expectations M x(t) and M y(t) are equal to M x(t) = x0 + (v0 cos θgl − V )t and M y(t) = y0 − v0 sin θgl t.

151

Applications in Aviation

Consequently, the trajectory corresponding to the mathematical expectations M x(t) and M y(t) is a straight line. For an observer of system Oxy, the angle between this line and the axis Ox is  θgl

 = arctan

v0 sin θgl v0 cos θgl − V

 .

The origin of coordinates O is on this line and coincides with the calculated point of the airplane’s landing. To estimate the probability of landing on the predetermined portion (x , x ) of the deck, we have defined the probability that the first achievement of zero level by process y(t) occurs on the interval (t , t ), where M x(t ) = x , M x(t ) = x . Restrictions on the relative vertical speed vy and restrictions on the relative pitch angle θn are considered as constraints on the phase coordinates at the time of landing. The calculation system of equations has the form dΔz = AΔzdt + BdQ(t),

(3.18)

where Δz is a 16-component vector of the extended phase space:  Δz =

Δy, Δvy , Δϑn , ωz , yk ,

dyk d2 yk d3 yk , , , dt dt2 dt3

ϑk ,

dϑk d2 ϑk d3 ϑk , , , w x , w y , x y , ε1 dt dt2 dt3

T ;

Q(t) is a 4-component vector of independent Wiener processes, A(t) is a 16×16 - matrix, B(t) is a 16×4 - matrix. System (3.18) is obtained by combining the following equations: • equations (3.12), where formulas (3.6)–(3.8) and selected control law (3.15) are taken into account; • equations (3.13) that describe forming filters for the longitudinal and vertical components of the atmospheric turbulence; • equations (3.14) that describe the forming filter for heaving motions; • equations (3.14) that describe the forming filter for pitching motions. If glissade I is tracked, then the non-zero elements of matrix A = ||aij || are as follows:

152

Chapter 3

a12 = 1,

a22 = −Y α − Y δ kα ,

a21 = Y δ v0 kh ,

a23 = g sin θ0 +v0 (Y α +Y δ kα ),

a29 = g sin θ0 + v0 (Y α + Y δ kα ) + vx0 (−Y α − Y δ kα ),

a27 = 1,

a2,10 = (−Y α − Y δ kα )(−x0 + vx0 t) − 2vx0 , a2,13 =

a26 = −Y α −Y δ kα ,

a24 = Y δ v0 kωz ,

ρv0 Cyn , m/S

a2,14 =

a3,10 = −1,

a34 = 1,

a43 = Mzα + Mzδ kα ,

a4,10 = −

ρv0 Cyα ρv0 Cxn + +Y δ kα , 2m/S 2m/S a41 = Mzδ kh ,

a4,16 = Mzδ kvy , a9,10 = 1, v0 , Lx

a56 = 1,

a86 = −a2hv , a10,11 = 1,

a12,10 = −a2pt , a13,13 = −

Mzα + Mzδ kα , v0

a46 = −

Mzα + Mzδ kα , v0

(Mzα + Mzδ kα )vx0 + Mzα + Mzδ kα , v0

(Mzα + Mzδ kα )(−x0 + vx0 t) , v0

a85 = −a1hv ,

a2,16 = Y δ v0 kvy ,

a42 = −

a44 = M ωz + Mzδ kωz ,

a49 = −

a2,11 = x0 − vx0 t,

a4,14 =

a67 = 1,

a87 = −a3hv , a11,12 = 1,

a12,11 = −a3pt ,

a14,15 = 1, a16,2 =

1 , T

a15,14 = −

Mzα + Mzδ kα , v0

a78 = 1, a88 = −a4hv ,

a12,9 = −a1pt , a12,12 = −a4pt , v02 , L2y

a15,15 = −

2v0 , Ly

1 a16,16 = − . T

When glissade II is tracked elements of matrix A = ||aij || are the same as in the case of tracking glissade I, except a29 = g sin θ0 +v0 (Y α+Y δ kα )+vx0 (−Y α−Y δ kα )+(−x0+vx0 t)Y δ v0 kh , a49 = −

(Mzα + Mzδ kα )vx0 + Mzα + Mzδ kα + (−x0 + vx0 t)Mzδ kh , v0 vx0 −x0 + vx0 t , a16,10 = . a16,9 = T T

153

Applications in Aviation

When glissade III is tracked elements of matrix A = ||aij || are the same as when tracking glissade II, except a25 = Y δ v0 kh ,

a45 = Mzδ kh ,

a16,6 =

1 . T

The elements of matrix B = ||bij || do not depend on the type of glissade that is tracked. The non-zero elements bij are b83 = −a4hv b0hv ,

b73 = b0hv ,

* b13,1 = σwx

b12,4 = −a4pt b0pt ,

b11,4 = b0pt ,

2v0 , Lx

' b14,2 = σwy '

√

b15,2 = (1 − 2 3)σwy

3v0 , Ly

v03 d + b14,2 (t). L3y dt

We present the calculation formula for estimation Pˆ of the probability that the initial contact happens on deck site (x , x ) and at the moment of this contact the relative vertical speed vy and the relative pitch angle ϑn of an airplane will be within the prescribed limits. In this case, formula (2.4) takes the form t Pˆ = −



0 dt

t

−vymax

ϑ nmax

du2

u2 ρt (0, u2 , u3 )du3 ,

(3.19)

ϑnmin

where ρt (u1 , u2 , u3 ) is the joint distribution density of values u1 (t) ≡ ≡ y(t), u2 (t) ≡ vy (t), and u3 (t) ≡ ϑn (t); vymax is the maximum permissible (as absolute value) relative vertical speed of an airplane at the time of landing; ϑnmin is the minimum permissible relative pitch angle of an airplane at the time of landing, ϑnmax is the maximum permissible relative pitch angle of an airplane at the time of landing. Using Gaussian’s view of the distribution density ρt , we will analytically carry out an internal integration in (3.19). For this purpose, we will consider the random vector z(t) with components z1 (t) = y(t), z2 (t) = vy (t), z3 (t) = ϑn (t), zi (t) = Δzi (t), i = 4, 5, . . . , 16; cross-correlation functions

154

Chapter 3

kij (t) = M {[zi (t) − mi (t)][zj (t) − mj (t)]}, i, j = 1, 2, . . . , 16, where mi (t) = M {zi (t)}; variances σi2 = kii (t), i = 1, 2, . . . , 16; and correlation coefficients rij (t) =

kij (t) . σi (t)σj (t)

The functions kij (t) are determined by numerical integration of the matrix differential equation dK = AK + KAT + BB T , dt

(3.20)

where K = ||kij ||. The distribution density ρt (u1 , u2 , u3 ) is   3  (u −m (t))(u −m (t)) 1 Rij (t) i σi i (t)σjj(t) j exp − 2R(t) i,j=1   , ρt (u1 , u2 , u3 ) = (2π)3 σ1 (t)σ2 (t)σ3 (t) R(t) (3.21) where Rij (t) is an algebraic complement of the element rij (t) of the determinant    1 r12 (t) r13 (t)   1 r23 (t)  . R(t) =  r21 (t)  r31 (t) r32 (t) 1  Using representation (3.21), we will analytically perform one integration in (3.19) and the formula (3.19) takes the form t Pˆ = −

ϑ nmax

dt t

Here



f (vymax , u3 , t)du3 .

(3.22)

ϑnmin

F1 + F 2  · f (vymax , u3 , t) =  3 (2π) σ1 (t)σ2 (t)σ3 (t) R(t)   2  1 m1 (t) (u3 −m3 )2 m1 (t)(u3 −m3 ) · exp − + +2r13 (t) , 2 (t)) 2(1−r13 σ12 (t) σ32 (t) σ1 (t)σ3 (t) (3.23)

155

Applications in Aviation

where   1 2 2 F1 = exp{−a(−vymax − b) } − exp{−ab } , 2a  √  √ √ b π F2 = √ φ(− ab) − φ( a(−vymax − b)) , 2 a 2 (t) 1 − r13 , 2R(t)σ22 (t)  σ2 (t) m1 (t) − b = b(u3 , t) = m2 + (r13 (t)r23 (t) − r12 (t)) 2 1 − r13 (t) σ1 (t)  u 3 − m3 −(r12 (t)r13 (t) − r23 (t)) , σ3 (t) z 2 exp{−x2 }dx. φ(z) = √ π

a = a(t) =

0

We will consider vymax = 7 m/s, ϑnmin = 1◦ , and ϑnmax = 14◦ : i.e., we estimate the probability that the airplane landing occurs on the site deck of point x to point x and, at the moment of landing, the conditions −7 m/s ≤ vy ≤ 0 m/s and 1◦ ≤ ϑn ≤ 14◦ are satisfied. The distance x − x is approximately 100 meters. The probability of the airplane landing on site (x , x ) without considering the restrictions on vy and ϑn is estimated by the value t



0 −u2 ρt (0, u2 )du2 .

dt t

(3.24)

−∞

Given that the  exp − 2(1−r12 =

12 (t))



ρt (u1 , u2 ) = (u1−m1 (t))2 σ12 (t)

1 (t))(u2−m2 ) 2) −2r12 (u1−m + (uσ2−m 2 σ1 (t)σ2 (t) 2 (t)  2 (t) 2πσ1 (t)σ2 (t) 1−r12

the inner integral in (3.24) can be calculated analytically:

2

 ,

156

Chapter 3

0 

−∞

 2 (t)σ (t) 1 − r12 2 · −u2 ρt (0, u2 )du2 = 2πσ1 (t)

m2 (t) · exp − 12 2σ1 (t) where



2

exp{−β (t)} +

√

 (3.25) πβ(t)[1 − φ(−β(t))] ,

m2 1 (t) r12 (t) m σ1 (t) − σ2 (t)  . β(t) = 2 (t)) 2(1 − r12

The maximum possible differences in the estimates of (3.19) and (3.24) from the exact values of the correponding probabilities do not exceed  t ∞ (3.26) ΔP = dt u2 ρt (0, u2 )du2 . t0

0

Here, as before, the inner integral is calculated analytically: 

∞ u2 ρt (0, u2 )du2 =

2 (t)σ (t) 1 − r12 2 · 2πσ1 (t)

 (3.27)  0 2  √ m1 (t) 2 · exp − 2 exp{−β (t)} − πβ(t)[1 − φ(β(t))] . 2σ1 (t) 3.3.7 Analysis of Calculations Results and Estimate of Efficiency of the Method The calculations were performed for sea force B = 0, 1, 2, 3, 4, 5, 6; for the three pairs, (kh , kvy )1 , (kh , kvy )2 , (kh , kvy )3 of coefficients kh and kvy in control law (3.15); and for three types of tracked glissade: I, II, and III. If we recall that glissade III is fixed in space; glissade II repeats vertical ship-rocking yk and remains parallel to glissade III; glissade I is also stabilized at an angle, it repeats the vertical shiprocking yk and it tracks the pitch ship-rocking. If, then, the pitch angle of ship at some moment is equal to ϑk and the distance from the airplane to the center of pitch rocking of the ship at this moment is equal to s, then glissade I is shifted in the vertical plane by the distance −ϑk s (see Fig. 3–6). We use the following notation:

157

Applications in Aviation

• σzI is the standard deviation of parameter z 1 caused by the combined influence of atmospheric turbulence wx , wy and ship’s motions yk , ϑk when tracking glissade I; • σzI,w is the standard deviation of parameter z caused by the influence of only atmospheric turbulence wx , wy when tracking glissade I; • σzI,u is the standard deviation of parameter z caused by the influence of only ship’s motions yk , ϑk when tracking glissade I. σy , m 6

2.5

2.0

1.5

1.0

I II .. .. .. III .... .... .... . . . .. .. .. .. .. .. ... .... .... . . ... ... ... (kh , kvy )1 .... .... ..... . . ... .. ... ... ..........  . . . .. ... .............................................. .... ........ .. I ..................................................... .... ............... . . ............................................... ... . . . ......................................... .. .................... . . . .................................... . . .. .................................................................... ...... I .................................................. ....... ........ ... ........ . . .. II . . ..... .. .. .......... . . . . . . ... ... ... ... III ....... ... .... ............ . . . . . . .. ... .... ... ... ..... ............ . . . . . . . . ... ... .......... .... ...... .......... . . . . . . . . . . ... .... ....... (kh , kvy )2 ..... ........ ................... . . . . . . . . . . . . . . . ......... .... ... ... .............. ...... ...... .....  .................. .............. .. ........ ............. . . . . . . ............................................................................................................. . . .... II . . ............................ .......................... .... ........ . . . . . . . . . . . . . . . . . . ....................... . . . . ... ... III ....................................................................................................... .... .... .... ... ..... ........ .......... ................................. . . . . (kh , kvy )3 . . . . . . . . . . . ......... ...... ..... ........... ....... ....... ............ ...................... . . . . . .  . . . . . . . . . . . . . . . . . .. .. .... ............ .......... ................. ........................................................................................................................................................................................................................................... -B

0.5 0

2

4

6

Fig. 3–8 Standard deviation of the height y above the deck surface at the moment t 1 The following values will be considered as z: the height y above the deck surface, the relative vertical speed vy , and the relative pitch angle ϑn of the airplane.

158

Chapter 3

Similar notations σzII , σzII,w , and σzII,u are used when tracking glissade II. Similar notations σzIII , σzIII,w , and σzIII,u are used when tracking glissade III. We will discuss the results presented in Fig. 3–8 — 3–25. Fig. 3–8 presents the results of the calculations for the root meansquare deviation σy of height above the deck surface at the moment t ; this depends on sea force B. The separate influence of atmospheric turbulence and the ship’s motions on σy at t = t is shown in Fig. 3–9. σy , m 6

2.5

.. ... ... .. . ... ... ... .. . h vy 1 ... ... ... .. . ... .. ............................. .... .......................... ... ........................ . . . ...................... .. .................... .... ................... .... .................. ... ................ ... ............... . . . . . .............. .............. ...... ...... ............. ........................................

(k , k )

 

2.0

1.5

1.0

0.5

0.0

.. I ... . . .. ... . . .. ... .I . . . ... . . . . (kh , kvy )3 . . ....... .. .............. ... . . @ . . . . .. .... .. (kh , kvy )2 @ ... ............. ... . . . . . ..... @... @ @ ... .............. . (kh , kvy )2 . . . . ...... (kh , kvy )1 @ .. .. ......  ... @ ..... ........................ ...I . . ... . ................... . @ ........... ................................. . @ . . . ........................... . . . . . . . .......................... . ......... . . ...................... ...................... . . . ....... ... @.................................... .................... ...................... . . . ..................... ................................................... . . . . @ (kh , kvy )3 .. .. . . . ..................... ... ..... @ ... ... . . @ ...... . . . .......  .. .... ........ .. . . (kh , kvy )1,3 ........ . . . . . . . . . . ... . ...... . ......... .............................................................................................................................................. ........ ... .......... ......................................... . . .............................................................................. . .  .. .. ... .. ... ..... . . . (k , k ) . . . . .. .. .. .. .. .. . . II . . . . . . . h vy 2 . . . . . . . . . . . . . . .  .. .. . . . . . III ... . . . . . .. . . . . .  . . . . .. .. .. .. .. .. .. . . . . .. . . . . . . . . . . . . . . ........ . . . . . . . . . . . . .. .. . . . .. . .. .. .. . . . . . . . . . . . . ........ . . . . . . .. .. .. ... ... .. .. .. . . . . . . . . .@ . . . . . . . . . . . (k , k ) .... . . . ..... ....... @ h vy 1,2,3 ......... . . .. . . .. .. .. .. .. .. .. .. .. . . . . . . . . . . . . ... . . . . . . . ..... . . . -B 0 2 4 6 Fig. 3–9 Separate influence of atmospheric turbulence (continuous line) and ship motions (broken line) on σy at the moment t

159

Applications in Aviation

We note the following. As you can see, σyI,w (B) = σyII,w (B) = = σyIII,w (B), and this is understandable because in the absence of the ship’s motions yk , ϑk glissades I, II, and III are the same. The value σyw (B) does not increase, and may even decrease, with increasing B for small values B. This is due to the atmospheric turbulence model. As seen from Fig. 3–4, the turbulence intensity decreases with the increasing B for small values B: the average wind increases but pulsations (characterized by values σwx and σwy ) wilt. An airplane with the pair of coefficients (kh , kvy )3 is the most resistant to the actions of atmospheric turbulence. This can be explained by the fact that an airplane with the pair of coefficients (kh , kvy )3 has the best quality of transitional processes (see Fig. 3–7). I. II. . III . .. .... .... . . ... .... .... . .. .. ... .. ... .. ... .... ... ... . ... (kh , kvy )1 .. .......... . . .. ... ... ... ...........  . . .. ... ... ............................................. .... .... ... ................................................... .................. . ................................................ . . . . .. .......................................... ..... ......... .................................... .I .......................................................................... .. .................................................. ... ... . .. . I ... ..... . . .. .. ... .. .. II ... ...... ..... . . . . . .... ... ... ... III .... ...... .......... . . . ... .. .. . ..... .... ... ... ....... ........ .............. . . . . . . ....... ....... ............. . . ........ (kh , kvy )2 ........ ........ ............... . . . . . . . . . . .... . . . . . . . . . . . . . . . . . . . . . ...... ...... .... .... ...............  ....... ....... .... ................... ....... ............................. . . . ................................................................................................................... . . . ... II ... ..... .... ............................ .... ................... .......... ....................... .......... III . ......................................................................................................... . . ....................................... ... .. ..... ..... .... (kh , kvy )3 ................. .......... . . . . . . . . . . . . . . . . . . . ....... ...... ........... .............. ......... ........  .................. .......................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ................................................................................................................................................................................................................................................... σy , m 6

2.5

2.0

1.5

1.0

-B

0.5 0

2

4

Fig. 3–10 Standard deviation of the height y above the deck surface at the moment tland

6

160

Chapter 3

σy , m 6

2.5

.. ... ... .. . ... ... ... .. . h vy 1 ... ... ... .. . ... ... ............................. ... .......................... .... ........................ . . . ...................... .. .................... .... ................... .... .................. ................ .... ............... ... . . . . .............. ..... .............. ...... ............. ........................................

(k , k )

 

2.0

(kh , kvy )3

1.5

. ... ... ... ... . . . .... .... ... h vy 2 ... . . . . .... .... .... h vy 2 .... . . . . . ...... ...... h vy 1 ...... ...... ...... . . . . . . . ................................. ....... ............................. ........ .......................... ........ ....................... .......... ...................... .... ........... . . . . . .................... . . . . . . ...... . ................................... ........................................ ...... ......... ...... . . . . . h vy 3 ...... ...... ....... ....... ...... . . . . . . ........ ........ ........ ......... ............................................................................................................................................................ ......... . . . ........................................ . . . . . . .....................................................................

(k , k )

@

@ @

.. ...

...

.

.. ...

... I

. @ @ @ ... ..I . @ @ . ... . @...  . @ @ ... ...  @ @ . ... . . @ @ . . @ @ ... @ .. .@ @@. . . . . 1.0 .@ (k , k ) . . . . @ .. @ @. . . . @ ... . .@  . . . . @ .. @ ... ....I  @ @ ... ... ..... . . . . . . @ . . @ @. . . . . .. ... ... . . . . . . . . . .@ @ . . . . . . . @ ... 0.5 .. . . . ..... @ . . .@ .... ..... ... . . . . . . . . . . . . . . .. .. .. .. .. .. II . . . . . . @ . . . . . . . . . . . . . . . . . . . . . . . . . .. . . .... . . . . . .. .. .. .. .. .. .. .. .. . . . . . .. .. .. .. III .... . . . . . . . . . . . . . .. .. ... ... ... ... ... .. .. .. .. .. . . . .. . . . .. .. .. . .. . . .. . . . . . .. . . . . . .... . . . . . . . . . . . . . . .. .. .. .. .. .. . . . . .. .. .. .. .. .. .. .. .. . . . . . .. .. .. .. .. .. . . .. ..... .. .. . .. .. .. .. . . . . ... @ (kh , kvy )1,2,3 . .. . . .. . .. @ . .... .. .. .. .. .. .. ......... . .. ... . . . -B .. 0.0 0 2 4 6 (k , k )

(k , k )

Fig. 3–11 Separate influence of atmospheric turbulence (continuous line) and ship motions (broken line) on σy at the moment tland

As you can see, σyu (B) varies linearly with B. Of course, this is not accidental because this is explained by the linearity of the ship’s motions model. We note that σyIII,u does not depend on values kh and kvy . This is easy to explain: glissade III is stabilized in space and in the absence of atmospheric turbulence the airplane will fly exactly according to this glissade, regardless of values kh and kvy . In this

161

Applications in Aviation

case, perturbations are absent in equations of airplane motion and ship motions will not affect the movement of airplane. σy , m 6 2.0

1.5

1.0

.... I ... ... . . .. .... . I .... . . ... . ... ... . . . . . ... ... at the moment tland ... .... ... .... . . . . . . . .. .... @ @ .... .... .... . . @ @ ......... . . .... .. .... @ @ .... at the moment t .... .... . . . . . . @........ .... @ .. @ @ .............. @ ....... @ . . . . . . . @ . . . . . . . . @ . . . . @ ... II ...... ...... ....... ........ ........ . . . @ . . . . . . . . . . . . . . . . @ ..... ..... II ....... @ ........ .. . ......... ......... @ .... ..... ......... ......... . . . . . . . . . . . . . . . ................. . . . . . . . @ . . . . . . . . . . . . . . . . . . . . . ...... ...... ......... ...................... @ ............... ............ .. @ ......................... ............................................. . . . . . . . . . . . . . . . . . . . . . . . . . . . .............. ........... .............. ...............................................................................................................................................................................................................................................................

-B

0.5 0

2

4

6

Fig. 3–12 Comparison of values σy at the moments t and tland for (kh , kvy )3

Note that σyI,u > σyIII,u . The spectral density of ship’s motions, both vertical and pitching, have a pronounced maximum, which corresponds to a certain period Tk of the ship’s motions. The airplane has time to track the movement of glissade I if the response time trsp of the transition process (see Fig. 3–7) is at least several times smaller than a quarter of the period of ship’s motions, i.e. , 14 Tk . In our case, 1 4 Tk < 3 s and this condition is not performed. As a result, the airplane did not have time to track the movement of glissade I and in order to achieve the lower values σy tracking the fixed glissade III is more favorable. Tracking glissade I only leads to more build up and increases σy . The difference σyI,u − σyIII,u decreases with the transition (kh , kvy )3 → (kh , kvy )2 → (kh , kvy )1 . To pair (kh , kvy )3 assign the

162

Chapter 3

trsp ≈ 4 s; to pair (kh , kvy )2 assign the trsp ≈ 7 s; and to pair (kh , kvy )1 assign the trsp ≈ 13 s. The condition trsp > 14 Tk means that the reaction time of the airplane is more than the time of moving glissades I and II from the extreme amplitude position to the middle position. The more this condition is satisfied during tracking glissade I, the closer the behavior of the airplane will be to its behavior when the tracked glissade occupies the middle position, i.e., coincides with glissade III, which is fixed in space. For this reason, the charts σyI,u (B) and σyIII,u (B) approach each other with the transition (kh , kvy )3 → (kh , kvy )2 → (kh , kvy )1 (see Fig. 3–9). σvy , m/s 6 2.0

1.5

1.0

0.5

.I .... .... . . . . .... .... .... . . . . .... .... .... . . . ... .... I ..... ....... ..... . . . ....... . . . . . . . . . .... ...... ...... ..... ...... . .... . . . . . . . . ... ....... ..... ....... .I ....... ..... . . . . . . .... . . . . ..... ....... . ..... . . . . . . . . . . . (kh , kvy )1 . . . ...... ..... .... ........ .... ........ .... ....... ....... . . . . . . . . . . . . . .  . . . . . . II ... ........ ....... ..... ..... ....... ..........  .......... ..... ............... .............. III . . . . . . . . . . . . . . . (k , k ) . . . . . .  h vy 2 ...... ...... .... ...... ............... ............ . ................... ....... ...... .............. .................................................................@ ....... ................. ............................... .   . . . . . . . . . . . . . . . . ..... ...... ............. ........... ......... ....... ....... ..... @II   ................................... .............. ......... ......... ........ .............................. .................................. ............................................... ............... .............. ........... ............................................................................................................................................................................................................................................................................................................................................................................ .......... .................................................................................................. @ ............... ................................................................................................@ ...........................................................................................................................................................................................................................@ @ @ ............................

@III

A (k , k ) A h vy 3

@II

@III

-B

0.0 0

2

4

6

Fig. 3–13 Standard deviation of the relative vertical speed vy at the moment t

163

Applications in Aviation

The comparative qualitative behavior of σyI (B), σyIII (B) is now explained by considering the joint effects of σyI,u (B) and σyI,w (B), σyIII,u (B) and σyIII,w (B). Like a decrease in the difference σyI,u (B)− −σyIII,u (B), the difference σyI (B) − σyIII (B) also decreases with the transition (kh , kvy )3 → (kh , kvy )2 → (kh , kvy )1 . We note that the action of atmospheric turbulence leads to the fact that this decrease is more pronounced. It is important to also note that, when tracking glissade I, for (kh , kvy )2 and (kh , kvy )3 the impact of ship’s motions is more powerful than the impact of atmospheric turbulence; in this case the graphs σyI (B) increase monotonically with the increasing sea force B. This starts with B = 0. In all other cases (see Fig. 3–8), σy first decreases with the increasing sea force B and only when B ≈ 4 begins to increase. σvy , m/s 6

2.0

1.5

1.0

0.5

0.0

.I ... . . .. (kh , kvy )3 ... . . @ .... (kh , kvy )2 .. @ ... . . @ ... . (kh , kvy )1 @ ... . . @ .... @ ... I . (kh , kvy )1 @ .... . . @ . . . .. ... @ @  ... @ .... . . . . . @ .... ..  . ... @ . ....  (kh , kvy )2 . . . . . . @ ... ..   .... .... I ... @ . . .   . . ..... . . . . . . @ . . . .   (kh , kvy )3 ... ... .. ..... .... .... ... ... ..... . . . . .... .    . . . . . . . . ...... ... .. ..... .. .....    ....... .... .... ... ....... ..... . .... . . . . . ....... . . ... ... . . . . . . . . . . . .......... ... ...... . . . . ........ . . . ............... . . . . ............................................................................ .. . . . . . . . . . . .  ............................................................................................................................................................................................................................ ................................................................... II   ... ...... ... .................. ................. . . ................ ................................................................ ......................................................................................................................................................................................................................................................................................................................................................................................... III ............................. ............................................ ........................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ......................................................................................................................................@ .... ........ .. ... ...... ......... . ........ .....@ ............ .... . . . . . . . (kh , kvy )2,3 . . . . . . . . .. . . . . . . .. . . . . . .. . . . . . .. . . . . . .......... @ (k , k ) @ ....... ........ ................................. . h v 1 . y . .. ..... ............................... @ ..................... (kh , kvy )1,2,3 .................. @ ....... ......... . .. . @ . .. . . -B ....... .. 0

2

4

6

Fig. 3–14 Separate influence of atmospheric turbulence (continuous line) and ship motions (broken line) on σvy at the moment t

164

Chapter 3

As you can see, the results of the calculations for σyII,u are only slightly changed when changing the pair (kh , kvy ) and, in addition, only marginally different from σyIII,u . The following inequality is satisfied: σyI,u − σyII,u > σyII,u − σyIII,u , i.e., a pitch ship-rocking contributes the major share to increasing the value σyu with the transition from tracking glissade III to tracking glissade I. We also note that the results of calculations for σyu are slightly changed with the transition from tracking glissade III to tracking glissade II even at B = 6. σvy , m/s 6

1.5

1.0

0.5

.... I .... .... . . . .. ..... ..... . . . . ... ..... .... . . . . .... .... I .... ........ ...... . . . . . ....... . . . . . . . . . . . . . (kh , kvy )1 I ........ ...... ........ ........ ...... ........ ........ . ...... . . . . . . . . . . . . . . . . . .  ....... ....... ...... ......... .......... ....... ... II ........ .......... ....... . . . .  . . . . . . . . . . . . . . . . . . ......... III . . . . . . . . . . . . . . . . . . . . . . . . . . . (k , k ) . . . . . . .  h vy 2 ....... ..... ....... ......... ............ ................... ... ....... ......... .............. ....... ............................. .................... ................................... . . . . .   . . . . . . . . . . . . . . . . . . . . . . . . . ..... ............... ............ .......... ........ ....... @ .........................................................   .............................................................................................. @II .. . . .................................. ............................................... ............................................................................................................................................................................................................................................................................. .................................................................................................. .......... ............. ...................................................................................... ............................ ...... .................................................................................................................................. @ ...............................................................................................................................................................................................................................@ @ ......... ....................................

@ @ @ @III @II @III

A (k , k ) A h vy 3

-B

0.0 0

2

4

6

Fig. 3–15 Standard deviation of the relative vertical speed vy at the moment tland

Since σyI,w = σyII,w = σyIII,w , we can suggest that the properties (noted in the previous paragraph) should be preserved, as well as graphs σy (B) that take into account the total effect of atmospheric turbulence and ship’s motions. This conclusion is confirmed by

165

Applications in Aviation

Fig. 3–8, where for each pair (kh , kvy ) the distinction of graphs σyIII (B) and σyII (B) is negligible compared with the distinction of graphs σyI (B) and σyIII (B). σvy , m/s 6

1.5

1.0

0.5

0.0

(kh , kvy )3

.I @ ...... @ . (kh , kvy )2 ... ... @ ..... . (kh , kvy )1 (kh , kvy )1 ... @ ... . @ . .  @ ... .. . @ .  @ .... I .. @ . . . . . (k , k ) . h vy 2 .  @ ..... ... . ...   .... . . ..@ . . . . . . .   @ ... .... .... ...   (kh , kvy )3 @ ....... . . . . . .. .. ... . . ... .    . . ....@ .. I .. ..... @ ... .. . . ...... ....... . . . .. . . . . . . .    ......... ........ ... .. . . .......... . . .... . . . . . . . . . . . . . . . . ....... .................... .. ........ .. ............... ........................................................................... .................................  . .. .............................. .... .... .. ................................................................................................................................................................................. ....................................................... .  .  . . .... ........... . . . ................ . . . . . ............................................................................ ... . . . . . . . ....... ... .. ....................................... ................. ............................. ..... ............... ................ ................. II ............................................................. ............ ... .... ............................................................................................................................................................................................................................................................................................................................................................................................. .. . .. . ...... III .. @ .. . . . .... .... .. .... .. .... .. .... .. .............. .... ....... ... ....... .. .. . .. .. . . . .. .. . .. .. . . .. .. .. . .. .. .... ...... . . . @ (k , k ) .... .... ... .... ........... ............ ..............@ h vy 2,3 .. .... .... .... .... .... .. .. ...................... .. .. , k )1@ (k .. .. . .. h v .. .. y .. .. . .. .. , k ) (k . . .. .. . . @ .. .. h v 1,2,3 . .. . . .. y . .. . .......... ..... ....... @ .... @ ...... .... .. -B 0 2 4 6 Fig. 3–16 Separate influence of atmospheric turbulence (continuous line) and ship motions (broken line) on σvy at the moment tland

Fig. 3–10 shows σy (B) at the prospective moment tland of landing. The moment tland is defined by the condition M {y(tland )} = 0. Fig. 3–11 shows the separate influence of atmospheric turbulence and the ship’s motions at this moment. A comparison of Fig. 3–10 with Fig. 3–8 indicates that the picture of dependencies σy (B) in Fig. 3–10 is qualitatively the same as in Fig. 3–8. We can only note that, when the values B are high, the absolute values σy (B) in Fig. 3–10 are somewhat smaller than in Fig. 3–8. This is due to lower values σyu (B) (to compare Fig. 3–9 and Fig. 3–11): the contribution of isolated ship motions at the moment tland is less than at the moment t . The

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expected landing point (corresponding to the moment tland ) coincides with the center of the keeled oscillations of the ship. Therefore, this point moves up and down due to heave only, while the up-and-down of the feed cut (corresponding to the moment t ) are invoked by the keel rolls. This fact leads to a decrease of the relative mean-square deviations σyu (B) when the airplane is close to the expected moment of landing. For greater visibility, Fig. 3–12 shows the value σy (B) of the moments t and tland . σvy , m/s 6 2.0

1.5

1.0

0.5

.... I .... .... . . . .... .... .... . . . . .I .... .... .... .... .... . . . . . . . ... ... at the moment tland ..... ..... .... .... . . . . . . . . ... @ @ .... ..... .... @ @.......... .......... ...@ ...... ..@ at the moment t ..... @.......... . . . . .... @ ... @ @......... ............... @ @ . . . . . . . . . ... @ ..... @ .... ....... ..... ............. @ . . . @ . . . ....... . . . . @ .... ............. . @ . . . . ...... ....... @ @ .............. II ....... .............. . . . . . . @ ... ... @ ....................... . . ........................ . . . . . . . . . @ ............................. ...................... . . . . . . . . . . . @ . . . . . . . . . . .......................... ........ ........ ........ ......... .................................. ............................ ....................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........................ ........... ........................................................................................................................................

-B

0.0 0

2

4

6

Fig. 3–17 Comparison of values σvy at the moments t and tland for (kh , kvy )3

Fig. 3–13 — 3–17 are similar to Fig. 3–8 — 3–12 with the only difference is that, instead of standard deviations σy , Fig. 3–13 — 3–17 shows standard deviations σvy of the airplane’s relative vertical

167

Applications in Aviation

velocity vy above the deck surface. We can repeat for Fig. 3–13 — 3–17 the arguments and the comments that were made during the discussion of Fig. 3–8 — 3–12. Let us only draw attention to the one qualitative difference. This difference is related to the behavior of dependencies σvIy (B) at transition (kh , kvy )3 → (kh , kvy )2 → → (kh , kvy )1 . For the large values B, the value σvIy (B) decreases at the transition (kh , kvy )3 → (kh , kvy )2 → (kh , kvy )1 while σyI (B) increases (with the exception of only a small region near the value B=5 in Fig. 3–8, where σyI (B) decreases slightly at transition (kh , kvy )3 → → (kh , kvy )2 ). This is explained by the fact that σvwy (B) slightly responds to changes (kh , kvy ) in contrast to the σyw (B). At the same time, the reaction σ I,u (B), which is caused by changes in (kh , kvy ), appears in vy to the same degree as in y. As a result of this, starting with σϑ n , 6

4.5

3.5

2.5

1.5

◦

I .... .... . . . .. .... .... . . . .. .... .... . . . .. .... .... . . . .. .... .... . . . ... (kh , kvy )3 .... .... . . . ...  .... .... .  (kh , kvy )2 . . ....  .... .... . .  .  .... ....   (kh , kvy )1 ... I .... . . . ....... .... ....... . .   . . . . . . . . ....... .....   ....... .... ....... . .... . . . .  . . . . .   ... ........ ... II ..... ........   ..... .... ......... . . . . . . . . . I . . ..... .. III . . . . . . . . .  . . . . . . .   II ............. ......... ....... ..... ......... ...... .. ..   ......... @ .......... ...... . ............................ . . . . . . . . . . . . . .........@ . ..........@ . . . . . . . . . . . . . . . . . . . . . . . . . .... .  . . . . . . . . . . . . . . .   ........... ...... ............ ........... .......... ...@ .......... ................................................................................. ............... III ........... . . . . . . . . . . . . . . . . . . . . . .  ........................ . . . . . . .......... ................. .............. .......... ........... .......... .................................................................................................................................................................................................................................................................................................................................... . ................................ ....... II .............. ...................... . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . ............. . . ........................................................................................................................................................................................................................................ .................................... III . . . . . . . . . . . . ................................................. . . . .  . . . ...................... .................... ............ .....................................................................................................................................................................................................................................................................................................................

0.5 0

2

4

6

Fig. 3–18 Standard deviation of the relative pitch angle ϑn at the moment t

-B

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B ≈ 1, the increase of σvI,u is more significant at transition (kh , kvy )1 → y →(kh , kvy )2 →(kh , kvy )3 , than the decrease of σvwy (B), and the value σvIy (B) (which takes into account the total impact of atmospheric turbulence and the ship’s motions) increases at transition (kh , kvy )1 → → (kh , kvy )2 → (kh , kvy )3 starting with B ≈ 1. . .. I .. . . .. .. . .. .. (kh , kvy )3 .. . .. @ @.... . .. .. . .. .. .. . . .. .. . (kh , kvy )3 .. .. (kh , kvy )2 ..  . .. I . . .... . . @ .  . @ . . . .. ....  (kh , kvy )2 .. .... . . . . . .   . .... .. ....   .. . . . . .   (kh , kvy )1 .... .... (kh , kvy )1 .... . .. .    .. . . ..... .. .. B .B........B................. .... (kh , kvy )2 .    .. . . . . ........ .. I .......... B B .B........ ....  .................................................... .... ........... ....... ................ ............ ....... ............. ........................................... .......B ....................................................................... . . . .........B . . . . . . . . . ................ . . . . . . . . . . . . . . . . . B B . . . . . . . . . . . . ...................................................................   .. ........ .................. .... II .... .. ........ ..................B.........B..........................................B...........B................ .... . .............. .......................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .................................................................................................................................................................................. B B ........................... B B ... II .. .... .... .. ...... .. ....................... ................................ . ...................................................................................................................................................................................................................... ...............................................................................B.....B ..........B B . . . . . . . . . . . B . II . . . . . . . . . . . . .. . . .. . . . . .. ......B... ..... ..... .................. ....................B......... .. .. ......... III ... .. ....... ..................................................... ... B ... ... ... ... ... ... ... . ... ... ... ... ... ... ................ .......... ...... .................. ..... ........................... ...... ...... ............ ...... ........ ................. ...... .... ....... .... ... ... ... -B ... ... 0 2 4 6 σϑ n , 6

4

3

2

1

0

◦

Fig. 3–19 Separate influence of atmospheric turbulence (continuous line) and ship motions (broken line) on σϑn at the moment t

Fig. 3–18 — 3–22 represents the calculation results from the standard deviation σϑn for the airplane’s relative pitch angle ϑn relative to the deck surface. As can be seen, the mutual disposition (B) at each pair (kh , kvy ) is of the curves σϑI n (B), σϑIIn (B), and σϑIII n

169

Applications in Aviation

the same as in the cases with σvy and σy . A similar observation (B), σϑII,u (B), and σϑIII,u (B) when can be made for straight lines σϑI,u n n n only the impact of ship’s motions is taken into account. Now note the contrast in the figures for σy and σvy . This difference relates to the dependences σϑwn (B) when only the impact of atmospheric turbulence is taken into account. The values σϑn (B) increase at transition (kh , kvy )1 → (kh , kvy )2 → (kh , kvy )3 , whereas in cases vy and y the corresponding standard deviations are reduced. This can be explained as follows. The control law for tracking a given glissade uses a linear deviation (the error signal ε in equation (3.15)) from the desired position in space and it does not respond directly to the difference from a nominal value for any angular parameter such as, for example, a pitch angle. In other words, the airplane control is carried out so as to eliminate the σϑ n , 6

3

2

1

◦

I ..... ..... . . . (kh , kvy )3 ..... .... .... . . . ..  ..... .....  . . . . ....  (kh , kvy )2 .... .... . . . .   ... .....   .... . . . . . ......   (kh , kvy )1 ....... ...... . .    . . . .......    ....... ..... I ....... . . . . . ........ .    ........ ....... . . . . . . . . . . . . . . . ....... .... II .......    ....... ......... ........ .......... ......... . III ........ . . . . . . . . . . .    . . . . . . . . . . . . . . ...... ......... ..... .......... .......... ........................ ............... ...    ........................ . . . . . . . . . . . . ..... ...... .... .. II ........... ........ ............. ..................  ............................. .............. ....................... .............. .......... . .......................................................................................................................................................................................... ........................................................................... .................. III ...............................................................................................................  ........ ..................................... I . .......... . .....................  .......................... ........................................................................................................................................................................... ............ . . . . . . . . . . . . . .  . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . .... ........................................................................................................................................................................................... ............................ ..... ......... . . . . . . . . . . ........ .. ..............  .......................... .................. III ............................................. ................................................................................................................................................................................................................................................................................................................................................................................................ -B

0 0

2

4

6

Fig. 3–20 Standard deviation of the relative pitch angle ϑn at the moment tland

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linear deviations without regard to changes of angular deviations. In these circumstances, when the atmospheric turbulence is taken into account, the pair (kh , kvy )3 (which corresponds to the best transition process (see. Fig. 3–7) and the lower values σyw (B) and σvwy (B)) leads to large values σϑwn (B). σϑ n , 6

3

2

1

0

◦

I ... ... . . ... (kh , kvy )3 ... . . . ... @ ... . . (kh , kvy )3 ... ... . .  ...  ... . . (k , k ) . h vy 2  ... ... (kh , kvy )1   . . .   ... . . BBB ...   (kh , kvy )1 .I B B B............. ... (k , k ) . h v 2 . y    . . .... . . . . . . . . . ....B B B ........ ...    B B.B.B......... ......B........B........B........ ... . . . . .  . .. .. II . ....... ................ ... B B ........................ ............... ......... ............................................................................ .. .............................................................................................................................................................................B.......B........................ .........B..........B..........B............... .   . . . ... ...... ........ ...................... B B B........ .............. ............................................................................  . ...............................................................................................................................................................................................B.......B............................... ..................B.........B.......B . . .... . I .... ............ ........B..... B....B............................. ... ...... ........ ...................... ....................... ...................................... ........B ..B ..... .... ........ ......... .. ................................................................................................................................................................................................................................................... ......... .......... . B B II . ... .. B .. ... .. ... .. .................... B . .. ... .... ....... ..... .. ....... .. ..... ..... .... .. ..... .... ... ...... ........ ........ ..... .... .... .... ..... BB .................... .... B ... III ..... .. ... .. ........................................ ... .. .... .. .. ... .... .. .... .. . .... ... .... ... .... ... . .... ... .... .... . ... . .. . .... . ... .. . .... . ... .. .... . . .. ... .. . .... . .. ... .... . ... .... .. . .... . ... .. .... . .. ... .... . .. ... . .... . .. . ... .. ........ ..... .... -B ... ... 0 2 4 6 Fig. 3–21 Separate influence of atmospheric turbulence (continuous line) and ship motions (broken line) on σϑn at the moment tland

Fig. 3–23 and 3–24 represent the results of calculations for the evaluation Pˆ of the probability of landing on a predetermined portion of the deck in compliance with the restrictions on vy and ϑn (thick line) and without these restrictions (thin line). Let us analyze these results. As can be seen, the minimum probability of a successful landing is obtained for (kh , kvy )1 regardless of the type of the tracked glissade. This is expected due to the fact that the pair (kh , kvy )1 results in the

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Applications in Aviation

largest values σy . At the moderate values B, in the case (kh , kvy )1 the increase of the probability with increasing B is explained by the corresponding decrease of σy . This decrease of σy is caused by decreasing the atmospheric turbulence intensity (see Fig. 3–4). Similar observations explain the increase of the probability for cases (kh , kvy )2 , (kh , kvy )3 when tracking glissades II and III (see Fig. 3–24). σϑ n , 6

4.5

3.5

2.5

1.5

◦

I .... .... . . . .. .... .... . . . .. .... .... . . . .. .... .... . . . .. I .... .... .... .... at the moment tland . . . . . . . . .... .... .... @ @ ......... ..... . . . ... ... @ @ .... .... ..... ....@ .  . . . . @ . . at the moment t .... .... .... @ @.......... .... @ . . . @ @ . . @..... ... @ ..... ... .... @ .......... @ . . . ... @ ..... @ ..... ....... . . . . . . . . . . . @ ...... .... @ . . . . . . . . @ ...... .... @ .... II ........ . ..... . . . . . . . . . @ @ ... .... ....... . . . . . . . .... . . . @ ..... .... ........ . . . . . . . . . . . . . . . . . . . ... ...... ............... @ @............ .. ....... .......... ....... .................. . . . . . . . . . . . . . . . ....... ......... ........... .............. ........ ............ ................. . . ............................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....................................................................................................................................... -B

0.5 0

2

4

6

Fig. 3–22 Comparison of values σϑn at the moments t and tland for (kh , kvy )3

We will compare the probability of a successful landing when tracking glissade I for cases (kh , kvy )2 and (kh , kvy )3 (see Fig. 3–23). If we look at the probability of landing on a predetermined portion of the deck without taking into account limitations on vy and ϑn , then

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the pair (kh , kvy )3 provides a greater probability for any value B. This is consistent with the fact that σyI (B) for (kh , kvy )3 is less than σyI (B) for (kh , kvy )2 at the expected landing time tland for any value B (see Fig. 3–10). Then we note that the rapprochement of the curves σyI (B) at (kh , kvy )2 and (kh , kvy )3 in Fig. 3–10 near values B ≈ 4 ÷ 5 leads to the rapprochement of thin curves in Fig. 3–23; this is close to the same values B ≈ 4÷5. If we now look at the probability of landing on a predetermined portion of the deck in compliance with the restrictions on vy and ϑn , then starting with B ≈ 4 the probability will be greater for (kh , kvy )2 than for (kh , kvy )3 . This is easily explained if we compare standard deviations σvIy and σϑI n and the values vymax − |vy0 |, |ϑnmax − ϑ0 |, and |ϑnmin − ϑ0 |, where vy0 = M {vy }, ϑ0 = M {ϑn }. If we consider the pair (kh , kvy )2 , then deviations σvIy and σϑI n are two to three times smaller then these values, even at B ≈ 5 ÷ 6. Pˆ

(k , k )

h vy 3 ....................................................................................................................................................................... ................................................................... .................................................... ........................................... (kh , kvy )2 ......... ......................... ....... ....................... ............ .. ....... ................................................................................................................................................... ... ... ... .... ... ... ... ... ... ... ... ... ... ..................... ....................... ... .... ... ... ... ... ... ... ... ... ... ... ... .... ... ... ... ......................................... .................. ........................................ ................... 0.9 ..... .......................... .................. ..... ....................... .................. .... ..................... .................. .... ............. .......... .... .............. I .... .............. ... ..... ... ........... ... I 0.8 .............. ... .... ..... .... ... ... .... ... ... .... .... .. ... ... ... ........................................................... ... ....... ..... ... .. ............ .... ... . . . . . . . . . . . . . . . . . . . . . . . . . . .................. ... .. .. ........ ............. ... (kh , kvy )1 ... ... ... ..... ... ........... ... ... ... ... .... .............. ............................ .. ... ....... .... ... ...... ... ... ... .... ... ...... ... ... ...... ..... .. ....... ... .. ... .... ... .... ..... ...... ... . ... . ... ... ... ... .... . ... ... ... ... ... .... ...... ..... ... . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............ ...  .... ... ... ............. ......................... .... ... ... ... ... ... ..... ... ... ... ... ... ... ... ... .... ... ... ..... ... .... ... ... ... ... ... ... ... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . ............ ....................... ... ................................ ............ ......................... .. I .......... ......... ....... ......... ................... 0.7 .......... .................. ......... ................... ......... .................. ......... .............. .......... .......... III .. . I ....... .. II

1

0.6

0

2

4

-B

6

Fig. 3–23 Estimation Pˆ of the probability of landing on a predetermined portion of the deck in compliance with (thick line) and without (thin line) restrictions on vy and ϑn

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Applications in Aviation

For this reason, the probability of landing on a predetermined portion of the deck with and without restrictions on vy and ϑn differ little up to B = 6. The same situation holds for tracking glissades II and III for all three pairs (kh , kvy )1 , (kh , kvy )2 , (kh , kvy )3 and for tracking glissade I for the pair (kh , kvy )1 . The situation is qualitatively different for the pair (kh , kvy )3 in the case of tracking glissade I at sufficiently large values B: deviations σvIy and σϑI n are comparable with the values vymax − |vy0 | and |ϑnmax − ϑ0 |, |ϑnmin − ϑ0 |. As a result, the probability of taking into account the restrictions is significantly less then the probability of excluding these restrictions (see Fig. 3–23); at the high B, the thick line that corresponds to tracking glissade I in the case (kh , kvy )3 passes considerably lower than the corresponding thin line, and even lower than the thick and thin lines that correspond to tracking glissade I in (kh , kvy )2 . Pˆ

(kh , kvy )3

.............................................................. ............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. ..................................................... ........................................... .................... .............. ............... .............. ............ ........ ............ III ............ 0.95 . . . . . . . . . . . . . . . . ... ............ ... ... .... ... ... ... ... .... ... ... ... . II ... ............. (kh , kvy )2 ... ... ... .... ... ... ... . ......... ... . ... ... . ... . ... ... ... ... ... ... ... ... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........................... ................... . ............................... ... ... ... ... .... ... ... .... ... .................... .... ... ... ... ... ... ... ... ... ... .... ... ... ... ... ... .... ... ... ... ... ... ... ... .... ... ... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............ .. ...... ................... ............... .......................................................................................................................................................................................... ................ .............. .............. ............. ............. ............. ............. ............ ............ .......... 0.90 ......... ......... ......... .......... .......... ......... ......... ......... ......... ......... ......... ........ .......... . III ......... 0.85 ......... ......... .......... .... . II

1

-B

0.80 0

2

4

6

Fig. 3–24 Estimation Pˆ of the probability of landing on a predetermined portion of the deck in compliance with (thick line) and without (thin line) restrictions on vy and ϑn

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Chapter 3

A comparison of all options considered (see Fig. 3–23, 3–24) shows that the best option that provides the greatest probability of successful landing at any value B is the case of tracking glissade III at (kh , kvy )3 . lg ΔP 6 -B

0 2

4

6

.... ...................... I ................. . . . . . . . . . . . . . ... ............. .................. I ........... ................ . .......... . . . . . . . . @ . . . . . . . . . . . . ........... ........... I ........ (kh , kvy )3 ............. ........... ........ @ @ ............. .......... ....... . . . . . . . . . . . . . . . . . . . . . . . . .... II ....... ........... @ .... @............. ......... ........... ....... ........ .......... . . . . . . . . . . . . . . . . . . . . . . . .. .. III ..... ........ ..... .... .... ....... ......... .... ............ ..... ............... .... . . . . . . . . . . . . .. .. ... ....... .... .... ..... ..... .......... .... ... II ............. .................... . ... . . . . . ..... III . . . . . ....... . . . . . . . . . . . . . . . . . ... ... ... .. (kh , kvy )1 ...... ....................... ..... ..... ... .. ........ ..... II ............ .. ................ . . . . . . . . . . . ... .. ... ... .. .......... ... ....... ......  .......... ..... ..... III .............. ...... ...................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..  ................... .......... ............ . .... .... .. .. ............................................................................................................................................................................................................................................... .... ...... ............ . . .. ... . ... .. .. . .. . .... .... .... .... .. ... ....... ........ ............. . . . . . .... .... . ...... ... .. .. .. .. ....... ..... .... ... .. .... ........ ....... ....... ...... . . . . . . . ........... . ...... .... .... ......... ........ (kh , kvy )2 ......... ............. ....... ....... . . . . . . . . . . . . .. .... .......... ......... ........ ....@ (kh , kvy )3 ......... ............. .......................... . . . . . .
. . . . . . . . . . . ..... ....... @ . . . . ......................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . ............. ............ ..... ...... ..................................................
........ . ........... ........... ... ............... ... .. @ (kh , kvy )3 . . . . . . . ... @ .. ........ .......... ... ............. ...

(kh , kvy )2

-4

-8

-12

-16

Fig. 3–25 Error of estimating the probability of a successful landing

Fig. 3–25 shows how the error ΔP when estimating the probability depends on B for different combinations of tracking glissade (I, II or III) and coefficients kh and kvy in the control law ((kh , kvy )1 ,

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175

(kh , kvy )2 or (kh , kvy )3 ). When tracking glissades III and II, the error ΔP is less than 10−10 at B = 0÷4 and this error is less than 10−5 at B = 0÷6. The greatest error is obtained when tracking the glissade I for (kh , kvy )3 ; this error is equal to 0.0001 at B = 4 and equal to 0.007 at B = 6. In accordance with the physical sense of the number ΔP its value increases with the increasing σy and σvy . Therefore, for each of the pairs (kh , kvy )1 , (kh , kvy )2 , (kh , kvy )3 an error increases in the transition III → II → I. Also, changing the pair (kh , kvy ) crossing the curves that correspond to tracking glissade I is easily explained by dependencies σyI (B) and σvIy (B). If B = 0, then the standard deviations σyI and σvIy decrease in the transition (kh , kvy )1 → (kh , kvy )2 → (kh , kvy )3 . Due to this, the value ΔP also decreases. If B > 0, then the value σyI continues to decrease in the transition (kh , kvy )1 → (kh , kvy )2 → → (kh , kvy )3 , but σvIy increases in this case, starting since some B and this increasing for sufficiently large B has a greater impact on ΔP than the reduction of σyI . In conclusion, we will now make some remarks concerning the effectiveness of the proposed method. First of all, we would like to draw attention to the negligible error between the true value of the unknown probability P {ZD } and its approximate calculated value Pˆ . This allows us to consider formula (2.4), which determines the value Pˆ as ready functional for the problem of optimal (in the probabilistic sense) control by an airplane during landing. It is also important to note that the small labor intensity of the method is important when solving these practical problems. The disadvantage of this method is the need to use the linearized description of airplane’s motion. However, as already mentioned, the method of linearization of the nonlinear motion’s equations, relative to a nominal landing trajectory, is legitimate and widely used in the investigations into landing. The only method that can be used to estimate the probability P {ZD } under consideration using a nonlinear description is the method of statistical testing. This method consists in the numerical modeling of disturbances and a subsequent integration of motion’s equations, and all of this is repeated many times. As a result of each integration, the moment of landing is determined and values of phase coordinates at this moment are also determined. If N is the total number of integrations and Nz is the number of integrations corresponding to the successful landing, then the estimation of probability P {ZD } is Nz /N . The error |P {ZD } − Nz /N | in this

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estimation is not determined. In any case, one can not say that it is less than 1/N . The probability of the successful landing on a ship is 0.97 or higher for real problems, but this is not to mention similar problems in civil aviation, where this probability is much closer to 1. In order to be able to compare various control laws under the criterion of probability of successful landing, the probability is necessary to estimate with an accuracy of at least 10−3 . This makes it difficult to use the method of statistical testing due to its labor intensity. If probabilities differ by less than 10−3 , then the method of statistical testing is not applicable due to high labor intensity of this method, and it is only possible to examine the possibilities theorectically. At the same time, the proposed method allows us to calculate and compare these probabilities, if the difference between them is more than ΔP . This speaks in favor of the proposed method, if we take into account that, as a rule, it is important to know not only the probability of the value itself, but also how the probability varies within parameters of the problem.

3.4 Implementation of the Method for Overland Landing Variant The purpose of this section is to study the proposed method for more gently sloping landing trajectories (compared to landing on a ship) that take place when landing on land. When considering this we need to recall the numerical example from section 2.9. In this example, we estimate the probability that the first achievement of the zero level by process y(x) = −a1 x + a0 + ζ(x), x ∈ (−∞, x ], where a1 > 0 and a0 are constants, ζ(x) is the Gaussian process , M {ζ(x)} ≡ 0,

(4.1)

˜ will occur at a given interval (x , x ); this will contain the point x ˜ + a0 = 0 of a zero-crossing by the line y = −a1 x + a0 , i.e., −a1 x and x ˜ ∈ (x , x ). The lower P1 and the upper P2 estimations of the required probability were investigated depending on the parameter a1 /σ1 , where σ12 is the variance of processes y  (x) and ζ  (x) (see the Table from section 2.9). If we fix the value σ1 , then the difference P2 − P1 increases with decreasing the value a1 , which characterizes the angle of inclination of the line y = −a1 x + a0 . The difference

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P2 − P1 increases so much for sufficiently small values a1 that the idea of estimating the required probability by using values P1 and P2 becomes meaningless. This numerical example was not accidently discussed in Chapter 2. As shown later in section 3.3, under a certain conditions, process (4.1) describes the change of altitude during an airplane’s landing. In this case, the required probability makes sense of the probability of landing the airplane on a predetermined portion of surface; the line y = −a1 x + a0 makes sense of a nominal landing trajectory, on which the airplane moves in the absence of disturbances; the point x ˜ makes sense of an expected landing point when the airplane moves along a nominal trajectory. When landing on a ship, the angle of the nominal landing trajectory (this angle is defined by value a1 ) is such that it leads to a nearly zero or negligible values ΔP = P2 − P1 (see Fig. 3–25). When landing on land, the trajectories are more gentle and, therefore, value a1 is smaller and the difference P2 −P1 is greater. Furthermore, when landing on land, the length x − x increases and this also leads to an increase in value P2 − P1 . Therefore, the question arises about the possibility of using the proposed method for landbased airplanes in the sense that the desired probability can only be estimated roughly due to the insufficient proximity of bottom P1 and top P2 estimates. The answer to this question is given in this section. 3.4.1 Equations of Motion, Pertutbation Model, and Control Law As in the case of landing on a ship, we will consider only the longitudinal motion of an airplane, so that the system of motion equations takes the form of (3.1). To apply the proposed method, equations (3.1) have to be linearized relative to the nominal landing trajectory. As a rule, the nominal trajectory is not straight at the final stage of air section of the trajectory when landing on land. However, to sufficiently answer the above question, we need to carry out calculations for the nominal trajectory, which is straight until the moment of landing. Then, as in the case of landing on a ship, the linearized system of motion’s equations will have the form (3.2). As in section 3.3, the angle of the inclination of the nominal trajectory will be denoted by θgl . However, this angle is not a fixed value and is, instead, a main variable parameter, which accepts significantly lower values than in section 3.3.

178

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Random perturbations are caused by the longitudinal wx and the vertical wy components of atmospheric turbulence. The spectral densities of random processes wx (t) and wy (t) are approximated by fractional-rational expressions (3.11) and (3.12), so that the system of equations for processes wx (t) and wy (t) has the form (3.13). As already mentioned in section 3.3, for intensities σwx , σwy and scales Lx , Ly of atmospheric turbulence there are many empirical formulas depending on the place of observation and the state of weather conditions. We used the following model of atmospheric turbulence over Earth’s solid surface:  h, 10 m ≤ h ≤ 300 m; Ly = Lx = 150 m, 300 m, h ≥ 300 m;  0.33 h , h ≥ 10 m; σwx = 2σwy = 0.19v 10 where h is the height of flight, and v is the module of horizontal velocity component of a mean wind. The value v is different for different realizations of landing and has the following density distribution:   v v2 p(v) = 2 exp − 2 , σv = 2.6 m/s. σv 2σv Characteristics of disturbances at h < 10 m are accepted the same as when h = 10 m. Control law for tracking the nominal trajectory is chosen as follows: dΔh , (4.2) Δδ = δ − δ0 = kα Δα + kωz ωz + kh Δh + kvy dt where δ is the deviation of longitudinal body of control, α is the angle of attack, h is the height of flight, ωz is the angular velocity; Δδ, Δα, Δh are deviations from values δ0 , α0 , h0 (t) that correspond to airplane’s motion along the nominal trajectory of landing in the case when there are no perturbations. The coefficients kα , kωz , kh , kvy in control law (4.2) were chosen in the same way as in the case of ship’s landing (see section 3.3). For selected combinations of coefficients kα , kωz , kh , kvy , the height’s transition processes were the same as in Fig. 3–7. This is because airplane’s specifications were used, like in the case of a ship’s landing, and the contrast of formulas (4.2) and (3.15) proved to be negligible.

179

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3.4.2 Calculation System of Equations and Calculation Formulas for Estimating the Probability We estimate the probability of landing an airplane on a predetermined portion of surface on land. Restrictions in the phase coordinates at the moment of landing have not been taken into account because the purpose of section 3.4 is the definition of the probability estimation error, and this error does not depend on whether these restrictions are taken into account or not. A predetermined portion is located symmetrically with respect to the intersection point of the level of the landing surface by the nominal trajectory. The calculation’s system of equations has the following form: dΔz = AΔzdt + BdQ(t),

(4.3)

where Δz is a 7-component vector of the extended phase space:  T dΔh Δz = , Δϑ, ωz , Δh, wx , αw , xy ; dt Q(t) is a 2-component vector of independent Wiener processes, A is a 7 × 7 - matrix, B is a 7 × 2 - matrix. System (4.3) is obtained by combining equations (3.2), where the selected control law (4.2) is taken into account, and equations (3.13) that describe the forming filters for the longitudinal and vertical components of atmospheric turbulence. The non-zero elements of matrix A = ||aij || and matrix B = ||bij || are as follows: a11 = −Y α + Y δ v0 kvy − Y δ kα , a13 = Y δ v0 kωz , a16 =

a12 = g sin θ0 + v0 (Y α + Y δ kα ),

a14 = Y δ v0 kh ,

ρv02 Cyα ρv 2 Cxn + Y δ v0 kα + 0 , 2m/S 2m/S

a31 = (−Mzα + Mzδ v0 kvy − Mzδ kα )/v0 , a33 = Mzω¯ z + Mzδ kωz , a55

v0 =− , Lx

a15 = −

a67

a34 = Mzδ kh , 1 = , v0

a23 = 1,

a32 = Mzα + Mzδ kα ,

a36 = Mzα + Mzδ kα , 

a76

ρv0 Cyn , m/S

v0 =− Ly

2 v0 ,

a77 = −

a41 = 1, 2v0 , Ly

180

Chapter 3

* b51 = σwx

'

2v0 , Lx

b62 = σwy

3v0 , Ly

* √ v0 v0 d b72 = (1 − 2 3)σwy + b62 (t). Ly Ly dt Accepted designations are similar to the designations from section 3.3. The calculation formula for estimating Pˆ the required probability has the same form as in the case of the ship’s landing (see formula (3.24)), with the only difference that an integration is added by the √ dimensionless speed v¯ = v/ 2σv : ∞ Pˆ =

Pˆv p˜(¯ v )d¯ v,

(4.4)

0

where p˜(¯ v ) = 2¯ v exp{−¯ v 2 }, and Pˆv is the value of estimation in accordance with formula (3.24) if we simplify the perturbation model and √ assume that the same speed v = 2σv v¯ is used for all realizations of landing. Formula (4.4) is reduced to the form ∞ Pˆ =

t p˜(¯ v )d¯ v

0



f (t, v¯)dt,

(4.5)

t

where (see formula (3.25))    2 (t)σ (t) 1 − r12 m2 (t) 2 exp − 21 2 · f (t, v¯) = 2πσ1 (t) 2σ1 (t)¯ v (4.6)  √ β(t) 2 2 [1 − φ(−β(t)/¯ v )] , v }+ π · exp{−β (t)/¯ v¯ m1 (t) = M h(t), σ12 = M (h(t) − m1 (t))2 = M (Δh(t))2 ,   2  dΔh 1 dΔh 2 σ2 (t) = M , r12 (t) = M Δh , dt σ1 σ2 dt the function β(t) is defined in section 3.3. The variances σ12 (t) and σ22 (t) and the correlation coefficient r12 (t) are determined by the numerical integration of the system

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Applications in Aviation

dK = AK + KAT + BB T , dt

(4.7)

√  where K = M Δz(t)(Δz(t))T when v = 2σv , i.e., when v¯ = 1. If the correlation matrices Kv1 (t) and Kv2 (t) are the solutions to system (4.7), when v = v1 and v = v2 respectively, then these matrices are related as  2 v1 Kv1 (t) = Kv2 (t) . v2 This circumstance has allowed us to distinguish the dependence on the parameter v¯ in the expression for f (t, v¯) from (4.5) in the explicit form and obtain formula (4.6).

Pˆ

1

0.9

0.8

0.7

0.6 0

x −x = 500 m

@ @ ............................. ................................. . . . . . . . . . . ...... ....... .......... .... . ........... . . . . . . . . . . .... ... @ x −x = 300 m ..... . . . . .. . @ .... . ..... . . ... ... . .... . ... . .. . .... x −x = 100 m . . .. @ .... @ ...... .... ...... .. .. . .. .. . .. .. . .. .. . . θgl , ◦ .. 0.5

1

1.5

Fig. 3–26 Estimation Pˆ of the probability of landing on a predetermined portion of surface on the land at (kh , kvy )2

182

Chapter 3

The error of probability estimation does not exceed ∞ ΔP =

ΔPv p˜(¯ v )d¯ v, 0

where (see formulas (3.26), (3.27))    t  2 (t)σ (t) 1 − r12 m21 (t) 2 exp − 2 ΔPv = · 2πσ1 (t) 2σ1 (t)¯ v2 t0   √ β(t) [1 − φ(β(t)/¯ v )] dt. v2 } − π · exp{−β 2 (t)/¯ v¯

Pˆ

1

0.95

0.90

0.85

(kh , kvy )3 @ @................ ................ ............ ......... . . .. . . . . . . . . . ............ .. . . ... . . . . . . . . . . . . . . ...... ... .... ... @ (kh , kvy )2 ............. . . ... . . ... ... ..... . . . ... .. . ... . .... @ (kh , kvy )1 . . .. . .. . ... . . ... . . . ... .. ... ... .. .. .. . ... .. .. ... ... .. .. ... ... .. .. ... ... . .. .. ... . ..

0.80 0

0.5

1

θgl , 1.5

◦

Fig. 3–27 Estimation Pˆ of the probability of landing on a predetermined portion of surface on the land at x −x = 300 m

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Applications in Aviation

3.4.3 Results of Calculations Calculations were made for x −x = 100, 300, and 500 m for the same combinations of coefficients kα , kωz , kh , and kvy that were used in the case of landing on a ship. The inclination angle θgl of nominal landing path was a main variable parameter. The calculation results for Pˆ and ΔP are presented in graphical form in Fig. 3–26, 3–27, and 3–28. lg ΔP 6 0 0.5

1

θgl , 1.5

◦

(k , k ) , x −x = 300 m

-2

-4

-6

-8

vy 2 h ......... .................... ............. ............   ........................ .............. (kh , kvy )2 , x −x = 500 m ..................................... ............... ......... ..... ....... ................ ......... ..... ........ ................ ......... (kh , kvy )1 , x −x = 300 m ..... ........ ................ ......... ..... ........ ................ . . . . . . . . . ......... ..... ....... ................ ......... ..... ......... ................ .......... ... ..... ..... .......... ..... ....................................... .......... ....... ............... ..... .......... . . . ....... ............. ..... .......... ....... ............. ..... ........... ....... .............. ........... ..... ....... ........... ........... ..... ....... ........ ............ ..... ....... ....... ............. ..... ....... ....... .... ....... ....... ..... ....... ........ ..... ....... ....... ..... ....... ....... ..... ....... ....... ..... ....... ...... .....   ....... ...... (kh , kvy )2 , x −x = 100 m ..... ....... ........ ....... ....... ..... ....... ....... ..... ....... ....... ..... ....... ....... ..... ....... ....... ..... ....... ...... ..... ....... ..   ..... ...... (kh , kvy )3 , x −x = 300 m ..... ..... ..... ..... ..... ..... ..... ..... ..... ....

Fig. 3–28 Error of estimating the probability of landing on a predetermined portion of surface on the land

As expected, the greater the size of the landing portion, the higher the probability, and this probability increases monotonically when

184

Chapter 3

increasing the angle θgl (see Fig. 3–26). Comparing Fig. 3–27 and Fig. 3–7 confirms the intuitively predictable result that the more sluggish transients lead to smaller values Pˆ , i.e., the more sluggish transients lead to a less accurate landing. Fig. 3–28 shows how the estimation error changes depending on parameters of the problem. It is seen that changing the length of landing portion barely affects the value ΔP . The negligible error of ΔP , even at very small angles of inclination in the nominal trajectory, allows you to answer the question posed at the beginning of section 3.4, and conclude that the proposed method can be used to compare the control laws according to the criterion of the probability of a successful landing on both a ship and on land. 3.4.4 Using the Method for Solving Problems of Civil Aviation Until now, the presentation and results of sections 3.3 and 3.4 are concerned with a statement on the problem of the probability of a successful landing being estimated before the flight with the help of the mathematical modeling of an airplane’s motion. However, these results can be used to solve this problem using another formulation because after the actual flight you need to analyze the realization of a random process of landing and accordingly you need to estimate the probability of a successful landing and the skill of the pilot. This formulation of the problem is particularly relevant in civil aviation (CA). As is well-known, multi-channel means that registration parameters (MCMRP) of the flight are installed on all CA airplanes and the ground service of each airport has a unit dealing with deciphering of MCMRP records. These records show how the flight parameters change during the motion. The entire flight can be decrypted if necessary but, as a rule, only the most important and responsible flight segments are decrypted, including the stage of landing. Deviations in flight parameters from their nominal values at given moments of motion can be found by means of MCMRP records. Often, however, a violation of some restrictions from a flight manual can reduce the probability of an unwanted event, if the situation as a whole is taken into account. Therefore, the integral indicator of landing process, such as the probability of a successful landing, is more important because according to this indicator we can objectively judge the quality of piloting and landing degree of security in general. Pilots also support

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this assertion1 . This assertion can be illustrated in the following way. We will assume that deviations Δh1 and Δh2 from the nominal value of height are defined for the two realizations of landing at some fixed moment (e.g., the moment of flight over the beginning of the runway), and estimations Pˆ1 and Pˆ2 for the probability of a successful landing are obtained using the same realizations. If Δh1 < Δh2 , then it is not necessary that Pˆ1 > Pˆ2 : it may well be that Pˆ1 < Pˆ2 . If, when deciphering MCMRP records, we show that Δh1 < Δh2 but the values Pˆ1 and Pˆ2 are unknown to us, then it is natural to conclude that, from the point of view of landing safety, realization 1 is preferable because at one of important moments of landing the nominal trajectory corresponds precisely to this realization. However, if the inequality Pˆ1 < Pˆ2 holds, then realization 2 is preferable because the degree of landing security is most fully characterized by the probability of a successful landing. Therefore, the defect of flight information processing methods consists in the fact that these methods allow you to evaluate only some characteristics of the random process of landing and only at fixed times, and these methods do not allow any evaluation of integrated indicators, in particular the probability of a successful landing. Based on the results from sections 3.3 and 3.4, we can offer the following scheme, which compensates this omission and allows us to estimate the probability of a successful landing by using the transcript of the concrete realization of landing. This will give an opportunity to evaluate the technique of piloting, and, if necessary, to make the necessary adjustments to the scheme and nature of piloting, thereby ensuring the prevention of airplane accidents. As discussed in sections 3.3 and 3.4, an aircraft motion while landing can be accurately described by a solution y(t) which uses the linear system of stochastic differential equations, and y(t) is a n-dimensional normal process. The characteristics of this process depend on the type of airplane, the nature of random disturbances, and the manner of piloting an airplane by pilot. Various manners of piloting correspond to different pilots or even to a single pilot, but at different times. Processes y(t) describing the landing of a particular airplane with the same random perturbations, but different manners piloting will have, generally speaking, different characteristics. Since the process y(t) is normal and its density distribution is completely 1 For example, see: V.E. Ovcharov, ...But the Instructor Better, Civil Aviation, 1992, no. 1, pp. 20-21 (in Russian)

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determined by moments of the first and the second orders, this difference in characteristics will only be manifested in the values of these moments. However, these moments can be evaluated using well-known methods and the realization of process y(t)1 if this realization is restored by using records of MCMRP. Therefore, in order to estimate the probability we must fulfill the following. First, using MCMRP records, we must decipher the realization of process y(t), which describes the motion of an airplane on landing. Then, using this realization, we need to define the moments of the first and the second orders of process y(t), where, as a rule, the first order moments are known in advance because these moments characterize the nominal trajectory. After that we uniquely define the distribution density ρt of components of process y(t). This makes it possible to estimate the required probability of a successful landing by the formula for Pˆ and the error of this estimating by the formula for ΔP .

1 See, for example, the book: J.S. Bendat and A.G. Piersol, Analysis and Measurement Procedures, New York-Chichester-Brisbane-Toronto-Singapore: Wiley & Sons, 1986.

Conclusion

In this book, the author’s original results are presented. 1. For a wide class of continuous random processes X(t), we have founda sufficient condition that limits of the type  n−1 ∞   P A2k (ti , ti+1 ) lim max

i=1,...,n−1

(ti+1 −ti )→0

i=1

k=1

are equal to zero, where tI =t1 < t2 < . . . < tn−1 < tn=tII is a partition of segment [tI , tII ], and A2k (ti , ti+1 ), i = 1, 2, . . . , n−1, is the event that process X(t) crossed a given level u on an interval (ti , ti+1 ) exactly 2k times, k = 1, 2, . . . .1 The obtained result is formulated below. Denote by Gu (tI , tII ) the set of scalar functions that are continuous on the segment [tI , tII ] and are not identically equal to u on any subinterval inside this segment. Denote by A− j (ta , tb ), j = 1, 2, . . . , the event that a number of crossings of the level u by the process X(t) on interval (ta , tb ) equals j, and the first crossing is a downcrossing. Let with probability 1 the sample functions of the process X(t) belong to the set Gu (tI , tII ) and do not touch the level u on the interval (tI , tII ). Let P {X(t)=u} = 0 for every point t ∈ [tI , tII ] except, perhaps, a finite number of points. Suppose that there exists a constant C > 0 such that the inequality       ti+1 − ti ) < u ∩ X(ti+1 ) > u ≤ P X(ti ) > u ∩ X(ti + 2 ≤ Cε(ti+1 − ti ),

i = 1, . . . , n−1,

holds for every small partition tI = t1 < t2 < . . . < tn−1 < tn = tII , where the function ε(τ ) satisfies the condition

max

lim

i=1,...,n−1

(ti+1 −ti )→0

∞ n−1 



ti+1 − ti 2 ε 2m i=1 m=0 m

 = 0.

1 The exact definitions of concepts: a crossing of a level, a downcrossing of a level, an upcrossing of a level, and a touching of a level are given in sections 2.2 and 2.3.

188

Conclusion

Then the equality

max

n−1 

lim

i=1,...,n−1

(ti+1 −ti )→0

i=1

P

 ∞

A− 2k (ti , ti+1 )

 =0

(1)

k=1

holds. In particular, equality (1) holds if ε(τ ) = τ 1+α , where α > 0. 2. For Gaussian processes X(t), we need to establish the requirements for the correlation function of the process X(t) when the above sufficient condition for equality (1) holds. By definition, put p(t, τ ) = P {(X(t) > u) ∩ (X(t + τ ) < u) ∩ (X(t + 2τ ) > u)}. We can then prove the following result. ◦

Let X(t) = m(t)+X(t), where m(t) = M {X(t)} is the mathemati◦

cal expectation of the process X(t), and X(t) is the stationary normal ◦ process with the variance σ 2 = M {X(t)}2 and the normalized correla◦

◦

tion function r(τ ) = M {X(t)σX2 (t+τ )} . Let I be a finite open or closed interval from a domain from the definition of process X(t). Suppose that there exist constants τ0 > 0, A > 0, α > 0, B > 0, and β > 0 such that 1) r(τ ) is twice differentiated on the segment [0, τ0 ], r (0)=0, r (0) < 0, and |r (τ1 ) − r (τ2 )| ≤ A|τ1 − τ2 |α if τ1 , τ2 ∈ (0, τ0 ); 2) m(t) is continuous on I and differentiated at every internal point t ∈ I, and |m (t1 ) − m (t2 )| ≤ B|t1 − t2 |β if |t1 − t2 | < τ0 . Then there exist the positive constants τ ∗ , C, and γ such that the inequality p(t, τ ) ≤ Cτ 1+γ holds for every τ ∈ (0, τ ∗ ) and every t ∈ I, where t + 2τ ∈ I. 3. We propose a scheme which can be used for finding the probability of any events related to the crossings of a level by a random process. This scheme involves the following.

189

Conclusion

1) By means of direct search, we will consider possible options of behavior for sample functions of a random process, in relation to crossing a given level. 2) Using any criterion (such as, the criterion regarding the number of upcrossings and downcrossings), we will find a convenient partition to separate the sample functions into bunches. 3) By introducing probabilities to these bunches (i.e., the probability that a sample function belongs to a certain bunch), we will use the probabilities of bunches to demonstarate those probabilities of interest to us. 4) The probabilities of bunches are estimated by means of any characteristics of a random process (for example, by means of an average number of upcrossings and downcrossings) and a joint distribution density of values in this process at fixed points. 4. The described scheme is used for estimating the probability of the event that the first achievement of a given level by the component of a multidimensional random process occurs at some moment from a given range within the independent variable and when, at this moment, other components of the process satisfy the given restrictions. Namely, the following result is obtained. Let X(t) be an n-dimensional random process, and let u be a given number. We will consider the processes X(t) = {X1 (t), . . . , Xn (t)} of two types: a) t ∈ [t0 , t ], and P {X1 (t0 ) > u} = 1;

(2)

b) t ∈ (t0 , t ], t0 ≥ −∞, and lim P {X1 (t) > u} = 1.

t→t0

(3)

Let t ∈(t0 , t ), and let D be a subset of the (n−1)-dimensional Euclidean space Rn−1 . We will introduce the event  ∗  ∃ t ∈ (t , t ) such that for every t < t∗ X1 (t) > u, ZD = . X1 (t∗ ) = u, (X2 (t∗ ), . . . , Xn (t∗ )) ∈ D Denote by N (t0 , t ) the average number of crossings of level u by component X1 (t) on interval (t0 , t ), denote by N + (t0 , t ) the average number of upcrossings of level u by component X1 (t) on interval (t0 , t ), denote by N − (t0 , t ) the average number of downcrossings

190

Conclusion

of level u by component X1 (t) on interval (t0 , t ), and denote by −   (t , t ) the average number of downcrossings of level u by compoND nent X1 (t) on interval (t , t ) so that at the moments of these downcrossings the condition (X2 , . . . , Xn ) ∈ D is satisfied for the other components X2 (t), . . . , Xn (t) of the process X(t). We have proved the following result. Suppose that 1) with probability 1 the sample functions of component X1 (t) belong to the set Gu (t0 , t ) and do not touch the level u on the interval (t0 , t ), N (t0 , t ) < ∞; 2) P {X1 (t ) = u} = 0; 3) condition (2) is satisfied if the variable t changes on the segment [t0 , t ], t0 > −∞, t < ∞, or condition (3) is satisfied if the variable t changes on the interval (t0 , t ], t0 ≥ −∞, t < ∞. Then for any m = 2, 3, . . . and any partition of the interval (t0 , t ) t0 < t1 < . . . < tm−1 < tm = t , where P {X1 (ti ) = u} = 0 for every i = 1, 2, . . . , m−1, the inequalities −   −   (t , t ) − N + (t0 , t ) + Δ ≤ P {ZD } ≤ ND (t , t ) ND

hold, where Δ = Δ(t1 , . . . , tm ) =

m−1 

 P

(4)

 - ! . m  X1 (ti ) < u ∩ . X1 (tj ) > u

i=1

j=i+1

If new points tk , such as P {X1 (tk ) = u} = 0, are added to the available points of the division ti , i = 1, 2, . . . , m−1, the value Δ can only increase. The number N + (t0 , t ) can be calculated using the Rice formula +

t



N (t0 , t ) =   where ft (u, y) = ft (x1 , y)

∞ dt

t0

x1 =u



yft (u, y)dy,

(5)

0

, ft (x1 , y) is the joint distribution den-

sity of random values X1 (t) and Y (t), where Y (t) ≡ dXdt1 (t) is the derivative in the mean-square of the process X1 (t). The number −   (t , t ) can be calculated by the Rice formula ND t

−   (t , t ) = − ND

t







dt . . . D

0 dx2 . . . dxn yft (u, y, x2 , . . . , xn )dy, −∞

(6)

191

Conclusion

where ft (x1 , y, x2 , . . . , xn ) is the joint distribution density of random values X1 (t), Y (t), X2 (t), . . . , Xn (t). 5. The following result concerns the question over the existence of lim Δ(t1 , . . . , tm ) as a cross-partition of the interval (t0 , t ). We have formulated the corresponding statement for type a) processes. For type b) processes, this statement is similarly formulated. Suppose that the conditions 1) and 2) from the previous statement are satisfied and the process of type a) is considered (i.e., the variable t changes on the segment [t0 , t ], t0 > −∞, t < ∞, and also equality (2) is satisfied ). Let t0 < t1 < . . . < ti . . . < tl = t and t0 = t˜0 < t˜1 < . . . < t˜j < . . . < t˜m = t be any partitions of segments [t0 , t ] and [t0 , t ]; P {X1 (ti ) = u} = 0 for every i = 1, . . . , l − 1 and P {X1 (t˜j ) = u} = 0 for every j = 1, . . . , m. In addition, we suppose that equality (1) holds. Then there exist the limits lim

Δ(t1 , . . . , tl ) = Δlim (t0 , t )

lim

Δ(t˜1 , . . . , t˜m ) = Δlim (t0 , t ),

max (ti −ti−1 )→0

i=1,...,l

and max

j=1,...,m

(t˜j −t˜j−1 )→0

and the exact equality P {Z} = N − (t , t ) − N + (t , t ) + Δlim (t0 , t ) − Δlim (t0 , t ) holds, where the notation Z is used for the event ZD when D = Rn−1 . The numbers Δlim (t0 , t ) and Δlim (t0 , t ) have the following sense: Δlim (t0 , t ) = P

 ∞ k=1

   ∞ −    A− (t , t ) , Δ (t , t ) = P A (t , t ) . lim 0 2k 0 2k 0 k=1

6. Bounds (4) were successfully applied to solving a very important problem in aviation: the problem of finding the probability of safe landing for an airplane. In this case, the process X(t) describes an airplane’s behavior during landing, component X1 (t) is the flight altitude, level u equals zero, the independent variable t is the flight length, and the event ZD denotes safe landing, i.e., the fact that the airplane touches the landing surface for the first time on a given segment and, at this moment, the phase coordinates of the airplane (elevation angle,

192

Conclusion

banking angle, vertical velocity and so on), which represent components of vector X(t), remain inside the admissible ranges that will exclude an emergensy. This safe landing probability was computed for specific real airplanes, including ship-based airplanes when they land on a real air carrier, and it takes into account the possibility of the rocking surface of the ship. The required probability P {ZD } was estimated using formula (6) and the maximum possible error of this estimate was calculated using formula (5). This method has proven to be extremly efficient and accurate; it is able to almost exactly compute the probability in question. This allows us to consider integral (6) as a ready function to synthesize optimal control laws in order to control an airplane during landing with respect to the criterion of safe landing probability.