Asymptotic Analyses for Complex Evolutionary Systems with Markov and Semi-Markov Switching Using Approximation Schemes E-Book
Yaroslav Chabanyuk
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- Herausgeber: John Wiley & Sons
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- Sprache: Englisch
This book analyzes stochastic evolutionary models under the impulse of diffusion, as well as Markov and semi-Markov switches. Models are investigated under the conditions of classical and non-classical (Levy and Poisson) approximations in addition to jumping stochastic approximations and continuous optimization procedures. Among other asymptotic properties, particular attention is given to weak convergence, dissipativity, stability and the control of processes and their generators. Weak convergence of stochastic processes is usually proved by verifying two conditions: the tightness of the distributions of the converging processes, which ensures the existence of a converging subsequence, and the uniqueness of the weak limit. Achieving the limit can be done on the semigroups that correspond to the converging process as well as on appropriate generators. While this provides the convergence of generators, a natural question arises concerning the uniqueness of a limit semigroup.
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Veröffentlichungsjahr: 2020
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Table of Contents
Cover
Title page
Copyright
Acronyms
Introduction
1 Average Scheme and Diffusion Approximation Scheme
1.1. Stability of stochastic systems in the average scheme
1.2. Stability of stochastic systems in the diffusion approximation scheme
2 Levy Approximation Scheme
2.1. Differential equations with small stochastic additions in the Levy approximation scheme
2.2. Asymptotic dissipativity of stochastic processes with impulse perturbations in the Levy approximation scheme
2.3. Double merging of phase space for differential equations with small stochastic supplements under Levy approximation conditions
3 Asymptotical Analysis of Random Evolutionary Systems Under Poisson Approximation Conditions
3.1. Differential equations with small stochastic additions under Poisson approximation conditions
3.2. Asymptotic dissipativity of stochastic processes with impulse perturbation in the Poisson approximation scheme
3.3. Double merging of the phase space for differential equations with small stochastic supplements under Poisson approximation conditions
4 Stochastic Approximation Procedure
4.1. Markov environment
4.2. Semi-Markov environment
4.3. Asymptotic normality of fluctuations of the procedure of stochastic approximation with diffusive perturbation in a Markov environment
4.4. Asymptotic normality of SAP in a semi-Markov environment
5 Stochastic Optimization Procedure
5.1. SOP in the average scheme
5.2. SOP under the diffusion approximation scheme
6 Combination of Approximations of Different Types
6.1. Asymptotic properties of a stochastic diffusion process with an equilibrium point of a quality criterion
6.2. Asymptotic properties of the impulse perturbation process with a control function under Levy approximation conditions
References
Index
End User License Agreement
Guide
Cover
Table of Contents
Title page
Copyright
Acronyms
Introduction
Begin Reading
References
Index
End User License Agreement
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To our teachers
Series EditorNikolaos Limnios
Asymptotic Analyses for Complex Evolutionary Systems with Markov and Semi-Markov Switching Using Approximation Schemes
Yaroslav Chabanyuk
Anatolii Nikitin
Uliana Khimka
First published 2020 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
ISTE Ltd27–37 St George’s RoadLondon SW19 4EUUK USA
www.iste.co.uk
John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USA
www.wiley.com
© ISTE Ltd 2020
The rights of Yaroslav Chabanyuk, Anatolii Nikitin and Uliana Khimka to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.
Library of Congress Control Number: 2020936535
British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-78630-556-5
Acronyms
SDE
Stochastic Differential Equation
IPP
Impulse Perturbation Process
MP
Markov Process
SMP
Semi-Markov Process
SAP
Stochastic Approximation Procedure
SOP
Stochastic Optimization Procedure
MRP
Markov Renewal Process
EMRP
Extended Markov Renewal Process
SSPP
Solving of Singular Perturbed Problem
Introduction
The emergence of the theory of stochastic differential equations played a prominent role in the works of S.N. Bernshtain, M.M. Bogolyubov and M.M. Krylov. A systematic study of stochastic differential equations was first carried out by Y.I. Gichman. The concept of random evolution was introduced by Griego and Hersh (1969) and Bellman (1957).
Applications of such a model were derived from the work of Feng (1999), Fleming and Soner (2006), Feng and Kurtz (2006), which were stimulated by the problems of the stability of stochastic systems.
In the 1960s and 1970s, the problems associated with the theory of random evolution were actively investigated by American mathematicians R. Hersch, M. Pinskii, G. Papanikolau, T. Kurtts, R. Griego, L. Horossey (Skorokhod 1989; Sviridenko 1998; Skorokhod et al. 2002; Samoilenko et al. 2017) and others. In particular, G. Papanikolaou, D. Stroock and S. Varadan proposed a martingale approach for the proof of boundary theorems (Papanicolaou et al. 1977), Stroock using methods similar to solving the singular perturbation problem.
An effective means to prove the limit theorems in the theory of random evolution was developed by Korolyuk and Turbin (1993). Thanks to their methods, A.F. Turbin, O.S. Hochhel and other authors proved important boundary theorems on regular approximations for singularly perturbed solutions of differential equations for different stochastic models, as well as for dynamical systems with coefficients dependent on Markov processes (Feng and Kurtz 2006). Skorokhod (1989), V.E. Shapiro and V.M. Login investigated several important problems of the theory of dynamical systems described by stochastic differential equations. The problem of the stability of dynamical systems with random disturbance of their parameters was investigated by Gorostiza (1972, 1973a, 1973b), E.F. Tsar'kov and V.S. Korolyuk. Problems of stability and of the stochastic approximation of evolution systems with switches were investigated by Y.M. Chabanyuk and in the works of Kurtz (1973, 1987), Donsker and Varadhan (1975a), Papanicolaou (1975), Korolyuk and Turbin (1993) and Koroliuk et al. (2010a). V.S. Korolyuk and A.V. Svishchuk also developed the theory of semi-Markovic random evolution, based on the theory of martingales. Many applications of switching processes to the analysis of network systems can be found in the works of V.V. Anisimov (Bainov and Simeonov 1992; Albeverio et al. 2008, 2009; Anisimov 2008). Samoilenko I.V. (Albeverio et al. 2008, 2009) studied the Markov processes with impulse disturbances in approximation schemes.
Methods of studying the asymptotic properties used in the dissertation have been described in the monographs of Korolyuk and Limnios (2005) and Skorokhod. These methods are mainly applied to models in averaging and diffusion approximation schemes (Chabanyuk et al. 2007). At the same time, with a growing time interval, the averaging scheme demonstrates the deterministic averaged behavior of the system, and the scheme of diffusion approximation is stochastic fluctuations around a deterministic averaged trajectory. These two schemes differ in the normalization of the switching process, namely, in the case of the averaging scheme, the acceleration of time is considered by the parameter ɛ-1, whereas in the case of diffusion approximation, it is considered by the parameter ɛ-2. An important element of the algorithm is the assumption that the switching process is ergodic.
Note that the main ideas of the random process have been presented in the works of Cramer (1938), Cole (1951), Chernoff (1952), Bellman (1957), Doléans-Dade and Meyer (1970), Papanicolaou (1971a, 1971b, 1973), Hersh (1974), Roth (1974), Cinlar et al. (1980) and Doob (1990). Works on the problem of large deviations should be singled out: Freidlin (1978), Ethier and Kurtz (1985), Deuschel and Stroock (1989), Bryc (1990), Dembo and Zeitouni (1993), Mogulskii (1993), O'Brien and Vervaat (1995), Dupuis and Ellis (1997) and Koroliuk and Samoilenko (2014).
Tasks of weak convergence of the random process have been presented in the works of Kushner (1990) and Kolesnik (2001, 2003).
The application of stochastic differential equations has been presented in the following works: Hopf (1950), Pinsky (1968), Papanicolaou and Kellek (1971), Papanicolaou and Hersh (1972), Nashed (1976), Papanicolaou et al. (1977), Crandall and Lions (1983), Fukushima and Stroock (1986), Protter (1990), Puhalskii (1991), Davis (1993), Dellacherie and Meyer (1993), Devooght (1997), Hillen (2001), Othmer and Hillen (2002), Hersh (2003), Jacod and Shiryaev (2003) and Orsingher and Sommella (2004).
In the monographs of Korolyuk and Korolyuk (1999) and Limnios and Oprişan (2001), the classification of stochastic models has been considered. In the works of Korolyuk and Chabanyuk (2002) and Chabanyuk (2007), stability conditions of the dynamic system with semi-Markov switchings have been established in the average scheme and the diffusion approximation scheme. The fluctuations of such system have been considered in the paper by Chabanyuk et al. (2007).
A natural generalization of the stochastic approximation procedure (Nevelson and Khasminsky 1972) is an examination of the case where the regression function is directly dependent on an external environment with Markov or semi-Markov switching (Chabanyuk 2004a, 2004b, 2019). For the stochastic optimization procedure (Khimka and Chabanyuk 2013), sufficient conditions are obtained for convergence with Markov switching. The properties of the compensating operator of random evolution are used in the case of semi-Markov switching (Sviridenko 1998).
The stochastic approximation procedure has been investigated in the case of impulse perturbing in the work of Khimka and Chabanyuk (2013).
The conditions of asymptotic normality of the stochastic approximation procedure have been established in the works of Chabanyuk (2004a, 2004b, 2006). This property has been considered for the control task in the work of Nikitin and Khimka (2017).
Korolyuk and Limnios (2005) also considered several models for stochastic evolutions by taking into account impulse influences in the Poisson and Levy approximation schemes.
They also presented another important direction of research, namely the study of the dissipative nature of random evolutions under the Levy and Poisson approximation conditions (Bertoin 1996; Koroliuk et al. 2010a, 2010b). In our research, an asymptotic dissipative analysis of stochastic differential equations with impulse influences is carried out under the conditions of nonclassical approximation schemes (i.e. Levy and Poisson).
Another important direction of the monograph research is the establishment of conditions for a weak convergence of the diffusion transfer process with Markov switches and the control of the function of the quality criterion with the equilibrium point for which the stochastic approximation procedure is constructed in the series scheme. Under these conditions, the normalized process is also constructed and its asymptotic normality is established in the Ornstein–Uhlenbeck process when the transfer process changes under the influence of the Markov switching on the trajectory of a new evolution from the state in which it was at the moment of switching. The problem of normalized control with a single point of equilibrium of the quality criterion of the diffusion transfer process is studied under the conditions of the stochastic approximation procedure. A similar statement of the problem is completely new and allows for some significant generalizations.
The thesis contains some applications of the developed methods to the analysis of the model of counteraction to information attacks. The monograph includes the following new results:
– under the Levy approximation conditions, a limiting generator was constructed and an asymptotic behavior analysis of stochastic evolution systems with impulse perturbation and Markov switches was carried out;
– under the Poisson approximation conditions, an asymptotic analysis of the behavior of stochastic evolution systems with impulse disturbance and Markov switches was carried out;
– for a Poisson approximation, the boundary process contains two components: deterministic shift and Poisson jump component. In contrast, for a Levy approximation, the Levy limiting process contains three components: deterministic shift, diffusion time and Poisson jump component. In this case, the function that defines a deterministic shift is determined by frequent small jumps of the process; the function, which sets the diffusion component, is determined by frequent large jumps of the process; finally, the high jumps of the boundary process are set by the measure of rare large jumps of the border process;
– an asymptotic dissipative analysis of stochastic differential equations with impulse effects under the conditions of nonclassical approximation schemes (Levy and Poisson) was carried out;
– the conditions of weak convergence of the diffusion transfer process with Markov switches and the control of the function of the quality criterion with the equilibrium point for which the stochastic approximation procedure was constructed in the series scheme;
– for the process of transfer with Markov switches and control under the conditions of existence of a single point of equilibrium of the quality criterion, a normalized process was constructed and its asymptotic normality was established in the Ornstein–Uhlenbeck process when the transfer process changed under the influence of the Markov switching on the trajectory of a new evolution from the state in which it was at the moment of switching;
– for the stochastic approximation procedure, a generalization to the case of the extrinsic environment dependent regression function with Markov and Semi-Markov switches is considered. Sufficient stochastic convergence conditions for Markov switches were obtained for the stochastic optimization procedure.
1Average Scheme and Diffusion Approximation Scheme
1.1. Stability of stochastic systems in the average scheme
In this section, we analyze the stability of a dynamical system with semi-Markov switchings using a compensating operator of the semi-Markov process. The problem of the stability of a semi-Markov stochastic system is similar to the problem with Markov switchings.
We study the stochastic evolutionary equation
which defines an initial dynamical system with semi-Markov switchings in the series scheme with a small parameter Euclidean space. The velocities of the dynamical system are determined by the function vectors , which satisfy the conditions of the existing global solutions for deterministic systems
where x is the semi-Markov process in the standard phase space (X, X) (Korolyuk and Limnios 2005). The average dynamical system is determined by the solution of the evolutionary deterministic equation
where the averaging velocity is given by the equality
Under the conditions of the stability of the averaged system [1.3], the problem is to establish additional conditions to ensure the stability of the initial system [1.1] with sufficiently small values of the parameter ε > 0.
THEOREM 1.1.– Assume that for the average dynamical system [1.3] and [1.4], the Lyapunov function exists, which satisfies the following conditions:
C1 : exponential stability
additional conditions
C4 : distribution functions , of the time of stay in the states of the semi-Markov process x(t),t > 0, satisfy the Cramer condition, uniformly at x ∈ X:
C5 : estimates hold true:
Then, for all ɛ < ɛ0 (where ɛ0 is small enough), the solution of the evolutionary equation [1.1] at all initial conditions(where u* is small enough) is asymptotically stable with probability 1:
Under random initial conditions, the inequality is satisfied, where u* is small enough.
The extended Markov renewal process (EMRP) has the form
DEFINITION 1.1.– The compensating operator of the EMRP [1.6] is determined by the relation Sviridenko (1998)
We assume that the family of semigroups is generated by the accompanying system [1.2], i.e. it is determined by the generator
LEMMA 1.1.– The compensating operator of the EMRP [1.6] has the analytical form
PROOF.– We begin with the calculation of the conditional expectation using semigroups Ct(x). Since semigroups Ct (x) on the Banach space C(Rd) of continuous real-valued functions , with the supremum norm is determined by the relation
under the condition we have
i.e.
From the last relation, we have the equality for the conditional expectation
Using the representation of a semi-Markov kernel
we obtain
Setting the last representation in [1.7], we have the equality that completes the proof of the lemma.
LEMMA 1.2.– The compensating operator on test functions has the asymptotic representation
and
where
The reminder operators have the representation
where
PROOF.– The proof is based on the following semigroup representation:
where
We note that for calculation of the remainder operators, we use the following relations:
We also note that
Let us consider the perturbed Lyapunov function:
where , is the Lyapunov function for the average system [1.3], under conditions C1—C3 of the theorem.
The perturbation , is determined by the solution of the singular perturbation problem for the operator
The solution of the singular perturbation problem for the operator [1.13] is determined by the relation
where and the operator is a potential of the operator Q. We know that under condition C4 of the theorem, the operator R0 is bounded.
LEMMA 1.3.– The solution of the singular perturbation problem for the compensating operator [1.10] on the perturbed Lyapunov function [1.12] is determined by the relations:
The reminder operator has the form:
PROOF.– We use two representations of the generator Lε of Lemma 1.2, respectively for [1.10]
and for [1.11]
We represent the generator Lε on the perturbed Lyapunov function [1.12] as
Since
and
then
From the representation of the remainder operators as well as the perturbation V1(u, x) in the form [1.13], we obtain the representation of the remainder term in the form [1.15].
We note that under conditions C1 – C5 of the theorem, we have the inequality
To prove inequality [1.16], we first estimate the remainder operator [1.15], using conditions C2 – C3 of the theorem
and then estimate the perturbation V1(u, x):
According to the exponential stability condition C1, the first term in [1.14] satisfies the inequality
Combining the estimates [1.17], [1.18] and [1.19], we obtain [1.16] for each ɛ ≤ ɛ0, where ɛ0 is small enough, at some value of c > 0.
The condition C2 of the theorem provides an estimate of the semigroups Ct (x), which is generated by the operator [1.8]:
The following inequalities also hold true (see [1.12], [1.17] and [1.18]):
LEMMA 1.4.– ExtendedRP [1.6] is characterized by a martingale
PROOF.– The martingale property of a consequence yields from the uniformity of the compensating operator:
Here, σ-algebras , are generated by the EMRP.
We denote
Note that is the Markovian moment for the flow
Furthermore, we use the martingale property of the process with continuous time:
We note that
and
