Asymptotic Analysis of Unstable Solutions of Stochastic Differential Equations

Asymptotic Analysis of Unstable Solutions of Stochastic Differential Equations PDF

Author: Grigorij Kulinich

Publisher: Springer Nature

Published: 2020-04-29

Total Pages: 240

ISBN-13: 3030412911

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This book is devoted to unstable solutions of stochastic differential equations (SDEs). Despite the huge interest in the theory of SDEs, this book is the first to present a systematic study of the instability and asymptotic behavior of the corresponding unstable stochastic systems. The limit theorems contained in the book are not merely of purely mathematical value; rather, they also have practical value. Instability or violations of stability are noted in many phenomena, and the authors attempt to apply mathematical and stochastic methods to deal with them. The main goals include exploration of Brownian motion in environments with anomalies and study of the motion of the Brownian particle in layered media. A fairly wide class of continuous Markov processes is obtained in the limit. It includes Markov processes with discontinuous transition densities, processes that are not solutions of any Itô's SDEs, and the Bessel diffusion process. The book is self-contained, with presentation of definitions and auxiliary results in an Appendix. It will be of value for specialists in stochastic analysis and SDEs, as well as for researchers in other fields who deal with unstable systems and practitioners who apply stochastic models to describe phenomena of instability.

Qualitative and Asymptotic Analysis of Differential Equations with Random Perturbations

Qualitative and Asymptotic Analysis of Differential Equations with Random Perturbations PDF

Author: Anatoliy M. Samoilenko

Publisher: World Scientific

Published: 2011

Total Pages: 323

ISBN-13: 981432907X

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1. Differential equations with random right-hand sides and impulsive effects. 1.1. An impulsive process as a solution of an impulsive system. 1.2. Dissipativity. 1.3. Stability and Lyapunov functions. 1.4. Stability of systems with permanently acting random perturbations. 1.5. Solutions periodic in the restricted sense. 1.6. Periodic solutions of systems with small perturbations. 1.7. Periodic solutions of linear impulsive systems. 1.8. Weakly nonlinear systems. 1.9. Comments and references -- 2. Invariant sets for systems with random perturbations. 2.1. Invariant sets for systems with random right-hand sides. 2.2. Invariant sets for stochastic Ito systems. 2.3. The behaviour of invariant sets under small perturbations. 2.4. A study of stability of an equilibrium via the reduction principle for systems with regular random perturbations. 2.5. Stability of an equilibrium and the reduction principle for Ito type systems. 2.6. A study of stability of the invariant set via the reduction principle. Regular perturbations. 2.7. Stability of invariant sets and the reduction principle for Ito type systems. 2.8. Comments and references -- 3. Linear and quasilinear stochastic Ito systems. 3.1. Mean square exponential dichotomy. 3.2. A study of dichotomy in terms of quadratic forms. 3.3. Linear system solutions that are mean square bounded on the semiaxis. 3.4. Quasilinear systems. 3.5. Linear system solutions that are probability bounded on the axis. A generalized notion of a solution. 3.6. Asymptotic equivalence of linear systems. 3.7. Conditions for asymptotic equivalence of nonlinear systems. 3.8. Comments and references -- 4. Extensions of Ito systems on a torus. 4.1. Stability of invariant tori. 4.2. Random invariant tori for linear extensions. 4.3. Smoothness of invariant tori. 4.4. Random invariant tori for nonlinear extensions. 4.5. An ergodic theorem for a class of stochastic systems having a toroidal manifold. 4.6. Comments and references -- 5. The averaging method for equations with random perturbations. 5.1. A substantiation of the averaging method for systems with impulsive effect. 5.2. Asymptotics of normalized deviations of averaged solutions. 5.3. Applications to the theory of nonlinear oscillations. 5.4. Averaging for systems with impulsive effects at random times. 5.5. The second theorem of M.M. Bogolyubov for systems with regular random perturbations. 5.6. Averaging for stochastic Ito systems. An asymptotically finite interval. 5.7. Averaging on the semiaxis. 5.8. The averaging method and two-sided bounded solutions of Ito systems. 5.9. Comments and references

Asymptotic Analysis for Functional Stochastic Differential Equations

Asymptotic Analysis for Functional Stochastic Differential Equations PDF

Author: Jianhai Bao

Publisher: Springer

Published: 2016-11-19

Total Pages: 151

ISBN-13: 3319469797

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This brief treats dynamical systems that involve delays and random disturbances. The study is motivated by a wide variety of systems in real life in which random noise has to be taken into consideration and the effect of delays cannot be ignored. Concentrating on such systems that are described by functional stochastic differential equations, this work focuses on the study of large time behavior, in particular, ergodicity.This brief is written for probabilists, applied mathematicians, engineers, and scientists who need to use delay systems and functional stochastic differential equations in their work. Selected topics from the brief can also be used in a graduate level topics course in probability and stochastic processes.

Asymptotic Methods in the Theory of Stochastic Differential Equations

Asymptotic Methods in the Theory of Stochastic Differential Equations PDF

Author: A. V. Skorokhod

Publisher: American Mathematical Soc.

Published: 2009-01-07

Total Pages: 339

ISBN-13: 9780821846865

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Written by one of the foremost Soviet experts in the field, this book is intended for specialists in the theory of random processes and its applications. The author's 1982 monograph on stochastic differential equations, written with Iosif Ilich Gikhman, did not include a number of topics important to applications. The present work begins to fill this gap by investigating the asymptotic behavior of stochastic differential equations. The main topics are ergodic theory for Markov processes and for solutions of stochastic differential equations, stochastic differential equations containing a small parameter, and stability theory for solutions of systems of stochastic differential equations.

Asymptotic Methods in the Theory of Stochastic Differential Equations

Asymptotic Methods in the Theory of Stochastic Differential Equations PDF

Author: A. V. Skorokhod

Publisher: American Mathematical Soc.

Published: 2009-01-07

Total Pages: 362

ISBN-13: 9780821898253

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Ergodic theorems: General ergodic theorems Densities for transition probabilities and resolvents for Markov solutions of stochastic differential equations Ergodic theorems for one-dimensional stochastic equations Ergodic theorems for solutions of stochastic equations in $R^d$ Asymptotic behavior of systems of stochastic equations containing a small parameter: Equations with a small right-hand side Processes with rapid switching Averaging over variables for systems of stochastic differential equations Stability. Linear systems: Stability of sample paths of homogeneous Markov processes Linear equations in $R^d$ and the stochastic semigroups connected with them. Stability Stability of solutions of stochastic differential equations Linear stochastic equations in Hilbert space. Stochastic semigroups. Stability: Linear equations with bounded coefficients Strong stochastic semigroups with second moments Stability Bibliography

Asymptotic Analysis

Asymptotic Analysis PDF

Author: Mikhail V. Fedoryuk

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 370

ISBN-13: 3642580165

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In this book we present the main results on the asymptotic theory of ordinary linear differential equations and systems where there is a small parameter in the higher derivatives. We are concerned with the behaviour of solutions with respect to the parameter and for large values of the independent variable. The literature on this question is considerable and widely dispersed, but the methods of proofs are sufficiently similar for this material to be put together as a reference book. We have restricted ourselves to homogeneous equations. The asymptotic behaviour of an inhomogeneous equation can be obtained from the asymptotic behaviour of the corresponding fundamental system of solutions by applying methods for deriving asymptotic bounds on the relevant integrals. We systematically use the concept of an asymptotic expansion, details of which can if necessary be found in [Wasow 2, Olver 6]. By the "formal asymptotic solution" (F.A.S.) is understood a function which satisfies the equation to some degree of accuracy. Although this concept is not precisely defined, its meaning is always clear from the context. We also note that the term "Stokes line" used in the book is equivalent to the term "anti-Stokes line" employed in the physics literature.

Asymptotic Behavior and Stability Problems in Ordinary Differential Equations

Asymptotic Behavior and Stability Problems in Ordinary Differential Equations PDF

Author: Lamberto Cesari

Publisher: Springer

Published: 2013-11-09

Total Pages: 278

ISBN-13: 3662403684

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In the last few decades the theory of ordinary differential equations has grown rapidly under the action of forces which have been working both from within and without: from within, as a development and deepen ing of the concepts and of the topological and analytical methods brought about by LYAPUNOV, POINCARE, BENDIXSON, and a few others at the turn of the century; from without, in the wake of the technological development, particularly in communications, servomechanisms, auto matic controls, and electronics. The early research of the authors just mentioned lay in challenging problems of astronomy, but the line of thought thus produced found the most impressive applications in the new fields. The body of research now accumulated is overwhelming, and many books and reports have appeared on one or another of the multiple aspects of the new line of research which some authors call "qualitative theory of differential equations". The purpose of the present volume is to present many of the view points and questions in a readable short report for which completeness is not claimed. The bibliographical notes in each section are intended to be a guide to more detailed expositions and to the original papers. Some traditional topics such as the Sturm comparison theory have been omitted. Also excluded were all those papers, dealing with special differential equations motivated by and intended for the applications.

Stochastic Differential Equations with Markovian Switching

Stochastic Differential Equations with Markovian Switching PDF

Author: Xuerong Mao

Publisher: Imperial College Press

Published: 2006

Total Pages: 430

ISBN-13: 1860947018

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This textbook provides the first systematic presentation of the theory of stochastic differential equations with Markovian switching. It presents the basic principles at an introductory level but emphasizes current advanced level research trends. The material takes into account all the features of Ito equations, Markovian switching, interval systems and time-lag. The theory developed is applicable in different and complicated situations in many branches of science and industry.

Applied Stochastic Differential Equations

Applied Stochastic Differential Equations PDF

Author: Simo Särkkä

Publisher: Cambridge University Press

Published: 2019-05-02

Total Pages: 327

ISBN-13: 1316510085

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With this hands-on introduction readers will learn what SDEs are all about and how they should use them in practice.

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