Random Motions in Markov and Semi-Markov Random Environments 1

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Anatoliy Swishchuk. Random Motions in Markov and Semi-Markov Random Environments 1

Table of Contents

List of Illustrations

List of Tables

Guide

Pages

Random Motions in Markov and Semi-Markov Random Environments 1. Homogeneous Random Motions and their Applications

Preface

Acknowledgments

Introduction. I.1. Overview

I.2. Description of the book

1. Preliminary Concepts. 1.1. Introduction to random evolutions

1.2. Abstract potential operators

1.3. Markov processes: operator semigroups

1.4. Semi-Markov processes

1.5. Lumped Markov chains

1.6. Switched processes in Markov and semi-Markov media

2. Homogeneous Random Evolutions (HRE) and their Applications

2.1. Homogeneous random evolutions (HRE) 2.1.1. Definition and classification of HRE

2.1.2. Some examples of HRE

2.1.3. Martingale characterization of HRE

Martingale problem for discontinuous RE over a jump Markov process

Martingale problem for discontinuous RE over semi-Markov process

Martingale problem for RE over Wiener process

Martingale problem for RE over diffusion process

2.1.4. Analogue of Dynkin’s formula for HRE

2.1.5. Boundary value problems for HRE

2.2. Limit theorems for HRE

2.2.1. Weak convergence of HRE

2.2.2. Averaging of HRE

2.2.3. Diffusion approximation of HRE

2.2.4. Averaging of REs in reducible phase space: merged HRE

2.2.5. Diffusion approximation of HRE in reducible phase space

2.2.6. Normal deviations of HRE

2.2.7. Rates of convergence in the limit theorems for HRE

3. Asymptotic Analysis for Distributions of Markov, Semi-Markov and Random Evolutions

3.1. Asymptotic distribution of time to reach a level that is infinitely increasing by a family of semi-Markov processes on the set ℕ

3.2. Asymptotic inequalities for the distribution of the occupation time of a semi-Markov process in an increasing set of states

3.3. Asymptotic analysis of the occupation time distribution of an embedded semi-Markov process (with increasing states) in a diffusion process

3.4. Asymptotic analysis of a semigroup of operators of the singularly perturbed random evolution in semi-Markov media

3.5. Asymptotic expansion for distribution of random motion in Markov media under the Kac condition

3.5.1. The equation for the probability density of the particle position performing a random walk in ℝn

3.5.2. Equation for the probability density of the particle position

3.5.3. Reduction of a singularly perturbed evolution equation to a regularly perturbed equation

3.6. Asymptotic estimation for application of the telegraph process as an alternative to the diffusion process in the Black–Scholes formula

3.6.1. Asymptotic expansion for the singularly perturbed random evolution in Markov media in case of disbalance

3.6.2. Application to an economic model of stock market

4. Random Switched Processes with Delay in Reflecting Boundaries

4.1. Stationary distribution of evolutionary switched processes in a Markov environment with delay in reflecting boundaries

4.2. Stationary distribution of switched process in semi-Markov media with delay in reflecting barriers

4.2.1. Infinitesimal operator of random evolution with semi-Markov switching

4.2.2. Stationary distribution of random evolution in semi-Markov media with delaying boundaries in balance case

4.2.3. Stationary distribution of random evolution in semi-Markov media with delaying boundaries

4.3. Stationary efficiency of a system with two unreliable subsystems in cascade and one buffer: the Markov case. 4.3.1. Introduction

4.3.2. Stationary distribution of Markov stochastic evolutions

4.3.3. Stationary efficiency of a system with two unreliable subsystems in cascade and one buffer

4.3.4. Mathematical model

4.3.5. Main mathematical results

4.3.6. Numerical results for the symmetric case

4.4 Application of random evolutions with delaying barriers to modeling control of supply systems with feedback: the semi-Markov switching process

4.4.1. Estimation of stationary efficiency of one-phase system with a reservoir

4.4.2. Estimation of stationary efficiency of a production system with two unreliable supply lines

5. One-dimensional Random Motions in Markov and Semi-Markov Media

5.1. One-dimensional semi-Markov evolutions with general Erlang sojourn times

5.1.1. Mathematical model

5.1.2. Solution of PDEs with constant coefficients and derivability of functions ranged in commutative algebras

5.1.3. Infinite-dimensional case

5.1.4. The distribution of one-dimensional random evolutions in Erlang media

5.2. Distribution of limiting position of fading evolution

5.2.1. Distribution of random power series in cases of uniform and Erlang distributions

5.2.2. The distribution of the limiting position

5.3. Differential and integral equations for jump random motions

5.3.1. The Erlang jump telegraph process on a line

5.3.2. Examples. 5.3.2.1. Exponential case

5.3.2.2. Erlang-2 case

5.4. Estimation of the number of level crossings by the telegraph process

5.4.1. Estimation of the number of level crossings for the telegraph process in Kac’s condition

References

Index. A, B

C, D

F, G, H

I, J, K

L, M

N, O

P, R

S

T, W

Summary of Volume 2

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Отрывок из книги

Series Editor Nikolaos Limnios

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Section 1.6 describes a stochastic switching process in Markov and semi-Markov environments. We define semigroup operators associated with this process and consider their infinitesimal operator. In addition, the concept of superposition of independent semi-Markov processes is considered.

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