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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
- Veröffentlichungsjahr: 2020
This book is the first of two volumes on random motions in Markov and semi-Markov random environments. This first volume focuses on homogenous random motions. This volume consists of two parts, the first describing the basic concepts and methods that have been developed for random evolutions. These methods are the foundational tools used in both volumes, and this description includes many results in potential operators. Some techniques to find closed-form expressions in relevant applications are also presented. The second part deals with asymptotic results and presents a variety of applications, including random motion with different types of boundaries, the reliability of storage systems and solutions of partial differential equations with constant coefficients, using commutative algebra techniques. It also presents an alternative formulation to the Black-Scholes formula in finance, fading evolutions and telegraph processes, including jump telegraph processes and the estimation of the number of level crossings for telegraph processes.
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Table of Contents
Cover
Title Page
Copyright
Preface
Acknowledgments
Introduction
PART 1: Basic Methods
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)
2.2. Limit theorems for HRE
PART 2: Applications to Reliability, Random Motions, and Telegraph Processes
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
3.6. Asymptotic estimation for application of the telegraph process as an alternative to the diffusion process in the Black–Scholes formula
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
4.3. Stationary efficiency of a system with two unreliable subsystems in cascade and one buffer: the Markov case
4.4. Application of random evolutions with delaying barriers to modeling control of supply systems with feedback: the semi-Markov switching process
5 One-dimensional Random Motions in Markov and Semi-Markov Media
5.1. One-dimensional semi-Markov evolutions with general Erlang sojourn times
5.2. Distribution of limiting position of fading evolution
5.3. Differential and integral equations for jump random motions
5.4. Estimation of the number of level crossings by the telegraph process
References
Index
Summary of Volume 2
End User License Agreement
List of Illustrations
Chapter 4
Figure 4.1.
A system of two unreliable subsystems, say
S
1
and
S
2
, connected in s...
Figure 4.2. Efficiency parameter K as a function of reservoir size V for differe...
Figure 4.3. Efficiency parameter K as a function of reservoir size V for differe...
Guide
Cover
Table of Contents
Title page
Copyright
Preface
Acknowledgments
Introduction
Begin Reading
References
Index
Summary of Volume 2
End User License Agreement
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Series EditorNikolaos Limnios
Random Motions in Markov and Semi-Markov Random Environments 1
Homogeneous Random Motions and their Applications
Anatoliy Pogorui
Anatoliy Swishchuk
Ramón M. Rodríguez-Dagnino
First published 2021 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 Ltd
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John Wiley & Sons, Inc.
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© ISTE Ltd 2021
The rights of Anatoliy Pogorui, Anatoliy Swishchuk and Ramón M. Rodríguez-Dagnino 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: 2020946634
British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-78630-547-3
Preface
Motion is an essential element in our daily life. Thoughts related to motion can be found in the ancient Greek philosophers; however, we have to look several centuries ahead for relevant mathematical models. Galileo Galilei, Isaac Newton and Johannes Kepler, between the years 1550–1650, made remarkable advances in the construction of mathematical models for deterministic motion. Further advances in this line were made by Leonhard Euler and William Rowan Hamilton, and, in 1778, Joseph-Louis Lagrange proposed a new formulation of classical mechanics. This new formulation is based on the optimization of energy functionals, and it allows us to solve more sophisticated motion problems in a more systematic manner. This new approach to mechanics is the basis for dealing with modern quantum mechanics and the physics of high-energy particles. Albert Einstein, in 1905, through his special theory of relativity, introduced fine corrections for high velocities, close to the maximum speed of light. All of these fundamental mathematical models deal with many sophisticated motions such as that of astronomical objects, satellites (natural and artificial), particles in intense electromagnetic fields, particles under gravitational forces, a better understanding of light, and so on. However, even though these might be very complicated problems, we should mention that all of them have deterministic paths of motion.
In 1827, the Scottish botanist Robert Brown described a special kind of random motion produced by the interaction of many particles, while looking at pollen of the plant Clarkia pulchella immersed in water, using a microscope, and it was recognized that this type of motion could not be fully explained by modeling the motion of each particle or molecule. Albert Einstein, 78 years later, in 1905, published a seminal paper where he modeled the motion of the pollen as being moved by individual water molecules. In this work, the diffusion equation was introduced as a convenient mathematical model for this random phenomenon. A related model in this direction was presented in 1906 by Marian Smoluchowski, and the experimental verification was done by Jean Baptiste Perrin in 1908.
A similar model for Brownian motion was proposed in 1900 by Louis Bachelier in his PhD thesis entitled The Theory of Speculation, where he presented a stochastic analysis for valuing stock options in financial markets. This novel application of a stochastic model faced criticism at the beginning, but Bachelier’s instructor Henri Poincaré was in full support of this visionary idea. This fact shows a close relationship between models to explain random phenomena in physics (statistical mechanics) and in financial analysis, and also in many other areas.
A notable contribution of the American mathematician Norbert Wiener was to establish the mathematical foundations for Brownian motion, and for that reason it is also known as the Wiener process. Great mathematicians such as Paul Lévy, Andrey Kolmogorov and Kiyosi Itô, among many other brilliant experts in the new field of probability and stochastic processes, set the basis of these stochastic processes. For example, the famous Black–Scholes formula in financial markets is based on both diffusion processes and Itô’s ideas.
In spite of its success in modeling many types of random motion and other random quantities, the Wiener process has some drawbacks when capturing the physics of many applications. For instance, the modulus of velocity is almost always infinite at any instant in time, it has a free path length of zero, the path function of a particle is almost surely non-differentiable at any given point and its Hausdorff dimension is equal to 1.5, i.e. the path function is fractal. However, the actual movement of a physical particle and the actual evolution of share prices are barely justified as fractal quantities. Taking into account these considerations, in this book we propose and develop other stochastic processes that are close to the actual physical behavior of random motion in many other situations. Instead of the diffusion process (Brownian motion), we consider telegraph processes, Poisson and Markov processes and renewal and semi-Markov processes.
Markov (and semi-Markov) processes are named after the Russian mathematician Andrey Markov, who introduced them in around 1906. These processes have the important property of changing states under certain rules, i.e. they allow for abrupt changes (or switching) in the random phenomenon. As a result, these models are more appropriate for capturing random jumps, alternate velocities after traveling a certain random distance, random environments through the formulation of random evolutions, random motion with random changes of direction, interaction of particles with non-zero free paths, reliability of storage systems, and so on.
In addition, we will model financial markets with Markov and semi-Markov volatilities as well as price covariance and correlation swaps. Numerical evaluations of variance, volatility, covariance and correlations swaps with semi-Markov volatility are also presented. The novelty of these results lies in the pricing of volatility swaps in the closed form, and the pricing of covariance and correlation swaps in a market with two risky assets.
Anatoliy POGORUI
Zhytomyr State University, Ukraine
Anatoliy SWISHCHUK
University of Calgary, Canada
Ramón M. RODRÍGUEZ-DAGNINO
Tecnologico de Monterrey, Mexico
October 2020
Acknowledgments
Anatoliy Pogorui was partially supported by the State Fund for Basic Research (Ministry of Education and Science of Ukraine 20.02.2017 letter no. 12).
Anatoliy Pogorui
I would like to thank NSERC for its continuing support, my research collaborators, and my current and former graduate students. I also give thanks to my family for their inspiration and unconditional support.
Anatoliy Swishchuk
I would like to thank Tecnologico de Monterrey for providing me the time and support for these research activities. I also appreciate the support given by Conacyt through the project no. SEP-CB-2015-01-256237. The time and lovely support given to me by my wife Saida, my three daughters Dunia, Melissa and R. Melina, as well as my son Ramón Martín, are invaluable.
Ramón M. Rodríguez-Dagnino
