Random Motions in Markov and Semi-Markov Random Environments 1. Anatoliy Swishchuk
et al. (1998), Cahoy (2007), Beghin and Orsingher (2010b), Orsingher and Beghin (2009), D’Ovidio et al. (2014), and others.
The set of particles with interaction, where each particle moves on a line according to a telegraph process, up to collision with another particle, was studied by Pogorui (2012b). During the collision, the particles exchange momentums. In this book, the author calculates the distribution of time of the first collision for two telegraph particles that started simultaneously from different points on a line and investigates the limit of this distribution under Kac’s condition. The author also investigates the system of particles with Markov switching, which is bounded with reflecting boundaries. The distribution for the position of particles of the system in a fixed time was also obtained. The limiting properties of these distributions and an estimate of the number of collisions in the system with reflecting boundaries, as well as without them, are also studied. Such a system of particles can be interpreted as a model of one-dimensional gas and it is a kind of one-dimensional generalization of the deterministic models of gas, of the billiard type, that were studied by Kornfeld et al. (1982), for example. The velocity of particles in these models is considered to be finite. This is a major difference from systems where the position of a particle is described by a diffusion process, such as in Arratia flow. We should note that models with finite speeds of particles moving under the influence of forces of mutual attraction were studied by Sinai (1992), Lifshits and Shi (2005), Giraud (2001, 2005), Bertoin (2002) and Vysotsky (2008).
I.2. Description of the book
The book is divided into two volumes, each containing two parts. Part 1 of Volume 1 consists of basic concepts and methods developed for random evolutions. These methods are the elementary tools for the rest of the book, and they include many results in potential operators and the description of some techniques to find closed-form expressions in relevant applications.
Part 2 of Volume 1 comprises three chapters (3, 4 and 5) dealing with asymptotic results (Chapter 3) and applications ranging from random motion with different types of boundaries, reliability of storage systems, telegraph processes, an alternative formulation to the Black–Scholes formula in finance, fading evolutions, jump telegraph processes and estimation of the number of level crossings for telegraph processes (Chapters 4 and 5).
Part 1 of Volume 2 extends many of the results of the latter part of Volume 1 to higher dimensions and consists of two chapters (1 and 2). Chapter 1 has the importance of presenting novel results of the random motion of the realistic three-dimensional case that has barely been mentioned in the literature. Chapter 2 deals with the interaction of particles in Markov and semi-Markov media, a topic many researchers have a strong interest in.
Part 2 of Volume 2 discusses applications of Markov and semi-Markov motions in mathematical finance across three chapters (3, 4 and 5). It includes applications of the telegraph process in modeling a stock price dynamic (Chapter 3), pricing of variance, volatility, covariance and correlation swaps with Markov volatility (Chapter 4), and the same pricing swaps with semi-Markov volatilities (Chapter 5).
The following is a general overview of the chapters and sections of the book. Chapters 1 and 2 of Volume 1 review the literature on the topic of random evolutions and outline the main areas of research. Many of these auxiliary results are used throughout the book.
Section 1.1 outlines research directions on the theory of telegraph processes and their generalizations.
In section 1.2, we introduce the notion of the projector operator and the generalized inverse operator or potential for an invertible reduced operator used in perturbation theory for linear operators. In turn, this theory is often used in the study of the asymptotic distribution of probability for reaching a “hard to reach domain”.
In section 1.3, we consider the notion of a semigroup of operators generated by a Markov process. We give the definitions of the infinitesimal operator, the stationary distribution and the potential of a Markov process. These concepts are used in Chapter 3 for the asymptotic analysis of large deviations of semi-Markov processes.
Section 1.4 provides a constructive definition of a semi-Markov process based on the concept of the Markov renewal process (MRP). The notion of the semi-Markov kernel, which is a key definition for MRP, is considered. For a semi-Markov process, we introduce some auxiliary processes, with which a semi-Markov process forms a two-component (or bivariate) Markov process, and for such a process the infinitesimal operator is presented.
In section 1.5, we consider the notion of a lumped Markov chain and describe a phase merging scheme.
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.
In Chapter 2 of Volume 1 we introduce homogeneous random evolutions (HRE), the elementary definitions, classification and some examples. We also present the martingale characterization and an analogue of Dynkin’s formula for HRE. Some other important topics covered in this chapter are limit theorems, weak convergence and diffusion approximations, which are useful for Part 2 of Volume 2.
In Chapter 3 of Volume 1 we consider the asymptotic distribution of a functional of the time for reaching “hard to reach” areas of the phase space by a semi-Markov process on the line.
Section 3.1 is devoted to the analysis of the asymptotic distribution of a functional related to the time to reach a level that is infinitely removed by a semi-Markov process on the set of natural numbers.
In section 3.2, we give asymptotic estimates for the distribution of residence times of the semi-Markov process in the set of states that expands when the condition of existence of the functional A is not fulfilled.
In section 3.3, we obtain the asymptotic expansion for the distribution of the first exit time from the extending subset of the phase space of the semi-Markov process embedded in the diffusion process.
In section 3.4, we obtain asymptotic expansions for the perturbed semigroups of operators of the respective three-variate Markov process (after the standard extension of the phase space of the perturbed random evolution uε (t, x) in the semi-Markov media), provided that the evolution uε (t, x) weakly converges to the diffusion process as ε > 0.
In section 3.5, we obtain asymptotic expansions under Kac’s condition in the diffusion approximation for the distribution of a particle position, which performs a random walk in a multidimensional space with Markov switching.
Section 3.6 describes a novel financial formula as an alternative to the well-known Black-Scholes formula for modeling the dynamic behavior of stock markets. This new formula is based on the asymptotic expansion for the singularly perturbed random evolution in Markov media.