Читать книгу Random Motions in Markov and Semi-Markov Random Environments 1 - Anatoliy Swishchuk - Страница 9

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.

Chapter 4 of Volume 1 is devoted to the computation of the stationary distributions for random switched processes with reflecting boundaries in Markov and semi-Markov environments. These results are used for the calculation of the efficiency of inventory control systems with feedback, and for some reliability problems of systems that are modeled by Markov and semi-Markov evolution.

In section 4.1, we derive the stationary distribution of transport processes with delaying in the boundaries in Markov media. These results are applicable to the study of multiphase systems with several reservoirs.

Section 4.2 deals with the transport process with semi-Markov switching. We find the stationary measure for this process and it is described by the differential equation with phase space on the corresponding interval and constant vector field values that depend on the switching semi-Markov process with a finite set of states.

In section 4.3, we give examples of applications of this method for the calculation of the stationary distributions for random switched processes with reflecting boundaries. In particular, we compute the efficiency coefficient of a single and two-phase inventory control system with feedback.

In section 4.4, we apply random evolutions with delaying barriers to modeling the control of supply systems with feedback, by considering the semi-Markov switching process.

In Chapter 5 of Volume 1 we study various models of stochastic evolutions that generalize the Goldstein–Kac telegraph process and investigate their distributions.

Section 5.1 is devoted to one-dimensional semi-Markov evolutions in an Erlang environment. In section 5.1.1, we derive a hyperbolic differential equation of the telegraph type for the pdf of the position of a particle moving at finite velocity. The interarrival times between two successive changes of velocity have an Erlang distribution.

In section 5.1.2, a method of solution of this partial differential equation is developed by using monogenic functions associated with a finite-dimensional commutative algebra. Section 5.1.3 extends the results of section 5.1.2 to infinite-dimensional commutative algebras. In section 5.1.4, by using the methods from sections 5.1.2 and 5.1.3, we obtain the distribution of a one-dimensional random evolution in Erlang media.

In section 5.2, we find the distribution of limiting positions of a particle moving according to a fading evolution. We assume that the interarrival times are Erlang or uniform distributed for the switching process.

In section 5.3, differential and integral equations for jump random motions are introduced and some examples have been developed to illustrate the method.

Section 5.4 is devoted to the estimation of the number of level crossings by the telegraph process in Kac’s condition.

In Chapter 1 of Volume 2 we study random motions in higher dimensions. In section 1.1, we study the random walk of a particle with constant absolute velocity, which changes its direction according to a uniform distribution on the unit sphere at the renewal epochs of a switching process.

Section 1.2 deals with random motion with uniformly distributed directions and random velocity for the motion of particles in one, two, three and four dimensions. Similar cases are considered in section 1.3 for the distribution of random motion at non-constant velocity in semi-Markov media, and in section 1.4 for Goldstein–Kac telegraph equations and random flights in higher dimensions, where the directions of the movement and the velocity change at the renewal epochs. The jump telegraph process in ℝⁿ is considered in section 1.5.

In Chapter 2 of Volume 2 we study a system of interacting particles with Markov and semi-Markov switching. Section 2.1 is devoted to an ideal gas model with finite velocity of molecules. In this section, we find the distribution of the first meeting time of two telegraph particles on the line, which started simultaneously from different points.

In section 2.1.2, we estimate the number of particle collisions. In section 2.1.4, we obtain an asymptotic estimate of the number of collisions of the system of telegraph particles when time goes to infinity. Unlike the previous section, the system has no boundaries and particles can move arbitrarily far away in a straight line. Section 2.2 is devoted to the generalization of results of section 2.1 to the case of semi-Markov switching processes. In section 2.2.1, we obtain a set of renewal-type equations for the Laplace transform of the first collision of two particles. In section 2.2.2, we consider an example regarding a particular case of semi-Markov switching of particle velocity, and we study the limiting properties of the distribution of the particle position. Section 2.2.3 is devoted to studying the case where the first time of collision of two particles has finite expectation. We should note that in all previous examples, including the Markov case, expectation of the first time of collision of two particles is infinite.

In Chapter 3 of Volume 2 the application of the telegraph process for option prices, as an alternative to the diffusion process in the Black-Scholes formula, is further studied through asymptotic estimation of the corresponding operators. Some numerical results and plots are also presented.

Pricing variance, volatility derivatives, covariance and correlation swaps for financial markets with Markov-modulated volatilities are the main topics of Chapter 4 of Volume 2. These results are extended to the case of semi-Markov modulated volatilities in Chapter 5 of Volume 2. Numerical results and some plots to illustrate these results are also presented in these last two chapters.