This book is the second of two volumes on random motions in Markovand semi-Markov random environments. This second volume focuseson high-dimensional random motions.This volume consists of two parts. The first expands many of the resultsfound in Volume 1 to higher dimensions. It presents new results on therandom motion of the realistic three-dimensional case, which has so farbeen barely mentioned in the literature, and deals with the interaction ofparticles in Markov and semi-Markov media, which has, in contrast,been a topic of intense study. The second part contains applications of Markov and semi-Markovmotions in mathematical finance. It includes applications of telegraphprocesses in modeling stock price dynamics and investigates the pricingof variance, volatility, covariance and correlation swaps with Markovvolatility and the same pricing swaps with semi-Markov volatilities.
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I.2. Description of the book
PART 1: Higher-dimensional Random Motions and Interactive Particles
1 Random Motions in Higher Dimensions
1.1. Random motion at finite speed with semi-Markov switching directions process
1.2. Random motion with uniformly distributed directions and random velocity
1.3. The distribution of random motion at non-constant velocity in semi-Markov media
1.4. Goldstein–Kac telegraph equations and random flights in higher dimensions
1.5. The jump telegraph process in Rn
2 System of Interactive Particles with Markov and Semi-Markov Switching
2.1. Description of the Markov model
2.2. Interaction of particles governed by generalized integrated telegraph processes: a semi-Markov case
PART 2: Financial Applications
3 Asymptotic Estimation for Application of the Telegraph Process as an Alternative to the Diffusion Process in the Black–Scholes Formula
3.1. Asymptotic expansion for the singularly perturbed random evolution in Markov media in the case of disbalance
3.2. Application: Black–Scholes formula
4 Variance, Volatility, Covariance and Correlation Swaps for Financial Markets with Markov-modulated Volatilities
4.1. Volatility derivatives
4.2. Martingale representation of a Markov process
4.3. Variance and volatility swaps for financial markets with Markov-modulated stochastic volatilities
4.4. Covariance and correlation swaps for two risky assets for financial markets with Markov-modulated stochastic volatilities
4.5. Example: variance, volatility, covariance and correlation swaps for stochastic volatility driven by two state continuous Markov chain
4.6. Numerical example
4.7. Appendix 1
5 Modeling and Pricing of Variance, Volatility, Covariance and Correlation Swaps for Financial Markets with Semi-Markov Volatilities
5.2. Martingale representation of semi-Markov processes
5.3. Variance and volatility swaps for financial markets with semi-Markov stochastic volatilities
5.4. Covariance and correlation swaps for two risky assets in financial markets with semi-Markov stochastic volatilities
5.5. Numerical evaluation of covariance and correlation swaps with semi-Markov stochastic volatility
Summary of Volume 1
End User License Agreement
Table 4.1. One-step transition probability matrix
Table 4.2. One-step transition probability matrix
Table of Contents
Summary of Volume 1
End User License Agreement
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:
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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
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A CIP record for this book is available from the British Library
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 POGORUIZhytomyr State University, Ukraine
Anatoliy SWISHCHUKUniversity of Calgary, Canada
Ramón M. RODRÍGUEZ-DAGNINOTecnologico de Monterrey, Mexico
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).
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.
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
The theory of dynamical systems is one of the fields of modern mathematics that is under intensive study. In the theory of stochastic processes, researchers are actively studying dynamical systems, operating under the influence of random factors. A good representative of such systems is the theory of random evolutions. The first results within this field were obtained by Goldstein (1951) and Kac (1974), who studied the movement of a particle on a line with a speed that changes its sign under the Poisson process. Subsequently, this process was called the telegraph process or the Goldstein–Kac process. Further developments of this theory have been presented in the works of Griego and Hersh (1969, 1971), Hersh and Pinsky (1972), and Hersh (1974, 2003), which gave a definition of stochastic evolutions in a general setting.
Important advances in the theory of stochastic evolutions have been made in the formulation of limit theorems and their refinement; these consist of obtaining asymptotic expansions. These problems are studied in the works of Korolyuk and Turbin (1993), Skorokhod (1989), Turbin (1972, 1981), Korolyuk and Swishchuk (1986), Korolyuk and Limnios (2009, 2005), Shurenkov (1989, 1986), Dorogovtsev (2007b), Anisimov (1977), Girko (1982), Hersh and Pinsky (1972), Papanicolaou (1971a,b), Kertz (1978), Watkins (1984, 1985), Balakrishnan et al. (1988), Yeleyko and Zhernovyi (2002), Pogorui (1989, 1994, 2009a) and Pogorui and Rodríguez-Dagnino (2006, 2010a).
Among the many methods in finding limiting theorems, we should mention a class that could be called an asymptotic average scheme. By using these methods, a semi-Markov evolution can be reduced to a random evolution in a lumping state space with Markov switching, and is thus studied as a Markov scheme. Most of the results in this field are presented in the following books and papers: Korolyuk and Swishchuk (1995), Korolyuk and Korolyuk (1999), Korolyuk and Turbin (1993), Korolyuk and Limnios (2005), Turbin (1972), Pogorui (2004, 2012a), and Rodríguez-Said et al. (2007). Furthermore, this theory was also developed in the study of asymptotic expansions for functionals of random evolution in the phase averaging and diffusion approximation. This topic was the main subject of the following works: Korolyuk and Limnios (2009, 2004), Turbin (1981), Samoilenko (2005), Albeverio et al. (2009), Pogorui (2010a), Pogorui and Rodríguez-Dagnino (2010a), and Nischenko (2001).
The asymptotic average scheme has been applied to a semi-Markov evolution for the computation of the effectiveness of a multiphase system with a couple of storage units by Pogorui (2003, 2004), Pogorui and Turbin (2002), and Rodríguez-Said et al. (2007).
Research on random evolutions has also been carried out by applying martingale methods. It seems that the founders of this approach were Stroock and Varadhan (1969, 1979), but further developments can be found in the works of Skorokhod (1989), Pinsky (1991), Korolyuk and Korolyuk (1999), Sviridenko (1986), Swishchuk (1989), Hersh and Papanicolaou (1972), Iksanov and Rösler (2006), and Griego and Korzeniowski (1989).
In addition to the successful development of abstract stochastic evolutions in the decade 1980–1990, several scholars studied various generalizations of the Goldstein–Kac telegraph process to multidimensional spaces. In connection to this, the results of Gorostiza (1973), Gorostiza and Griego (1979), Orsingher (1985), Orsingher and Somella (2004), Turbin (1998), Orsingher and Ratanov (2002), Samoilenko (2001) and Lachal (2006) should be noted. In most of these works, the authors considered a finite number of directions of a particle movement and obtained differential equations for the probability density functions. In the works of Pogorui (2007) and Pogorui et al. (2014), the authors proposed a method for solving such equations by using monogenic functions, after associating a particular commutative algebra with them.
Masoliver et al. (1993a) studied the telegraph process with reflecting or partially reflecting boundaries and its distribution in a fixed interval of time. In papers by Pogorui (2005, 2006), and Pogorui and Rodríguez-Dagnino (2006, 2010b), the authors also study the stationary distribution of some Markov and semi-Markov evolution with delaying boundaries.
De Gregorio et al. (2005), and Stadje and Zacks (2004) considered the generalizationof the telegraph process on a line, where there is a discrete set of particle velocities, then at Poisson epochs a velocity is chosen from this set.
Pogorui (2010b), Pogorui and Rodríguez-Dagnino (2009b) and Samoilenko (2002) investigated fading evolutions, where the velocity of a particle tends to zero as the number of switches grows at infinite.
Pogorui and Rodríguez-Dagnino (2005) also studied a generalization of the telegraph process to the case of Erlang interarrivals between successive switches of particle velocities. For such processes, a differential equation for the probability density function (pdf) of the particle position on the line was obtained. In addition, in Pogorui (2011) a method for solving such equations, by using monogenic functions associated with this equation on a commutative algebra, was developed.
Orsingher and De Gregorio (2007), Stadje (2007) and others studied the motion of a particle in multidimensional spaces with constant absolute velocity and directions uniformly distributed on a unit sphere that change at Poisson points. The authors obtained explicit formulas for the position of the particle distributions for two- and four-dimensional space and investigated the “explosive effect” for the pdf of the position of a particle that approaches the singularity sphere for the plane and the three-dimensional space.
In recent years, much attention has been paid to the Pearson random walk with Gamma distributed steps and associated walks of Pearson–Dirichlet type, whose steps have the Dirichlet distribution. Franceschetti (2007) developed explicit formulas for the conditional pdf of the position of a particle at any number of steps for a walk in (n = 1, 2) with uniformly distributed directions and steps with the Dirichlet distribution with parameter q = 1. Beghin and Orsingher (2010a) obtained an expression for the conditional distribution of the position of a particle of the Pearson–Dirichlet walk with parameter q = 2 in the plane. Le Caër (2010, 2011) generalized these results to the case of the multidimensional Pearson–Dirichlet random walk with arbitrary parameter q, where the author introduces the concept of a “Hyperspherical Uniform” (HU) random walk. The HU walk is a motion, the endpoint distribution of which is identical to the distribution of the projection in the walk space of a point, with a position vector, randomly chosen on the surface of the unit hypersphere of some hyperspace in higher dimensions. By using properties of the HU random walk, Le Caër found walks for which the conditional probability density can be expressed in a closed form. We should also mention the recent papers by De Gregorio (2014) and Letac and Piccioni (2014), where they obtained a generalization and simplification of proofs of the results stated by Le Caër.
In all of the above-mentioned papers, the authors study the conditional distributions of the particle position at renewal epochs of the switching direction process. For a non-Markov switching process, it is possible to find the corresponding results. The mathematical technique is to replace the distribution of the non-Markov process by the Markov chain walk embedded in this process. By using this approach, Pogorui and Rodríguez-Dagnino (2011) obtained a recursive expression for the conditional characteristic functions of a random walk with Erlang switching, considering a non-Markov switching process. Namely, they studied the changes in the conditional characteristic functions of the particle position, not only at instants of the direction changing, but at all Poisson times. Pogorui (2011) studied the particular case of the Erlang distributed stay of the switching process in the states. Further results in this direction were obtained by Pogorui and Rodríguez-Dagnino (2012) where the authors also studied multidimensional random motion at random velocities. For some distributions of random velocity, they observed an “explosive effect” for the pdf of the position of a particle when it is approaching the singularity sphere in four-dimensional space.
Other directions of random walk theory, which have been studied intensively during recent years, are the fractal Brownian motion and the fractal generalization of the telegraph process. These processes have been studied by Qian 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 (2008a).
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.
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