Anatoliy Pogorui�s main research interests include probability, stochastic processes, mathematical modeling of an ideal gas using multi-dimensional random motions and the interaction of telegraph particles in semi-Markov environments and the application of random evolutions in the reliability theory of storage systems. Anatoliy Swishchuk is Professor of mathematical finance at the University of Calgary, Canada. His research areas include financial mathematics, random evolutions and their applications, stochastic calculus and biomathematics. Ramón M. Rodríguez-Dagnino has investigated applied probability aimed at modeling systems with stochastic behavior, random motions in wireless networks, video trace modeling and prediction, information source characterization, performance analysis of networks with heavytail traffic, generalized Gaussian estimation and spectral analysis.

• Preface ixAcknowledgments xiiiIntroduction xvPart 1. Higher-dimensional Random Motions and Interactive Particles 1Chapter 1. Random Motions in Higher Dimensions 31.1. Random motion at finite speed with semi-Markov switching directions process 51.1.1. Erlang-K-distributed direction alternations 71.1.2. Some properties of the random walk in a semi-Markov environment and its characteristic function 151.2. Random motion with uniformly distributed directions and random velocity 171.2.1. Renewal equation for the characteristic function of isotropic motion with random velocity in a semi-Markov media 171.2.2. One-dimensional case 201.2.3. Two-dimensional case 231.2.4. Three-dimensional case 231.2.5. Four-dimensional case 311.3. The distribution of random motion at non-constant velocity in semi-Markov media 321.3.1. Renewal equation for the characteristic function 341.3.2. Two-dimensional case 351.3.3. Three-dimensional case 371.3.4. Four-dimensional case 401.4. Goldstein–Kac telegraph equations and random flights in higher dimensions 431.4.1. Preliminaries about our modeling approach 451.4.2. Two-dimensional case 481.4.3. Three-dimensional case 511.4.4. Five-dimensional case 591.5. The jump telegraph process in Rn 621.5.1. The jump telegraph process in R3 631.5.2. Conclusions and final remarks 64Chapter 2. System of Interactive Particles with Markov and Semi-Markov Switching 672.1. Description of the Markov model 682.1.1. Distribution of the first meeting time of two telegraph processes 692.1.2. Estimate of the number of particle collisions 742.1.3. Free path times of a family of particles 762.1.4. Estimation of the number of particle collisions for systems with boundaries 782.1.5. Estimation of the number of particle collisions for systems without boundaries 832.2. Interaction of particles governed by generalized integrated telegraph processes: a semi-Markov case 872.2.1. Laplace transform of the distribution of the first collision of two particles 882.2.2. Semi-Markov case 912.2.3. Distribution of the first collision of two particles with finite expectation 95Part 2. Financial Applications 99Chapter 3. Asymptotic Estimation for Application of the Telegraph Process as an Alternative to the Diffusion Process in the Black–Scholes Formula 1013.1. Asymptotic expansion for the singularly perturbed random evolution in Markov media in the case of disbalance 1013.2. Application: Black–Scholes formula 106Chapter 4. Variance, Volatility, Covariance and Correlation Swaps for Financial Markets with Markov-modulated Volatilities 1114.1. Volatility derivatives 1114.1.1. Types of volatilities 1114.1.2. Models for volatilities 1134.1.3. Variance and volatility swaps 1154.1.4. Covariance and correlation swaps 1164.1.5. A brief literature review 1174.2. Martingale representation of a Markov process 1184.3. Variance and volatility swaps for financial markets with Markov-modulated stochastic volatilities 1224.3.1. Pricing variance swaps 1244.3.2. Pricing volatility swaps 1244.4. Covariance and correlation swaps for two risky assets for financial markets with Markov-modulated stochastic volatilities 1284.4.1. Pricing covariance swaps 1284.4.2. Pricing correlation swaps 1304.4.3. Correlation swap made simple 1304.5. Example: variance, volatility, covariance and correlation swaps for  stochastic volatility driven by two state continuous Markov chain 1324.6. Numerical example 1344.6.1. S&P 500: variance and volatility swaps 1344.6.2. S&P 500 and NASDAQ-100: covariance and correlation swaps 1354.7. Appendix 1 1384.7.1. Correlation swaps: first-order correction 138Chapter 5. Modeling and Pricing of Variance, Volatility, Covariance and Correlation Swaps for Financial Markets with Semi-Markov Volatilities 1435.1. Introduction 1435.2. Martingale representation of semi-Markov processes 1485.3. Variance and volatility swaps for financial markets with semi-Markov stochastic volatilities 1515.3.1. Pricing of variance swaps 1535.3.2. Pricing of volatility swaps 1555.3.3. Numerical evaluation of variance and volatility swaps with semi-Markov volatility 1585.4. Covariance and correlation swaps for two risky assets in financial markets with semi-Markov stochastic volatilities 1595.4.1. Pricing of covariance swaps 1605.4.2. Pricing of correlation swaps 1625.5. Numerical evaluation of covariance and correlation swaps with semi-Markov stochastic volatility 1645.6. Appendices 1655.6.1. Appendix 1. Realized correlation: first-order correction 1655.6.2. Appendix 2. Discussions of some extensions 169References 177Index 191Summary of Volume 1 193

Anatoliy Pogorui�s main research interests include probability, stochastic processes, mathematical modeling of an ideal gas using multi-dimensional random motions and the interaction of telegraph particles in semi-Markov environments and the application of random evolutions in the reliability theory of storage systems. Anatoliy Swishchuk is Professor of mathematical finance at the University of Calgary, Canada. His research areas include financial mathematics, random evolutions and their applications, stochastic calculus and biomathematics. Ramón M. Rodríguez-Dagnino has investigated applied probability aimed at modeling systems with stochastic behavior, random motions in wireless networks, video trace modeling and prediction, information source characterization, performance analysis of networks with heavytail traffic, generalized Gaussian estimation and spectral analysis.

• Preface ixAcknowledgments xiiiIntroduction xvPart 1. Higher-dimensional Random Motions and Interactive Particles 1Chapter 1. Random Motions in Higher Dimensions 31.1. Random motion at finite speed with semi-Markov switching directions process 51.1.1. Erlang-K-distributed direction alternations 71.1.2. Some properties of the random walk in a semi-Markov environment and its characteristic function 151.2. Random motion with uniformly distributed directions and random velocity 171.2.1. Renewal equation for the characteristic function of isotropic motion with random velocity in a semi-Markov media 171.2.2. One-dimensional case 201.2.3. Two-dimensional case 231.2.4. Three-dimensional case 231.2.5. Four-dimensional case 311.3. The distribution of random motion at non-constant velocity in semi-Markov media 321.3.1. Renewal equation for the characteristic function 341.3.2. Two-dimensional case 351.3.3. Three-dimensional case 371.3.4. Four-dimensional case 401.4. Goldstein–Kac telegraph equations and random flights in higher dimensions 431.4.1. Preliminaries about our modeling approach 451.4.2. Two-dimensional case 481.4.3. Three-dimensional case 511.4.4. Five-dimensional case 591.5. The jump telegraph process in Rn 621.5.1. The jump telegraph process in R3 631.5.2. Conclusions and final remarks 64Chapter 2. System of Interactive Particles with Markov and Semi-Markov Switching 672.1. Description of the Markov model 682.1.1. Distribution of the first meeting time of two telegraph processes 692.1.2. Estimate of the number of particle collisions 742.1.3. Free path times of a family of particles 762.1.4. Estimation of the number of particle collisions for systems with boundaries 782.1.5. Estimation of the number of particle collisions for systems without boundaries 832.2. Interaction of particles governed by generalized integrated telegraph processes: a semi-Markov case 872.2.1. Laplace transform of the distribution of the first collision of two particles 882.2.2. Semi-Markov case 912.2.3. Distribution of the first collision of two particles with finite expectation 95Part 2. Financial Applications 99Chapter 3. Asymptotic Estimation for Application of the Telegraph Process as an Alternative to the Diffusion Process in the Black–Scholes Formula 1013.1. Asymptotic expansion for the singularly perturbed random evolution in Markov media in the case of disbalance 1013.2. Application: Black–Scholes formula 106Chapter 4. Variance, Volatility, Covariance and Correlation Swaps for Financial Markets with Markov-modulated Volatilities 1114.1. Volatility derivatives 1114.1.1. Types of volatilities 1114.1.2. Models for volatilities 1134.1.3. Variance and volatility swaps 1154.1.4. Covariance and correlation swaps 1164.1.5. A brief literature review 1174.2. Martingale representation of a Markov process 1184.3. Variance and volatility swaps for financial markets with Markov-modulated stochastic volatilities 1224.3.1. Pricing variance swaps 1244.3.2. Pricing volatility swaps 1244.4. Covariance and correlation swaps for two risky assets for financial markets with Markov-modulated stochastic volatilities 1284.4.1. Pricing covariance swaps 1284.4.2. Pricing correlation swaps 1304.4.3. Correlation swap made simple 1304.5. Example: variance, volatility, covariance and correlation swaps for  stochastic volatility driven by two state continuous Markov chain 1324.6. Numerical example 1344.6.1. S&P 500: variance and volatility swaps 1344.6.2. S&P 500 and NASDAQ-100: covariance and correlation swaps 1354.7. Appendix 1 1384.7.1. Correlation swaps: first-order correction 138Chapter 5. Modeling and Pricing of Variance, Volatility, Covariance and Correlation Swaps for Financial Markets with Semi-Markov Volatilities 1435.1. Introduction 1435.2. Martingale representation of semi-Markov processes 1485.3. Variance and volatility swaps for financial markets with semi-Markov stochastic volatilities 1515.3.1. Pricing of variance swaps 1535.3.2. Pricing of volatility swaps 1555.3.3. Numerical evaluation of variance and volatility swaps with semi-Markov volatility 1585.4. Covariance and correlation swaps for two risky assets in financial markets with semi-Markov stochastic volatilities 1595.4.1. Pricing of covariance swaps 1605.4.2. Pricing of correlation swaps 1625.5. Numerical evaluation of covariance and correlation swaps with semi-Markov stochastic volatility 1645.6. Appendices 1655.6.1. Appendix 1. Realized correlation: first-order correction 1655.6.2. Appendix 2. Discussions of some extensions 169References 177Index 191Summary of Volume 1 193

Anatoliy Pogorui�s main research interests include probability, stochastic processes, mathematical modeling of an ideal gas using multi-dimensional random motions and the interaction of telegraph particles in semi-Markov environments and the application of random evolutions in the reliability theory of storage systems. Anatoliy Swishchuk is Professor of mathematical finance at the University of Calgary, Canada. His research areas include financial mathematics, random evolutions and their applications, stochastic calculus and biomathematics. Ramón M. Rodríguez-Dagnino has investigated applied probability aimed at modeling systems with stochastic behavior, random motions in wireless networks, video trace modeling and prediction, information source characterization, performance analysis of networks with heavytail traffic, generalized Gaussian estimation and spectral analysis.

• Preface ixAcknowledgments xiiiIntroduction xvPart 1. Higher-dimensional Random Motions and Interactive Particles 1Chapter 1. Random Motions in Higher Dimensions 31.1. Random motion at finite speed with semi-Markov switching directions process 51.1.1. Erlang-K-distributed direction alternations 71.1.2. Some properties of the random walk in a semi-Markov environment and its characteristic function 151.2. Random motion with uniformly distributed directions and random velocity 171.2.1. Renewal equation for the characteristic function of isotropic motion with random velocity in a semi-Markov media 171.2.2. One-dimensional case 201.2.3. Two-dimensional case 231.2.4. Three-dimensional case 231.2.5. Four-dimensional case 311.3. The distribution of random motion at non-constant velocity in semi-Markov media 321.3.1. Renewal equation for the characteristic function 341.3.2. Two-dimensional case 351.3.3. Three-dimensional case 371.3.4. Four-dimensional case 401.4. Goldstein–Kac telegraph equations and random flights in higher dimensions 431.4.1. Preliminaries about our modeling approach 451.4.2. Two-dimensional case 481.4.3. Three-dimensional case 511.4.4. Five-dimensional case 591.5. The jump telegraph process in Rn 621.5.1. The jump telegraph process in R3 631.5.2. Conclusions and final remarks 64Chapter 2. System of Interactive Particles with Markov and Semi-Markov Switching 672.1. Description of the Markov model 682.1.1. Distribution of the first meeting time of two telegraph processes 692.1.2. Estimate of the number of particle collisions 742.1.3. Free path times of a family of particles 762.1.4. Estimation of the number of particle collisions for systems with boundaries 782.1.5. Estimation of the number of particle collisions for systems without boundaries 832.2. Interaction of particles governed by generalized integrated telegraph processes: a semi-Markov case 872.2.1. Laplace transform of the distribution of the first collision of two particles 882.2.2. Semi-Markov case 912.2.3. Distribution of the first collision of two particles with finite expectation 95Part 2. Financial Applications 99Chapter 3. Asymptotic Estimation for Application of the Telegraph Process as an Alternative to the Diffusion Process in the Black–Scholes Formula 1013.1. Asymptotic expansion for the singularly perturbed random evolution in Markov media in the case of disbalance 1013.2. Application: Black–Scholes formula 106Chapter 4. Variance, Volatility, Covariance and Correlation Swaps for Financial Markets with Markov-modulated Volatilities 1114.1. Volatility derivatives 1114.1.1. Types of volatilities 1114.1.2. Models for volatilities 1134.1.3. Variance and volatility swaps 1154.1.4. Covariance and correlation swaps 1164.1.5. A brief literature review 1174.2. Martingale representation of a Markov process 1184.3. Variance and volatility swaps for financial markets with Markov-modulated stochastic volatilities 1224.3.1. Pricing variance swaps 1244.3.2. Pricing volatility swaps 1244.4. Covariance and correlation swaps for two risky assets for financial markets with Markov-modulated stochastic volatilities 1284.4.1. Pricing covariance swaps 1284.4.2. Pricing correlation swaps 1304.4.3. Correlation swap made simple 1304.5. Example: variance, volatility, covariance and correlation swaps for  stochastic volatility driven by two state continuous Markov chain 1324.6. Numerical example 1344.6.1. S&P 500: variance and volatility swaps 1344.6.2. S&P 500 and NASDAQ-100: covariance and correlation swaps 1354.7. Appendix 1 1384.7.1. Correlation swaps: first-order correction 138Chapter 5. Modeling and Pricing of Variance, Volatility, Covariance and Correlation Swaps for Financial Markets with Semi-Markov Volatilities 1435.1. Introduction 1435.2. Martingale representation of semi-Markov processes 1485.3. Variance and volatility swaps for financial markets with semi-Markov stochastic volatilities 1515.3.1. Pricing of variance swaps 1535.3.2. Pricing of volatility swaps 1555.3.3. Numerical evaluation of variance and volatility swaps with semi-Markov volatility 1585.4. Covariance and correlation swaps for two risky assets in financial markets with semi-Markov stochastic volatilities 1595.4.1. Pricing of covariance swaps 1605.4.2. Pricing of correlation swaps 1625.5. Numerical evaluation of covariance and correlation swaps with semi-Markov stochastic volatility 1645.6. Appendices 1655.6.1. Appendix 1. Realized correlation: first-order correction 1655.6.2. Appendix 2. Discussions of some extensions 169References 177Index 191Summary of Volume 1 193