Intro
Preface to the Second Edition
Preface to the First Edition
Contents
About the Authors
1 Preliminaries
1.1 Hypergeometric Functions
1.2 Modified Bessel Functions
1.3 Generalised Functions and Integral Transforms
1.4 One-Dimensional Markov Processes
1.5 Brownian Motion and Diffusion on ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper R) /StPNE pdfmark [/StBMC pdfmarkRps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
1.6 Stochastic Integrals and Itô's Formula
1.7 Poisson Process, Exponential, and HypoexponentialDistributions

2 Symmetric Telegraph Process on the Line
2.1 Definition of Process and the Structure of Distribution
2.2 Kolmogorov Equations
2.3 Telegraph Equation
2.4 Characteristic Function
2.5 Transition Density
2.6 Probability Distribution Function
2.7 Convergence to Brownian Motion
2.8 Laplace Transforms
Notes
3 Asymmetric Jump-Telegraph Processes
3.1 Asymmetric Continuous Telegraph Process
3.1.1 Generator and Transition Densities
3.1.2 Moment Generating Functions
3.1.3 Moments
3.2 Self-Exciting Piecewise Linear Continuous Processes

3.2.1 Piecewise Constant Process
3.2.2 Piecewise Linear Process
3.2.3 First Passage Time
3.3 Telegraph Processes with Alternating Deterministic Jumps
3.3.1 Transition Densities
3.3.2 Expectations and Variances, Jump-TelegraphMartingales
3.3.3 Change of Measure for Jump-Telegraph Processes
3.4 Telegraph Processes with Alternating Random Jumps
3.5 Short Memory Telegraph Processes with Random Jumps
3.5.1 Jump-Telegraph Processes with Parameters Depending on the Past
3.5.2 Martingales
3.6 Piecewise Linear Process with Renewal Starting Points

3.6.1 Distributions of X(t)
3.6.2 First Passage Time
3.6.2.1 Positive Velocities
3.6.2.2 Velocities of Opposite Signs
3.7 Double Telegraph Processes
3.7.1 Doubly Stochastic Poisson Process
3.7.2 Doubly Stochastic Telegraph Process with Jumps
3.7.3 Girsanov Transformation
3.8 Jump-Telegraph Processes with Poisson-Modulated Exponential Switching Times
3.8.1 Building a Model
3.8.2 Poisson-Modulated Exponential Distribution
3.8.3 Example: Poisson-Modulated Exponential Distributions with Linearly Increasing Switching Intensities

3.8.4 Piecewise Linear Process with Two Alternating Poisson Modulated Patterns and a Double Jump Component
3.9 Piecewise Deterministic Processes Following Two Alternating Patterns
3.9.1 Piecewise Linear Processes in the Linear NormedSpace
3.9.2 Time-Homogeneous Piecewise Deterministic Process
3.9.3 Examples
3.9.3.1 Squared Telegraph Process
3.9.3.2 A Random Motion in a Plane and Polar Coordinates
3.9.4 Self-Similarity
4 Jump-Diffusion Processes with Regime Switching
4.1 The Basic Model: Markov Modulated Jump-DiffusionProcesses