Diffusion Processes, Jump Processes, and Stochastic Differential Equations

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Chapter 1 Random Variables, Vectors, Processes, and Fields
1.1 RANDOM VARIABLES, VECTORS, AND THEIR DISTRIBUTIONS—A GLOSSARY
1.1.1 Basic Concepts
1.1.2 Absolutely Continuous, Discrete, Mixed, and Singular Probability Distributions
1.1.3 Characteristic Functions,Laplace Transforms, and Moment- Generating Functions
1.1.4 Examples
1.2 LAW OF LARGE NUMBERS AND THE CENTRAL LIMIT THEOREM
1.3 STOCHASTIC PROCESSES AND THEIR FINITE-DIMENSIONAL DISTRIBUTIONS
1.4 PROBLEMS AND EXERCISES
Chapter 2 From Random Walk to Brownian Motion
2.1 SYMMETRIC RANDOM WALK; PARABOLIC RESCALING AND RELATED FOKKER-PLANCK EQUATIONS
2.1.1 Brownian Motion as Hydrodynamic Limit of Random Walks
2.1.2 Brownian Motion via the Central Limit Theorem and the Invariance Principle
2.2 BASIC PROPERTIES OF BROWNIAN MOTION
2.3 ALMOST SURE CONTINUITY OF SAMPLE PATHS
2.4 NOWHERE DIFFERENTIABILITY OF BROWNIAN MOTION
2.5 HITTINGTIMES, AND OTHER SUBTLE PROPERTIES OF BROWNIAN MOTION
2.6 PROBLEMS AND EXERCISES
Chapter 3 Poisson Processes and Their mixtures
3.1 WHY POISSON PROCESS?
3.2 COVARIANCE STRUCTURE AND FINITE DIMENSIONAL DISTRIBUTIONS
3.3 WAITING TIMES AND INTER-JUMP TIMES
3.4 EXTENSIONS AND GENERALIZATIONS
3.5 FRACTIONAL POISSON PROCESSES (fP[sub(p)])
3.5.1 FP[sub(p)] Interarrival Time
3.6 PROBLEMS AND EXERCISES
Chapter 4 Lévy Processes and the Lévy-Khinchine Formula: Basic Facts
4.1 PROCESSES WITH STATIONARY AND INDEPENDENT INCREMENTS
4.2 FROM POISSON PROCESSES TO LÈVY PROCESSES
4.3 INFINITESIMAL GENERATORS OF LÈVY PROCESSES
4.4 SELF-SIMILAR LÈVY PROCESSES
4.5 PROPERTIES OF α-STABLE MOTIONS
4.6 INFINITESIMAL GENERATORS OF α-STABLE MOTIONS
4.7 PROBLEMS AND EXERCISES
Chapter 5 General Processes with Independent Increments
5.1 NONSTATIONARY PROCESSES WITH INDEPENDENT INCREMENTS
5.2 STOCHASTIC CONTINUITY AND JUMP PROCESSES
5.3 ANALYSIS OF JUMP STRUCTURE
5.4 RANDOM MEASURES AND RANDOM INTEGRALS ASSOCIATED WITH JUMP PROCESSES
5.4.1 Random Measures and Random Integrals
5.5 STRUCTURE OF GENERAL I.I. PROCESSES
Chapter 6 Stochastic Integrals for Brownian Motion and General Levy Processes
6.1 WIENER RANDOM INTEGRAL
6.2 ITO’S STOCHASTIC INTEGRAL FOR BROWNIAN MOTION
6.3 AN INSTRUCTIVE EXAMPLE
6.4 ITO’S FORMULA
6.5 MARTINGALE PROPERTY OF ITÔ INTEGRALS
6.6 WIENER AND ITÔ-TYPE STOCHASTIC INTEGRALS FOR α -STABLE ´ MOTION AND GENERAL LÈVY PROCESSES
Chapter 7 Itô Stochastic Differential Equations
7.1 DIFFERENTIAL EQUATIONS WITH NOISE
7.2 STOCHASTIC DIFFERENTIAL EQUATIONS: BASIC THEORY
7.3 SDEs WITH COEFFICIENTS DEPENDING ONLY ON TIME
7.4 POPULATION GROWTH MODEL AND OTHER EXAMPLES
7.4.1 Population Growth Model
7.4.2 Ornstein–Uhlenbeck Process
7.5 SYSTEMS OF SDEs AND VECTOR-VALUED ITÔS FORMULA
7.6 NUMERICAL SOLUTION OF STOCHASTIC DIFFERENTIAL EQUATIONS
Chapter 8 Asymmetric Exclusion Processes and Their Scaling Limits
8.1 ASYMMETRIC EXCLUSION PRINCIPLES
8.2 SCALING LIMIT
8.3 OTHER QUEUING REGIMES RELATED TO NON-NEAREST NEIGHBOR SYSTEMS
8.4 NETWORKS WITH MULTISERVER NODES AND PARTICLE SYSTEMS WITH STATE-DEPENDENT RATES
8.5 SHOCK AND RAREFACTION WAVE SOLUTIONS FOR THE RIEMANN PROBLEM FOR CONSERVATION LAWS
Chapter 9 Nonlinear Diffusion Equations
9.1 HYPERBOLIC EQUATIONS
9.2 NONLINEAR DIFFUSION APPROXIMATIONS
9.3 NONLINEAR PROCESSES
9.4 INTERACTING DIFFUSIONS AND MONTE-CARLO METHODS
Appendix A The Remarkable Bernoulli Family
Bibliography
Index