1. Random variables, vectors, processes and fields. 1.1. Random variables, vectors, and their distributions – a glossary. 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. 2. From Random Walk to Brownian Motion. 2.1. Symmetric random walk; parabolic rescaling and related Fokker-Planck equations. 2.2 Almost sure continuity of sample paths. 2.3 Nowhere differentiability of Brownian motion. 2.4 Hitting times, and other subtle properties of Brownian motion. 2.5. Problems and Exercises. 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 (fPp). 3.6. Problems and Exercises. 4. Levy processes and the Levy-Khinchine formula: basic facts. 4.1. Processes with stationary and independent increments. 4.2. From Poisson processes to Levy processes. 4.3. Infinitesimal generators of Levy processes. 4.4. Selfsimilar Levy processes. 4.5. Properties of ?-stable motions. 4.6. Infinitesimal generators of ?-stable motions. 4.7. Problems and Exercises. 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. 5.5. Structure of general I.I. processes. 5.6. Problems and Exercises. 6. Stochastic integrals for Brownian motion and general Levy Processes. 6.1. Wiener random integral. 6.2. Itô's stochastic integral for Brownian motion. 6.3. An instructive example. 6.4. Itô's formula. 6.5. Martingale property of Itô integrals. 6.6. Wiener and Itô-type stochastic integrals for ?-stable motion and general Levy processes. 6.7. Problems and Exercises. 7. Itô stochastic differential equations. 7.1. Differential equations with random 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. 7.5. Ornstein-Uhlenbeck process. 7.6. Systems of SDEs and vector-valued Itô's formula. 7.7. Kalman-Bucy filter. 7.8. Numerical solution of stochastic differential equations. 7.9. Problems and Exercises. 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. 8.6. Problems and Exercises. 9. Nonlinear diffusion equations. 9.1. Hyperbolic equations. 9.2. Nonlinear diffusion approximations. 9.3. Problems and Exercises
Diffusion Processes, Jump Processes, and Stochastic Differential Equations provides a compact exposition of the results explaining interrelations between di?usion stochastic processes, stochastic di?erential equations and the fractional in?nitesimal operators. The draft of this book has been extensively classroom tested by the author at Case Western Reserve University in a course that enrolled seniors and graduate students majoring in mathematics, statistics, engineering, physics, chemistry, economics and mathematical ?nance. The last topic proved to be particularly popular among students looking for careers on Wall Street and in research organizations devoted to ?nancial problems.
• Quickly and concisely builds from basic probability theory to advanced topics
• Suitable as a primary text for an advanced course in diffusion processes and stochastic differential equations
• Useful as supplementary reading across a range of topics.