Diffusion Processes, Jump Processes, and Stochastic Differential Equations
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Diffusion Processes, Jump Processes, and Stochastic Differential Equations provides a compact exposition of the results explaining interrelations between diﬀusion stochastic processes, stochastic diﬀerential equations and the fractional inﬁnitesimal 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 ﬁnance. The last topic proved to be particularly popular among students looking for careers on Wall Street and in research organizations devoted to ﬁnancial 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.
Table of Contents
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 ɑ-stable motions. 4.6. Infinitesimal generators of ɑ-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 ɑ-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
Wojbor A. Woyczyński earned his PhD in Mathematics in 1968 from Wroclaw University, Poland. He moved to the U.S. in 1970, and since 1982, has been the Professor of Mathematics and Statistics at Case Western Reserve University in Cleveland, where he served as chairman of the department from 1982 to 1991, and from 2001 to 2002. He has held tenured faculty positions at Wroclaw University, Poland, and at Cleveland State University, and visiting appointments at Carnegie-Mellon University, and Northwestern University. He has also given invited lecture series on short-term research visits at University of North Carolina, University of South Carolina, University of Paris, Gottingen University, Aarhus University, Nagoya University, University of Tokyo, University of Minnesota, the National University of Taiwan, Taipei, Humboldt University in Berlin, Germany, and the University of New South Wales in Sydney. He is also (co)author and/or editor of more than fifteen books on probability theory, harmonic and functional analysis, and applied mathematics, and currently serves as a member of the editorial board of the Applicationes Mathematicae, Springer monograph series UTX, and as a managing editor of the journal Probability and Mathematical Statistics. His research interests include probability theory, stochastic models, functional analysis and partial differential equations and their applications in statistics, statistical physics, surface chemistry, hydrodynamics and biomedicine in which he has published about 200 research papers. He has been the advisor of more than 40 graduate students. Among other honors, in 2013 he was awarded Paris Prix la Recherche, Laureat Mathematiques, for work on mathematical evolution theory. He is currently Professor of Mathematics, Applied Mathematics and Statistics, and Director of the Case Center for Stochastic and Chaotic Processes in Science and Technology at Case Western Reserve University, in Cleveland, Ohio, U.S.A.