Diffusion processes, jump processes, and stochastic differential equations
著者
書誌事項
Diffusion processes, jump processes, and stochastic differential equations
（A Chapman & Hall book）
CRC Press, Taylor & Francis Group, 2022
- : hbk
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注記
Bibliography: p. 119-121
Includes index
内容説明・目次
内容説明
Features
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.
目次
- 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. Ito's stochastic integral for Brownian motion. 6.3. An instructive example. 6.4. Ito's formula. 6.5. Martingale property of Ito integrals. 6.6. Wiener and Ito-type stochastic integrals for -stable motion and general Levy processes. 6.7. Problems and Exercises. 7. Ito 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 Ito'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
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