Markov Processes, Semigroups and Generators

by Kolokoltsov, Vassili N.

• ISBN13: 9783110250107
• ISBN10: 3110250101
• Format: Hardcover
• Copyright: 2011-03-17
• Publisher: De Gruyter

Summary

This work offers a highly useful, well developed reference on Markov processes, the universal model for random processes and evolutions. The wide range of applications, in exact sciences as well as in other areas like social studies, require a volume that offers a refresher on fundamentals before conveying the Markov processes and examples for applications. This work does just that, and with the necessary mathematical rigor.

Table of Contents

Preface	p. vii
Notations	p. xi
Standard abbreviations	p. xiv
Introduction to stochastic analysis	p. 1
Tools from probability and analysis	p. 3
Essentials of measure and probability	p. 3
Characteristic functions	p. 13
Conditioning	p. 16
Infinitely divisible and stable distributions	p. 21
Stable laws as the Holtzmark distributions	p. 27
Unimodality of probability laws	p. 29
Compactness for function spaces and measures	p. 35
Fractional derivatives and pseudo-differential operators	p. 42
Propagators and semigroups	p. 48
Brownian motion (BM)	p. 58
Random processes: basic notions	p. 58
Definition and basic properties of BM	p. 62
Construction via broken-line approximation	p. 66
Construction via Hilbert-space methods	p. 69
Construction via Kolmogorov's continuity	p. 71
Construction via random walks and tightness	p. 73
Simplest applications of martingales	p. 76
Skorohod embedding and the invariance principle	p. 78
More advanced Hilbert space methods: Wiener chaos and stochastic integral	p. 81
Fock spaces, Hermite polynomials and Malliavin calculus	p. 87
Stationarity: OU processes and Holtzmark fields	p. 91
Markov processes and martingales	p. 94
Definition of Lévy processes	p. 94
Poisson processes and integrals	p. 96
Construction of Lévy processes	p. 103
Subordinators	p. 108
Markov processes, semigroups and propagators	p. 110
Feller processes and conditionally positive operators	p. 115
Diffusions and jump-type Markov processes	p. 125
Markov processes on quotient spaces and reflections	p. 130
Martingales	p. 132
Stopping times and optional sampling	p. 138
Strong Markov property; diffusions as Feller processes with continuous paths	p. 143
Reflection principle and passage times	p. 147
SDE, ¿DE and martingale problems	p. 152
Markov semigroups and evolution equations	p. 152
The Dirichlet problem for diffusion operators	p. 158
The stationary Feynman-Kac formula	p. 162
Diffusions with variable drift, Ornstein-Uhlenbeck processes	p. 165
Stochastic integrals and SDE based on Lévy processes	p. 167
Markov property and regularity of solutions	p. 172
Stochastic integrals and quadratic variation for square-integrable martingales	p. 178
Convergence of processes and semigroups	p. 187
Weak convergence of martingales	p. 193
Martingale problems and Markov processes	p. 195
Stopping and localization	p. 199
Markov processes and beyond	p. 203
Processes in Euclidean spaces	p. 204
Direct analysis of regularity and well-posedness	p. 205
Introduction to sensitivity analysis	p. 213
The Lie-Trotter type limits and T-products	p. 213
Martingale problems for Lévy type generators: existence	p. 221
Martingale problems for Lévy type generators: moments	p. 226
Martingale problems for Lévy type generators: unbounded coefficients	p. 228
Decomposable generators	p. 231
Sdes driven by nonlinear Lévy noise	p. 240
Stochastic monotonicity and duality	p. 250
Stochastic scattering	p. 255
Nonlinear Markov chains, interacting particles and deterministic processes	p. 257
Comments	p. 262
Processes in domains with a boundary	p. 270
Stopped processes and boundary points	p. 270
Dirichlet problem and mixed initial-boundary problem	p. 274
The method of Lyapunov functions	p. 280
Local criteria for boundary points	p. 282
Decomposable generators in R+d	p. 286
Gluing boundary	p. 290
Processes on the half-line	p. 292
Generators of reflected processes	p. 293
Application to interacting particles: stochastic LLN	p. 295
Application to evolutionary games	p. 304
Application to finances: barrier options, credit derivatives, etc.	p. 307
Comments	p. 308
Heat kernels for stable-like processes	p. 310
One-dimensional stable laws: asymptotic expansions	p. 310
Stable laws: asymptotic expansions and identities	p. 314
Stable laws: bounds	p. 319
Stable laws: auxiliary convolution estimates	p. 323
Stable-like processes: heat kernel estimates	p. 328
Stable-like processes: Feller property	p. 335
Application to sample-path properties	p. 336
Application to stochastic control	p. 340
Application to Langevin equations driven by a stable noise	p. 345
Comments	p. 348
CTRW and fractional dynamics	p. 351
Convergence of Markov semigroups and processes	p. 351
Diffusive approximations for random walks and CLT	p. 354
Stable-like limits for position-dependent random walks	p. 355
Subordination by hitting times and generalized fractional evolutions	p. 361
Limit theorems for position dependent CTRW	p. 369
Comments	p. 371
Complex Markov chains and Feynman integral	p. 373
Infinitely-divisible complex distributions and complex Markov chains	p. 373
Path integral and perturbation theory	p. 380
Extensions	p. 385
Regularization of the Schrödinger equation by complex time or mass, or continuous observation	p. 390
Singular and growing potentials, magnetic fields and curvilinear state spaces	p. 393
Fock-space representation	p. 398
Comments	p. 400
Bibliography	p. 403
Index	p. 425
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