Markov processes, semigroups, and generators /

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Bibliographic Details
Author / Creator:Kolokolʹt͡sov, V. N. (Vasiliĭ Nikitich)
Imprint:Berlin ; New York : De Gruyter, c2011.
Description:xviii, 430 p. ; 25 cm.
Language:English
Series:De Gruyter studies in mathematics ; 38
De Gruyter studies in mathematics 38.
Subject:Markov processes.
Semigroups.
Group theory -- Generators.
Group theory -- Generators.
Markov processes.
Semigroups.
Format: Print Book
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/8363406
Hidden Bibliographic Details
ISBN:9783110250107 (alk. paper)
3110250101 (alk. paper)
Notes:Includes bibliographical references and index.
Table of Contents:
  • Preface
  • Notations
  • Standard abbreviations
  • I. Introduction to stochastic analysis
  • 1. Tools from probability and analysis
  • 1.1. Essentials of measure and probability
  • 1.2. Characteristic functions
  • 1.3. Conditioning
  • 1.4. Infinitely divisible and stable distributions
  • 1.5. Stable laws as the Holtzmark distributions
  • 1.6. Unimodality of probability laws
  • 1.7. Compactness for function spaces and measures
  • 1.8. Fractional derivatives and pseudo-differential operators
  • 1.9. Propagators and semigroups
  • 2. Brownian motion (BM)
  • 2.1. Random processes: basic notions
  • 2.2. Definition and basic properties of BM
  • 2.3. Construction via broken-line approximation
  • 2.4. Construction via Hilbert-space methods
  • 2.5. Construction via Kolmogorov's continuity
  • 2.6. Construction via random walks and tightness
  • 2.7. Simplest applications of martingales
  • 2.8. Skorohod embedding and the invariance principle
  • 2.9. More advanced Hilbert space methods: Wiener chaos and stochastic integral
  • 2.10. Fock spaces, Hermite polynomials and Malliavin calculus
  • 2.11. Stationarity: OU processes and Holtzmark fields
  • 3. Markov processes and martingales
  • 3.1. Definition of Lévy processes
  • 3.2. Poisson processes and integrals
  • 3.3. Construction of Lévy processes
  • 3.4. Subordinators
  • 3.5. Markov processes, semigroups and propagators
  • 3.6. Feller processes and conditionally positive operators
  • 3.7. Diffusions and jump-type Markov processes
  • 3.8. Markov processes on quotient spaces and reflections
  • 3.9. Martingales
  • 3.10. Stopping times and optional sampling
  • 3.11. Strong Markov property; diffusions as Feller processes with continuous paths
  • 3.12. Reflection principle and passage times
  • 4. SDE, ¿DE and martingale problems
  • 4.1. Markov semigroups and evolution equations
  • 4.2. The Dirichlet problem for diffusion operators
  • 4.3. The stationary Feynman-Kac formula
  • 4.4. Diffusions with variable drift, Ornstein-Uhlenbeck processes
  • 4.5. Stochastic integrals and SDE based on Lévy processes
  • 4.6. Markov property and regularity of solutions
  • 4.7. Stochastic integrals and quadratic variation for square-integrable martingales
  • 4.8. Convergence of processes and semigroups
  • 4.9. Weak convergence of martingales
  • 4.10. Martingale problems and Markov processes
  • 4.11. Stopping and localization
  • II. Markov processes and beyond
  • 5. Processes in Euclidean spaces
  • 5.1. Direct analysis of regularity and well-posedness
  • 5.2. Introduction to sensitivity analysis
  • 5.3. The Lie-Trotter type limits and T-products
  • 5.4. Martingale problems for Lévy type generators: existence
  • 5.5. Martingale problems for Lévy type generators: moments
  • 5.6. Martingale problems for Lévy type generators: unbounded coefficients
  • 5.7. Decomposable generators
  • 5.8. Sdes driven by nonlinear Lévy noise
  • 5.9. Stochastic monotonicity and duality
  • 5.10. Stochastic scattering
  • 5.11. Nonlinear Markov chains, interacting particles and deterministic processes
  • 5.12. Comments
  • 6. Processes in domains with a boundary
  • 6.1. Stopped processes and boundary points
  • 6.2. Dirichlet problem and mixed initial-boundary problem
  • 6.3. The method of Lyapunov functions
  • 6.4. Local criteria for boundary points
  • 6.5. Decomposable generators in R + d
  • 6.6. Gluing boundary
  • 6.7. Processes on the half-line
  • 6.8. Generators of reflected processes
  • 6.9. Application to interacting particles: stochastic LLN
  • 6.10. Application to evolutionary games
  • 6.11. Application to finances: barrier options, credit derivatives, etc.
  • 6.12. Comments
  • 7. Heat kernels for stable-like processes
  • 7.1. One-dimensional stable laws: asymptotic expansions
  • 7.2. Stable laws: asymptotic expansions and identities
  • 7.3. Stable laws: bounds
  • 7.4. Stable laws: auxiliary convolution estimates
  • 7.5. Stable-like processes: heat kernel estimates
  • 7.6. Stable-like processes: Feller property
  • 7.7. Application to sample-path properties
  • 7.8. Application to stochastic control
  • 7.9. Application to Langevin equations driven by a stable noise
  • 7.10. Comments
  • 8. CTRW and fractional dynamics
  • 8.1. Convergence of Markov semigroups and processes
  • 8.2. Diffusive approximations for random walks and CLT
  • 8.3. Stable-like limits for position-dependent random walks
  • 8.4. Subordination by hitting times and generalized fractional evolutions
  • 8.5. Limit theorems for position dependent CTRW
  • 8.6. Comments
  • 9. Complex Markov chains and Feynman integral
  • 9.1. Infinitely-divisible complex distributions and complex Markov chains
  • 9.2. Path integral and perturbation theory
  • 9.3. Extensions
  • 9.4. Regularization of the Schrödinger equation by complex time or mass, or continuous observation
  • 9.5. Singular and growing potentials, magnetic fields and curvilinear state spaces
  • 9.6. Fock-space representation
  • 9.7. Comments
  • Bibliography
  • Index