Preface -- Notations -- Standard abbreviations -- Contents -- 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, ΨDE 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 RdC -- 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.