This book is an introduction to probability theory covering laws of large numbers, central limit theorems, random walks, martingales, Markov chains, ergodic theorems, and Brownian motion. It is a comprehensive treatment concentrating on the results that are the most useful for applications. Its philosophy is that the best way to learn probability is to see it in action, so there are 200 examples and 450 problems.

Probability: Theory and Examples

* Measure Theory-Basic Notions * Measure Theory-Key Results * Processes, Distributions, and Independence * Random Sequences, Series, and Averages * Characteristic Functions and Classical Limit Theorems * Conditioning and Disintegration * Martingales and Optional Times * Markov Processes and Discrete-Time Chains * Random Walks and Renewal Theory * Stationary Processes and Ergodic Theory * Special Notions of Symmetry and Invariance * Poisson and Pure Jump-Type Markov Processes * Gaussian Processes and Brownian Motion * Skorohod Embedding and Invariance Principles * Independent Increments and Infinite Divisibility * Convergence of Random Processes, Measures, and Sets * Stochastic Integrals and Quadratic Variation * Continuous Martingales and Brownian Motion * Feller Processes and Semigroups * Ergodic Properties of Markov Processes * Stochastic Differential Equations and Martingale Problems * Local Time, Excursions, and Additive Functionals * One-Dimensional SDEs and Diffusions * Connections with PDEs and Potential Theory * Predictability, Compensation, and Excessive Functions * Semimartingales and General Stochastic Integration * Large Deviations * Appendix 1: Advanced Measure Theory * Appendix 2: Some Special Spaces * Historical and Bibliographical Notes * Bibliography * Indices

http://cyber.sibsutis.ru:82/Monarev/docs/nauka/PROBABILITY/Kallenberg%20O.%20Foundations%20of%20Modern%20Probability%20(Springer,1997)(ISBN%200387949577)(535s).pdf

Foundations of modern probability

Levy processes form a wide and rich class of random process, and have many applications ranging from physics to finance Stochastic calculus is the mathematics of systems interacting with random noise Here, the author ties these two subjects together, beginning with an introduction to the general theory of Levy processes, then leading on to develop the stochastic calculus for Levy processes in a direct and accessible way This fully revised edition now features a number of new topics These include: regular variation and subexponential distributions; necessary and sufficient conditions for Levy processes to have finite moments; characterisation of Levy processes with finite variation; Kunita's estimates for moments of Levy type stochastic integrals; new proofs of Ito representation and martingale representation theorems for general Levy processes; multiple Wiener-Levy integrals and chaos decomposition; an introduction to Malliavin calculus; an introduction to stability theory for Levy-driven SDEs

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Lévy Processes and Stochastic Calculus

An introduction to stochastic partial differential equations

This celebrated book has been prepared with readers' needs in mind, remaining a systematic treatment of the subject whilst retaining its vitality. The second volume follows on from the first, concentrating on stochastic integrals, stochastic differential equations, excursion theory and the general theory of processes. Much effort has gone into making these subjects as accessible as possible by providing many concrete examples that illustrate techniques of calculation, and by treating all topics from the ground up, starting from simple cases. Many of the examples and proofs are new; some important calculational techniques appeared for the first time in this book. Together with its companion volume, this book helps equip graduate students for research into a subject of great intrinsic interest and wide application in physics, biology, engineering, finance and computer science.

Diffusions, Markov processes, and martingales

Semisimple subalgebras of semisimple Lie algebras

Controlled Markov processes

Markov Processes: Theorems and Problems

We start from a model of a branching particle system with immigration and with death rate and branching mechanism depending on time and location. Then we consider a limit case when the mass of particles and their life times are small and their density is high. This way, we construct a measure-valued process $X_t$ which we call a superprocess. Replacing the underlying Markov process $\xi_t$ by the corresponding "historical process" $\xi_{\leq t}$, we construct a measure-valued process $M_t$ in functional spaces which we call a historical superprocess. The moment functions for superprocesses are evaluated. Linear positive additive functionals are studied. They are used to construct a continuous analog of a random tree obtained by stopping every particle at a time depending on its path (say, at the first exit time from a domain). A related special Markov property for superprocesses is proved which is useful for applications to certain nonlinear partial differential equations. The concluding section is devoted to a survey of the literature, and the terminology on Markov processes used in the paper is explained in the Appendix.

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Branching particle systems and superprocesses

The subject of this article is a class of measure-valued Markov processes. A typical example is super-Brownian motion. The Laplacian $\Delta$ plays a fundamental role in the theory of Brownian motion. For super-Brownian motion, an analogous role is played by the operator $\Delta u - \psi(u)$, where a nonlinear function $\psi$ describes the branching mechanism. The class of admissible functions $\psi$ includes the family $\psi(u) = u^\alpha, 1 < \alpha \leq 2$. Super-Brownian motion belongs to the class of continuous state branching processes investigated in 1968 in a pioneering work of Watanabe. Path properties of super-Brownian motion are well known due to the work of Dawson, Perkins, Le Gall and others. Partial differential equations involving the operator $\Delta u - \psi(u)$ have been studied independently by several analysts, including Loewner and Nirenberg, Friedman, Brezis, Veron, Baras and Pierre. Connections between the probabilistic and analytic theories have been established recently by the author.

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E. B. Dynkin

Papers

Semisimple subalgebras of semisimple Lie algebras

Controlled Markov processes

Markov Processes: Theorems and Problems

Branching particle systems and superprocesses

Superprocesses and Partial Differential Equations