Daniel W. Stroock - An Introduction to Markov Processes

Finite state space, a trial run -- Moving to Euclidean space, the real thing -- Itos approach in the Euclidean setting -- Further considerations -- Itos theory of stochastic integration -- Applications of stochastic integration to Brownian motion -- The Kunita-Watanabe extension -- Stratonovishs theory.;Kiyosi Its greatest contribution to probability theory may be his introduction of stochastic differential equations to explain the Kolmogorov-Feller theory of Markov processes. Starting with the geometric ideas that guided him, this book gives an account of Its program. The modern theory of Markov processes was initiated by A.N. Kolmogorov. However, Kolmogorovs approach was too analytic to reveal the probabilistic foundations on which it rests. In particular, it hides the central role played by the simplest Markov processes: those with independent, identically distributed increments. To remedy this defect, It interpreted Kolmogorovs famous forward equation as an equation that describes the integral curve of a vector field on the space of probability measures. Thus, in order to show how Its thinking leads to his theory of stochastic integral equations, Stroock begins with an account of integral curves on the space of probability measures and then arrives at stochastic integral equations when he moves to a pathspace setting. In the first half of the book, everything is done in the context of general independent increment processes and without explicit use of Its stochastic integral calculus. In the second half, the author provides a systematic development of Its theory of stochastic integration: first for Brownian motion and then for continuous martingales. The final chapter presents Stratonovichs variation on Its theme and ends with an application to the characterization of the paths on which a diffusion is supported. The book should be accessible to readers who have mastered the essentials of modern probability theory and should provide such readers with a reasonably thorough introduction to continuous-time, stochastic processes.