This text is a continuation to Varadhan’s Probability Theory published by the Courant Institute in 2001 and it is again an excellent introductory text. The book deals with elementary continuous time processes assuming a reader who might be in need of a soft and perhaps more intuitive introduction to the mathematical complexity of the contemporary theory of stochastic processes as presented, for example, in O. Kallenberg’s Foundations of Modern Probability (Springer 2002). The topics are chosen carefully to cover all essential stochastic dynamic models in dimension one. The main topics are Martingales, processes with independent increments, the Poisson process, point processes, jump Markov processes (semigroups of operators, recurrence and transience, and invariant distributions), Brownian motion, stochastic integrals and calculus (including the Girzanov theorem and the Feynman-Kac formula), stochastic differential equations and 1-dimensional diffusions. The volume covers contents of the author’s courses given at the Courant Institute. The reader is assumed to be familiar with the basics of probability and measure theory but a deeper knowledge of mathematical analysis is also crucial. The text is one of those that may be strongly recommended to all young mathematicians as a starter to precede a deeper study of probability and stochastic processes.

Reviewer:

jste