Essentials of Stochastic Processes (Springer Texts in Statistics)

This book is an introduction to stochastic processes written for undergraduates or beginning grad. students who have had a previous course in probability. Durrett has sketched a 25-page review of probability at the beginning of the book, which includes many examples and some challenging exercises. The rest of the book covers discrete and continuous time Markov chains, Poisson processes, Brownian motion, and some renewal theory, including material on queuing networks, spatial Poisson processes, and a fine chapter on martingales which treats optional sampling and forms a good basis for later study of Brownian motion and applications to option pricing and the Black-Scholes formula. Durrett wants his readers to be able to use stochastic processes to solve problems. He presents numerous examples to motivate and develop skills. Examples are explained in detail, sometimes including more than one solution. After stating a result, he frequently asks "Why is this true?" and then sketches a proof or offers an intuitive answer in order to develop the reader's insight (and to allow application-oriented readers to skip the details of a formal proof). He sometimes postpones or skips rigorous proofs so he can develop and apply the theory more quickly. A reader with applications in mind, especially one already familiar with the theory, will appreciate these features. --Mathematical Reviews