Brownian Motion and Stochastic Calculus (Springer-Verlag New York Inc.)

Designed as a text for graduate courses in stochastic processes. Written for readers familiar with measure-theoretic probability and discrete-time processes who wish to explore stochastic processes in continuous time. The vehicle chosen for this exposition is Brownian motion, which is presented as the canonical example of both a martingale and a Markov process with continuous paths. In this context, the theory of stochastic integration and stochastic calculus is developed. The power of this calculus is illustrated by results concerning representations of martingales and change of measure on Wiener space, and this in turn permits a presentation of recent advances in financial economics (options pricing and consumption/investment optimization). The book contains a detailed discussion of weak and strong solutions of stochastic differential equations and a study of local time for semimartingales, with special emphasis on the theory of Brownian local time. The t ext is complemented by a large number of problems and exercises.