A graduate-course text, written for readers familiar with
measure-theoretic probability and discrete-time processes, wishing 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, illustrated by results concerning representations
of martingales and change of measure on Wiener space, which in turn
permit a presentation of recent advances in financial economics. 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 whole is backed by a large number of problems and exercises.

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