This book is designed for postgraduate course in stochastic analysis. It require readers familiar with measure-theoretic probability and discrete-time processes, it continues to explore stochastic process in a continuous time set-up. The stochastic process 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.

The theory of stochastic integration and stochastic calculus is developed in this book. The power of Stochastic calculus is illustrated by results concerning the martingales representation and change of measure on Wiener space, and these development furthered application in financial economics, such as option pricing and consumption optimization. This book also contains a detailed discussion on weak and strong solutions of SdE and a study of local time for semimartingale, with special emphasis on the theory of Brownian local time.

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