Ebook: Brownian Motion and Stochastic Calculus

This book is designed for a graduate course in stochastic processes. It is written for the reader who is familiar with measure-theoretic probability and the theory of discrete-time processes who is now ready to explore continuous-time stochastic processes. The vehicle chosen for this exposition is Brownian motion, which is presented as the canonical example of both a Markov process and a martingale in continuous time. The authors show how, by means of stochastic integration and random time change, all continuous martingales and many continuous Markov processes can be represented in terms of Brownian motion. The text is complemented by a large number of exercises.

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This book is designed for a graduate course in stochastic processes. It is written for the reader who is familiar with measure-theoretic probability and the theory of discrete-time processes who is now ready to explore continuous-time stochastic processes. The vehicle chosen for this exposition is Brownian motion, which is presented as the canonical example of both a Markov process and a martingale in continuous time. The authors show how, by means of stochastic integration and random time change, all continuous martingales and many continuous Markov processes can be represented in terms of Brownian motion. The text is complemented by a large number of exercises.
Content:
Front Matter....Pages i-xxiii
Martingales, Stopping Times, and Filtrations....Pages 1-46
Brownian Motion....Pages 47-127
Stochastic Integration....Pages 128-238
Brownian Motion and Partial Differential Equations....Pages 239-280
Stochastic Differential Equations....Pages 281-398
P. L?vy’s Theory of Brownian Local Time....Pages 399-446
Back Matter....Pages 447-470

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This book is designed for a graduate course in stochastic processes. It is written for the reader who is familiar with measure-theoretic probability and the theory of discrete-time processes who is now ready to explore continuous-time stochastic processes. The vehicle chosen for this exposition is Brownian motion, which is presented as the canonical example of both a Markov process and a martingale in continuous time. The authors show how, by means of stochastic integration and random time change, all continuous martingales and many continuous Markov processes can be represented in terms of Brownian motion. The text is complemented by a large number of exercises.
Content:
Front Matter....Pages i-xxiii
Martingales, Stopping Times, and Filtrations....Pages 1-46
Brownian Motion....Pages 47-127
Stochastic Integration....Pages 128-238
Brownian Motion and Partial Differential Equations....Pages 239-280
Stochastic Differential Equations....Pages 281-398
P. L?vy’s Theory of Brownian Local Time....Pages 399-446
Back Matter....Pages 447-470
....