Brownian Motion and Stochastic Calculus

Nonfiction, Science & Nature, Science, Physics, Mechanics, Mathematics, Statistics
Cover of the book Brownian Motion and Stochastic Calculus by Ioannis Karatzas, Steven Shreve, Springer New York
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Author: Ioannis Karatzas, Steven Shreve ISBN: 9781461209492
Publisher: Springer New York Publication: March 27, 2014
Imprint: Springer Language: English
Author: Ioannis Karatzas, Steven Shreve
ISBN: 9781461209492
Publisher: Springer New York
Publication: March 27, 2014
Imprint: Springer
Language: English

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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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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