Brownian Motion and Stochastic Calculus, 2/e (Paperback)

Description

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.

 

Martingales, Stopping Times, and Filtrations.- Brownian Motion.- Stochastic Integration.- Brownian Motion and Partial Differential Equations.- Stochastic Differential Equations.- Lévy's Theory of Brownian Local Time.

描述

這是一本研究生課程教材，針對熟悉測度論概率和離散時間過程的讀者，希望探索連續時間的隨機過程。本書以布朗運動作為講解的主要例子，將其呈現為一個馬丁格爾和馬可夫過程的典型示例，並具有連續路徑。在這個背景下，發展了隨機積分和隨機微積分的理論，並通過關於馬丁格爾的表示和測度變換的結果進行了說明，進而呈現了金融經濟學的最新進展。本書詳細討論了隨機微分方程的弱解和強解，以及半馬丁格爾的局部時間的研究，特別強調布朗局部時間的理論。全書配有大量問題和練習。

目錄

馬丁格爾、停時和過濾器。- 布朗運動。- 隨機積分。- 布朗運動和偏微分方程。- 隨機微分方程。- 萊維的布朗局部時間理論。