Electronic ISBN:  9781470404277 
Product Code:  MEMO/175/826.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 175; 2005; 107 ppMSC: Primary 35;
This paper studies two types of integral transformation associated with fractional Brownian motion. They are applied to construct approximation schemes for fractional Brownian motion by polygonal approximation of standard Brownian motion. This approximation is the best in the sense that it minimizes the mean square error. The rate of convergence for this approximation is obtained. The integral transformations are combined with the idea of probability structure preserving mapping introduced in [48] and are applied to develop a stochastic calculus for fractional Brownian motions of all Hurst parameter \(H\in (0, 1)\). In particular we obtain RadonNikodym derivative of nonlinear (random) translation of fractional Brownian motion over finite interval, extending the results of [48] to general case. We obtain an integration by parts formula for general stochastic integral and an Itô type formula for some stochastic integral. The conditioning, Clark derivative, continuity of stochastic integral are also studied. As an application we study a linear quadratic control problem, where the system is driven by fractional Brownian motion.

Table of Contents

Chapters

1. Introduction

2. Linear stability: the model case

3. Pieces of paradifferential calculus

4. $L^2$ and conormal estimates near the boundary

5. Linear stability

6. Nonlinear stability


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This paper studies two types of integral transformation associated with fractional Brownian motion. They are applied to construct approximation schemes for fractional Brownian motion by polygonal approximation of standard Brownian motion. This approximation is the best in the sense that it minimizes the mean square error. The rate of convergence for this approximation is obtained. The integral transformations are combined with the idea of probability structure preserving mapping introduced in [48] and are applied to develop a stochastic calculus for fractional Brownian motions of all Hurst parameter \(H\in (0, 1)\). In particular we obtain RadonNikodym derivative of nonlinear (random) translation of fractional Brownian motion over finite interval, extending the results of [48] to general case. We obtain an integration by parts formula for general stochastic integral and an Itô type formula for some stochastic integral. The conditioning, Clark derivative, continuity of stochastic integral are also studied. As an application we study a linear quadratic control problem, where the system is driven by fractional Brownian motion.

Chapters

1. Introduction

2. Linear stability: the model case

3. Pieces of paradifferential calculus

4. $L^2$ and conormal estimates near the boundary

5. Linear stability

6. Nonlinear stability