Memoirs of the American Mathematical Society 2005; 127 pp; softcover Volume: 175 ISBN10: 0821837044 ISBN13: 9780821837047 List Price: US$68 Individual Members: US$40.80 Institutional Members: US$54.40 Order Code: MEMO/175/825
 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  Introduction
 Representations
 Induced transformation I
 Approximation
 Induced transformation II
 Stochastic calculus of variation
 Stochastic integration
 Nonlinear translation (Absolute continuity)
 Conditional expectation
 Integration by parts
 Composition (Itô formula)
 Clark type representation
 Continuation
 Stochastic control
 Appendix
 Bibliography
