Integral Transformations and Anticipative Calculus for Fractional Brownian Motions

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 Radon-Nikodym 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 Ito 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.