Stochastic analysis and differential equations with respect to fractional Brownian motion.

摘要

Stochastic analysis with respect to fractional Brownian motion. Fractional Brownian motion (fBM for short) { BHt , t ∈ R } with a Hurst index H ∈ (0, 1) is a centered Gaussian process. We study different underlying probability spaces, reproducing kernel Hilbert spaces and chaos expansions with respect to fBM. Malliavin calculus is used then to define the Skorohod integral, which is the main object throughout this thesis.;Transformations on Wiener space. We study the generalization of a fundamental theorem by Kusuoka [23] on anticipating Girsanov transformations. Roughly speaking, we study a transformations T on canonical space W into itself and we prove that the probability measure induced by such transformation is equivalent to the original one under some conditions of boundness and smoothness. Furthermore, similar to the Brownian case, one can also explicitly identify the Radon-Nikodym derivative of the two equivalent probability measures.;Stochastic differential equations driven by fBM. We study the class of one-dimensional stochastic differential equations (SDEs) driven by fractional Brownian motions with arbitrary Hurst parameter H ∈ (0, 1). In particular, the stochastic integrals appearing in the equation are defined in the Skorohod sense on fractional Wiener spaces, and the coefficients are allowed to be random, and even anticipating. By using the anticipating Girsanov transformation to transfer the original SDE into a much simpler one on the new probability space, we prove the existence and uniqueness of the solution.