Malliavin calculus for backward stochastic differential equations and stochastic differential equations driven by fractional Brownian motion and numerical schemes

Abstract

In this dissertation, I investigate two types of stochastic differential equations driven by fractional Brownian motion and backward stochastic differential equations. Malliavin calculus is a powerful tool in developing the main results in this dissertation. This dissertation is organized as follows. In Chapter 1, I introduce some notations and preliminaries on Malliavin Calculus for both Brownian motion and fractional Brownian motion. In Chapter 2, I study backward stochastic differential equations with general terminal value and general random generator. In particular, the terminal value has not necessary to be given by a forward diffusion equation. The randomness of the generator does not need to be from a forward equation neither. Motivated from applications to numerical simulations, first the Lp-H¨older continuity of the solution is obtained. Then, several numerical approximation schemes for backward stochastic differential equations are proposed and the rate of convergence of the schemes is established based on the obtained Lp-H¨older continuity results. Chapter 3 is concerned with a singular stochastic differential equation driven by an additive one-dimensional fractional Brownian motion with Hurst parameter H > 1 2 . Under some assumptions on the drift, we show that there is a unique solution, which has moments of all orders. We also apply the techniques of Malliavin calculus to prove that the solution has an absolutely continuous law at any time t > 0. In Chapter 4, I am interested in some approximation solutions of a type of stochastic differential equations driven by multi-dimensional fractional Brownian motion BH with Hurst parameter H > 1 2 . In order to obtain an optimal rate of convergence, some techniques are developed in the deterministic case. Some work in progress is contained in this chapter. The results obtained in Chapter 2 are accepted by the Annals of Applied Probability, and the material contained in Chapter 3 has been published in Statistics and Probability Letters 78 (2008) 2075-2085.