Approximation of heat equation and backward SDEs using random walk : convergence rates

Abstract

This thesis addresses questions related to approximation arising from the fields of stochastic analysis and partial differential equations. Theoretical results regarding convergence rates are obtained by using discretization schemes where the limiting process, the Brownian motion, is approximated by a simple discrete-time random walk. The rate of convergence is derived for a finite-difference approximation of the solution of a terminal value problem for the backward heat equation. This weak approximation result is proved for a terminal function which has bounded variation on compact sets. The sharpness of the according rate is achieved by applying some new results related to the first exit time behavior of Brownian bridges. In addition, convergence rates in the L2-norm are proved for Markovian forward-backward stochastic differential equations, where the underlying forward process is either Brownian motion or a more general Itô diffusion.