On forward-backward stochastic differential equations and related numerical methods
Abstract
A. On numerical methods of forward-backward SDEs. The main component of this dissertation is the numerical method for a class of forward-backward stochastic differential equations (FBSDEs). The method is designed around the Four Step Scheme (Douglas-Ma-Protter, 1996) but with a Hermite-spectral method to approximate the solution to the decoupling quasilinear PDE on the whole space. We carry out a rigorous synthetic error analysis for a fully discretized scheme, namely a first-order scheme in time and a Hermite-spectral scheme in space, of the FBSDEs. Equally important, we made a systematical numerical comparison between several schemes for the resulting decoupled forward SDE, including a stochastic version of the Adams-Bashforth scheme. It is shown that the stochastic version of the Adams-Bashforth scheme coupled with the Hermite-spectral method leads to a convergence rate of [special characters omitted] (in time) which is better than those in all other published work for the FBSDEs. B. Chebyshev-Galerkin method using lattice rule for high-dimensional elliptic equations. A problem that is closely related to the numerical FBSDEs, especially in the high dimensional case, is how to solve high dimensional PDEs for which the conventional numerical method is limited by the “curse of dimensionality”. We propose a new algorithm which is based on (i) choosing frequencies from Hyperbolic cross, and (ii) sampling input function on the node set of a suitable integration lattice. A Chebyshev-Galerkin spectral method is used to solve Poisson type equation. C. General forward-backward SDEs of Markovian type. The third project in this dissertation is the study of the general theory of FBSDE. In particular, we consider a class of forward-backward stochastic differential equations in a general Markovian framework. Using the idea of Four Step Scheme, we show that we can remove the most crucial restriction in the theory of BSDEs: the filtration being Brownian. In fact, in such a setting the martingale representation theorem is no longer necessary. However, we are confined to the cases where all the coefficients are deterministic. With appropriate regularity assumptions on the coefficients, we prove that the adapted solution exists and is unique over any prescribed time duration; and the backward components are determined explicitly by the forward component via the classical solution to a system of parabolic integro-partial differential equations. Our forward SDE is rather general, representing a large class of strong Markov processes. As an application, we prove a new type of integral representation theorem against a Lévy-type martingale.
Degree
Ph.D.
Advisors
Shen, Purdue University.
Subject Area
Mathematics
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