Some fine properties of backward stochastic differential equations

Jianfeng Zhang, Purdue University

Abstract

In this thesis we investigate various properties of the martingale part, usually denoted by Z, of the solution to a class of Backward Stochastic Differential Equations (BSDEs, for short), with path-dependent terminals. We first establish some Feynman-Kac type representation formulae for the process Z for BSDEs with simple terminals. The main feature of these formulae is that they do not involve the derivatives of the coefficients of the BSDEs, and the main device is the Malliavin Calculus. We also provide a probabilistic approach towards the classical solution to a nonlinear PDE. By extending our representation formulae and using some approximating techniques, we prove that, for a large class of BSDEs with path-dependent terminals, the process Z is pathwisely càdlàg (right continuous with left limits), or even continuous. Our proof of convergence relies heavily on the Meyer-Zhang tightness criterion. Based on the above results, we propose a “two-step scheme” to numerically solve BSDEs with path-dependent terminals. Our scheme (strongly) converges in L2, under mild conditions, with rate of convergence [special characters omitted]. Finally, with the same spirit but different techniques, we extend our representation formulae and path regularity results to models driven by Levy processes, motivated by questions arising in financial asset pricing theory where the market is incomplete.

Degree

Ph.D.

Advisors

Douglas, Purdue University.

Subject Area

Mathematics|Finance

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