How to Solve PDEs with BSDEs? An Introduction.
by
Philippe Briand
IRMAR, Université Rennes 1, 35042 Rennes cedex, FRANCE

This talk is an introduction to (nonlinear) backward stochastic differential equations (BSDEs for short) introduced by Étienne Pardoux and Shige Peng in 1990 with a special emphasize on relations between BSDEs and partial differential equations (PDEs in the sequel).

The starting point is the following: if u stands for the solution to the PDE

∂t u + 1 2 Du + f(u, ∇u) = 0,        u(T, ·)=g,        (t, x) ∈ [0, T]×Rn,

then, if (t, x) is fixed in [0, T]×Rⁿ, the couple of processes

Yt=u(t, x+Bt-Bt),        Zt = ∇u(t, x+Bt-Bt),        t ∈ [t, T],

where B is a Brownian motion, solves the BSDE

Yt = g(x+BT-Bt) + ó õ T t  f(Ys, Zs) ds - ó õ T t  Zs·dBs,        t ∈ [t, T].

In particular, the function u is given by

u(t, x) = Yt

which is known as the nonlinear Feynman-Kac formula.

The main goal of this talk is to explain how to use BSDEs and the Feynman-Kac formula to construct solutions, usually in the viscosity sense, to PDEs in different situations including parabolic and elliptic semilinear PDEs, quasilinear PDEs, quadratic PDEs, integral-PDEs, ... We will also see that it is possible to use BSDEs to solve homogenization problems or to construct numerical schemes for PDEs.

Date received: June 8, 2007

Copyright © 2007 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caum-84.