Stochastic representations for nonlinear parabolic pdes

Abstract

We discuss several different representations of nonlinear parabolic partial differential equations in terms of Markov processes. After a brief introduction of the linear case, different representations for nonlinear equations are discussed. One class of representations is in terms of stochastic control and differential games. An extension to geometric equations is also discussed. All of these representations are through the appropriate expected values of the data. Different type of representations are also available through backward stochastic differential equations. A recent extension to second-order backward stochastic differential equations allow us to represent all fully nonlinear scalar parabolic equations.