Probabilistic methods for semilinear partial differential equations. Applications to finance

Department of Mathematics, Imperial College London, 180 Queen's Gate, London, SW7 2AZ, UK. d.crisan@imperial.ac.uk; km3@imperial.ac.uk

Abstract

With the pioneering work of [Pardoux and Peng, Syst. Contr. Lett. 14 (1990) 55–61; Pardoux and Peng, Lecture Notes in Control and Information Sciences 176 (1992) 200–217]. We have at our disposal stochastic processes which solve the so-called backward stochastic differential equations. These processes provide us with a Feynman-Kac representation for the solutions of a class of nonlinear partial differential equations (PDEs) which appear in many applications in the field of Mathematical Finance. Therefore there is a great interest among both practitioners and theoreticians to develop reliable numerical methods for their numerical resolution. In this survey, we present a number of probabilistic methods for approximating solutions of semilinear PDEs all based on the corresponding Feynman-Kac representation. We also include a general introduction to backward stochastic differential equations and their connection with PDEs and provide a generic framework that accommodates existing probabilistic algorithms and facilitates the construction of new ones.