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Probabilistic methods for semilinearpartial differential equations.Applications to finance

Published online by Cambridge University Press:  26 August 2010

Dan Crisan and
Konstantinos Manolarakis
Dan Crisan
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen's Gate, London, SW7 2AZ, UK. d.crisan@imperial.ac.uk; km3@imperial.ac.uk
Konstantinos Manolarakis
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen's Gate, London, SW7 2AZ, UK. d.crisan@imperial.ac.uk; km3@imperial.ac.uk
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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 Sciences176 (1992) 200–217]. We have at our disposalstochastic processes which solve the so-called backward stochasticdifferential equations. These processes provide us with a Feynman-Kacrepresentation for the solutions of a class of nonlinear partial differential equations (PDEs) which appearin many applications in the field of Mathematical Finance. Therefore thereis a great interest among both practitioners and theoreticians to developreliable numerical methods for their numerical resolution. In this survey, we present a number of probabilistic methods forapproximating solutions of semilinear PDEs all based on the correspondingFeynman-Kac representation. We also include a general introduction tobackward stochastic differential equations and their connection with PDEsand provide a generic framework that accommodates existing probabilisticalgorithms and facilitates the construction of new ones.

Keywords

Probabilistic methodssemilinear PDEsBSDEsMonte Carlo methodsMalliavin calculuscubature methods

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