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Probabilistic methods for semilinear partial differential equations. Applications to finance
ESAIM: Mathematical Modelling and Numerical Analysis , Special Issue on Probabilistic methods and their applications, Tome 44 (2010) no. 5, pp. 1107-1133

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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.

DOI : 10.1051/m2an/2010054
Classification : 65C30, 65C05, 60H07, 62G08
Keywords: probabilistic methods, semilinear PDEs, BSDEs, Monte Carlo methods, Malliavin calculus, cubature methods 
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     title = {Probabilistic methods for semilinear partial differential equations. {Applications} to finance},
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Crisan, Dan; Manolarakis, Konstantinos. Probabilistic methods for semilinear partial differential equations. Applications to finance. ESAIM: Mathematical Modelling and Numerical Analysis , Special Issue on Probabilistic methods and their applications, Tome 44 (2010) no. 5, pp. 1107-1133. doi: 10.1051/m2an/2010054

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