Probabilistic methods for semilinear partial differential equations. Applications to finance
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 5, p. 1107-1133

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 : https://doi.org/10.1051/m2an/2010054
Classification:  65C30,  65C05,  60H07,  62G08
Keywords: probabilistic methods, semilinear PDEs, BSDEs, Monte Carlo methods, Malliavin calculus, cubature methods
@article{M2AN_2010__44_5_1107_0,
     author = {Crisan, Dan and Manolarakis, Konstantinos},
     title = {Probabilistic methods for semilinear partial differential equations. Applications to finance},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {44},
     number = {5},
     year = {2010},
     pages = {1107-1133},
     doi = {10.1051/m2an/2010054},
     zbl = {pre05798945},
     mrnumber = {2731405},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2010__44_5_1107_0}
}
Crisan, Dan; Manolarakis, Konstantinos. Probabilistic methods for semilinear partial differential equations. Applications to finance. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 5, pp. 1107-1133. doi : 10.1051/m2an/2010054. http://www.numdam.org/item/M2AN_2010__44_5_1107_0/

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