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.
Mots-clés : 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: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {1107--1133},
publisher = {EDP-Sciences},
volume = {44},
number = {5},
year = {2010},
doi = {10.1051/m2an/2010054},
mrnumber = {2731405},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2010054/}
}
TY - JOUR AU - Crisan, Dan AU - Manolarakis, Konstantinos TI - Probabilistic methods for semilinear partial differential equations. Applications to finance JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2010 SP - 1107 EP - 1133 VL - 44 IS - 5 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2010054/ DO - 10.1051/m2an/2010054 LA - en ID - M2AN_2010__44_5_1107_0 ER -
%0 Journal Article %A Crisan, Dan %A Manolarakis, Konstantinos %T Probabilistic methods for semilinear partial differential equations. Applications to finance %J ESAIM: Modélisation mathématique et analyse numérique %D 2010 %P 1107-1133 %V 44 %N 5 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2010054/ %R 10.1051/m2an/2010054 %G en %F M2AN_2010__44_5_1107_0
Crisan, Dan; Manolarakis, Konstantinos. Probabilistic methods for semilinear partial differential equations. Applications to finance. ESAIM: Modélisation mathématique et analyse numérique, Special Issue on Probabilistic methods and their applications, Tome 44 (2010) no. 5, pp. 1107-1133. doi : 10.1051/m2an/2010054. http://www.numdam.org/articles/10.1051/m2an/2010054/
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