Probabilistic methods for semilinear partial differential equations. Applications to finance

Dan Crisan; Konstantinos Manolarakis

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 44, Issue: 5, page 1107-1133
  • ISSN: 0764-583X

Abstract

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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 Sciences176 (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.

How to cite

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Crisan, Dan, and Manolarakis, Konstantinos. "Probabilistic methods for semilinear partial differential equations. Applications to finance." ESAIM: Mathematical Modelling and Numerical Analysis 44.5 (2010): 1107-1133. <http://eudml.org/doc/250847>.

@article{Crisan2010,
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 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. },
author = {Crisan, Dan, Manolarakis, Konstantinos},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Probabilistic methods; semilinear PDEs; BSDEs; Monte Carlo methods; Malliavin calculus; cubature methods; probabilistic methods; BSDEs Monte Carlo methods},
language = {eng},
month = {8},
number = {5},
pages = {1107-1133},
publisher = {EDP Sciences},
title = {Probabilistic methods for semilinear partial differential equations. Applications to finance},
url = {http://eudml.org/doc/250847},
volume = {44},
year = {2010},
}

TY - JOUR
AU - Crisan, Dan
AU - Manolarakis, Konstantinos
TI - Probabilistic methods for semilinear partial differential equations. Applications to finance
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/8//
PB - EDP Sciences
VL - 44
IS - 5
SP - 1107
EP - 1133
AB - 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 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.
LA - eng
KW - Probabilistic methods; semilinear PDEs; BSDEs; Monte Carlo methods; Malliavin calculus; cubature methods; probabilistic methods; BSDEs Monte Carlo methods
UR - http://eudml.org/doc/250847
ER -

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