Feynman-Kac representation of fully nonlinear PDEs and applications
نویسندگان
چکیده
The classical Feynman-Kac formula states the connection between linear parabolic partial differential equations (PDEs), like the heat equation, and expectation of stochastic processes driven by Brownian motion. It gives then a method for solving linear PDEs by Monte Carlo simulations of random processes. The extension to (fully)nonlinear PDEs led in the recent years to important developments in stochastic analysis and the emergence of the theory of backward stochastic differential equations (BSDEs), which can be viewed as nonlinear Feynman-Kac formulas. We review in this paper the main ideas and results in this area, and present implications of these probabilistic representations for the numerical resolution of nonlinear PDEs, together with some applications to stochastic control problems and model uncertainty in finance. MSC Classification (2000): 60H30, 65C99, 93E20
منابع مشابه
Wellposedness of Second Order Backward SDEs
We provide an existence and uniqueness theory for an extension of backward SDEs to the second order. While standard Backward SDEs are naturally connected to semilinear PDEs, our second order extension is connected to fully nonlinear PDEs, as suggested in [4]. In particular, we provide a fully nonlinear extension of the Feynman-Kac formula. Unlike [4], the alternative formulation of this paper i...
متن کاملDiscrete-time approximation of BSDEs and probabilistic schemes for fully nonlinear PDEs
The aim of this paper is to provide a survey on recent advances on probabilistic numerical methods for nonlinear PDEs, which serve as an alternative to classical deterministic schemes and allow to handle a large class of multidimensional nonlinear problems. These probabilistic schemes are based on the stochastic representation of semilinear PDEs by means of backward SDEs, which can be viewed as...
متن کاملBackward doubly stochastic differential equations and systems of quasilinear SPDEs
A new kind of backward stochastic differential equations (in short BSDE), where the solution is a pair of processes adapted to the past of the driving Brownian motion, has been introduced by the authors in [63. It was then shown in a series of papers by the second and both authors (see [8, 7, 9, 103), that this kind of backward SDEs gives a probabilistic representation for the solution of a lar...
متن کاملSharp Propagation of Chaos Estimates for Feynman-kac Particle Models
This article is concerned with the propagation-of-chaos properties of genetic type particle models. This class of models arises in a variety of scientific disciplines including theoretical physics, macromolecular biology, engineering sciences, and more particularly in computational statistics and advanced signal processing. From the pure mathematical point of view, these interacting particle sy...
متن کاملMarkov processes and parabolic partial differential equations
In the first part of this article, we present the main tools and definitions of Markov processes’ theory: transition semigroups, Feller processes, infinitesimal generator, Kolmogorov’s backward and forward equations and Feller diffusion. We also give several classical examples including stochastic differential equations (SDEs) and backward SDEs (BSDEs). The second part of this article is devote...
متن کامل