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A Probabilistic Numerical Method for Fully Nonlinear Parabolic PDEs

Fahim, Arash; Touzi, Nizar; Warin, Xavier (2011), A Probabilistic Numerical Method for Fully Nonlinear Parabolic PDEs, The Annals of Applied Probability, 21, 4, p. 1322-1364. http://dx.doi.org/10.1214/10-AAP723

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Type
Article accepté pour publication ou publié
Date
2011
Journal name
The Annals of Applied Probability
Volume
21
Number
4
Publisher
Institute of Mathematical Statistics
Pages
1322-1364
Publication identifier
http://dx.doi.org/10.1214/10-AAP723
Metadata
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Author(s)
Fahim, Arash
Touzi, Nizar
Warin, Xavier
Abstract (EN)
We consider the probabilistic numerical scheme for fully nonlinear PDEs suggested in [12], and show that it can be introduced naturally as a combination of Monte Carlo and finite differences scheme without appealing to the theory of backward stochastic differential equations. Our first main result provides the convergence of the discrete-time approximation and derives a bound on the discretization error in terms of the time step. An explicit implementable scheme requires to approximate the conditional expectation operators involved in the discretization. This induces a further Monte Carlo error. Our second main result is to prove the convergence of the latter approximation scheme, and to derive an upper bound on the approximation error. Numerical experiments are performed for the approximation of the solution of the mean curvature flow equation in dimensions two and three, and for two and five-dimensional (plus time) fully-nonlinear Hamilton-Jacobi-Bellman equations arising in the theory of portfolio optimization in financial mathematics.
Subjects / Keywords
second order backward stochastic differential equations; Viscosity Solutions; monotone schemes; Monte Carlo approximation
JEL
C15 - Statistical Simulation Methods: General

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