
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
View/ Open
Type
Article accepté pour publication ou publiéDate
2011Journal name
The Annals of Applied ProbabilityVolume
21Number
4Publisher
Institute of Mathematical Statistics
Pages
1322-1364
Publication identifier
Metadata
Show full item recordAbstract (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 approximationRelated items
Showing items related by title and author.
-
Bouchard, Bruno; Elie, Romuald; Touzi, Nizar (2009) Chapitre d'ouvrage
-
Tan, Xiaolu (2013) Article accepté pour publication ou publié
-
Henry-Labordère, Pierre; Oudjane, Nadia; Tan, Xiaolu; Touzi, Nizar; Warin, Xavier (2019) Article accepté pour publication ou publié
-
Matoussi, Anis; Possamaï, Dylan; Sabbagh, Wissal (2018) Article accepté pour publication ou publié
-
Second order BSDEs with jumps: existence and probabilistic representation for fully-nonlinear PIDEs Kazi-Tani, Mohamed Nabil; Possamaï, Dylan; Zhou, Chao (2015) Article accepté pour publication ou publié