A Probabilistic Scheme for Fully Non-linear Non-local Parabolic PDEs with singular Levy measures

Arash Fahim (1)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

We introduce a Monte Carlo scheme for the approximation of the solutions of fully non--linear parabolic non--local PDEs. The method is the generalization of the method proposed by [Fahim-Touzi-Warin,2011] for fully non--linear parabolic PDEs. As an independent result, we also introduce a Monte Carlo Quadrature method to approximate the integral with respect to Lévy measure which may appear inside the scheme. We consider the equations whose non--linearity is of the Hamilton--Jacobi--Belman type. We avoid the difficulties of infinite Levy measures by truncation of the Levy integral by some $\kappa>0$ near $0$. The first result provides the convergence of the scheme for general parabolic non--linearities. The second result provides bounds on the rate of convergence for concave non--linearities from above and below. For both results, it is crucial to choose $\kappa$ appropriately dependent on $h$.

Mots clés: Viscosity solution; non-local PDE; Monte Carlo approximation

Domaines: Probabilités [math.PR]

Arash Fahim. A Probabilistic Scheme for Fully Non-linear Non-local Parabolic PDEs with singular Levy measures. 2011. ⟨hal-00530432v2⟩