A Probabilistic Scheme for Fully Nonlinear Nonlocal Parabolic PDEs with singular Lévy measures

Abstract

We introduce a Monte Carlo scheme for fully nonlinear parabolic nonlocal PDE's whose nonlinearity in of Hamilton-Jacobi-Bellman-Isaacs (HJBI for short). We avoid the difficulties of infinite Lévy measure by truncation of the Lévy integral. The first result provides the convergence of the scheme for general parabolic nonlinearities. The second result provides bounds on the rate of convergence for concave (or equivalently convex) nonlinearities. For both results, it is crucial to choose truncation of the infinite Lévy measure appropriately dependent on the time discretization. We also introduce a Monte Carlo Quadrature method to approximate the nonlocal term in the HJBI nonlinearity.

Keywords: Viscosity solution, nonlocal PDE, Monte Carlo approximation