A Probabilistic Scheme for Fully Non-linear Non-local Parabolic Pdes with Singular Lévy Measures

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 Lévy measures by truncation of the Lévy integral by some κ > 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 κ appropriately dependent on h.