A Probabilistic Scheme for Fully Nonlinear Nonlocal Parabolic PDEs with singular L\'evy measures

TL;DR

This paper presents a Monte Carlo scheme for solving fully nonlinear nonlocal parabolic PDEs with singular Lévy measures, addressing convergence and rate of convergence issues through truncation and quadrature methods.

Contribution

It introduces a novel Monte Carlo scheme that handles infinite Lévy measures via truncation and provides convergence analysis for nonlinear PDEs with nonlocal terms.

Findings

Scheme converges for general nonlinearities

Bounds on convergence rate for concave/convex nonlinearities

Effective truncation dependent on time discretization

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 L\'evy measure by truncation of the L\'evy 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 L\'evy measure appropriately dependent on the time discretization. We also introduce a Monte Carlo Quadrature method to approximate the nonlocal term in the HJBI nonlinearity.