A Probabilistic Scheme for Fully Nonlinear Nonlocal Parabolic PDEs with singular Lévy measures
This work provides a novel numerical method for a class of PDEs that are challenging due to singular Lévy measures, which is important for applications in stochastic control and finance.
The paper introduces a Monte Carlo scheme for solving fully nonlinear parabolic nonlocal PDEs with singular Lévy measures, achieving convergence for general parabolic nonlinearities and providing rate-of-convergence bounds for concave/convex cases. The method uses truncation of the infinite Lévy measure dependent on time discretization and a Monte Carlo quadrature for the nonlocal term.
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.