Probabilistic Representations of Nonlocal Nonlinear PDEs via Branching Diffusions with Jumps

Abstract

We extend the branching diffusion Monte Carlo method of Henry-Labordere e.a. [2019] to the case of parabolic PDEs with mixed local-nonlocal analytic nonlinearities. We investigate branching diffusion representations of classical solutions, and we provide sufficient conditions under which the branching diffusion representation solves the PDE in the viscosity sense. Our theoretical setup directly leads to a Monte Carlo algorithm, whose applicability is showcased in a stylized high-dimensional example. As our main application, we demonstrate how the methodology can be used to value financial positions with defaultable, systemically important counterparties.