Branching diffusions with jumps, and valuation with systemic counterparties

We extend the branching diffusion Monte Carlo method of Henry-Labordère et al to the case of parabolic partial differential equations 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 partial differential equation in the viscosity sense. Our theoretical setup directly leads to a Monte Carlo algorithm, whose applicability is showcased in the valuation of financial positions with defaultable, systemically important counterparties and a high-dimensional underlying.