Branching diffusions with jumps, and valuation with systemic counterparties

Abstract

We extend the branching di usion Monte Carlo method of Henry-Labordère e.a.[11] to the case of parabolic PDEs with mixed local-nonlocal analytic nonlinearities. We investigate branching di usion representations of classical solutions, and we provide su cient conditions under which the branching di usion 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 highdimensional example. As our main application, we demonstrate how the methodology can be used to value nancial positions with defaultable, systemically important counterparties.

Existence and probabilistic representation of the solutions of semilinear parabolic PDEs with fractional Laplacians

“…This approach applies in principle to continuous Itô diffusion generators, provided that the corresponding Malliavin weight can be successfully estimated. In the absence of gradient nonlinearities, the tree-based approach has been recently implemented for nonlocal semilinear PDEs in Belak et al (2020).…”

Probabilistic representations of solutions of nonlinear PDEs

“…This approach applies in principle to continuous Itô diffusion generators, provided that the corresponding Malliavin weight can be successfully estimated. In the absence of gradient nonlinearities, the tree-based approach has been recently implemented for nonlocal semilinear PDEs in [8].…”

“…In [63], this approach has been extended to semilinear parabolic PDEs with pseudo-differential operators of the form ´ηp´Δ{2q and fractional Laplacians, using a branching process T x starting at x P R d and carrying a symmetric p2sq-stable process. In the absence of gradient nonlinearities, the tree-based approach has been recently implemented for nonlocal semilinear parabolic PDEs in [8].…”