Difference-quadrature schemes for nonlinear degenerate parabolic integro-PDE

TL;DR

This paper develops monotone difference-quadrature schemes for nonlinear degenerate parabolic integro-PDEs associated with controlled Levy processes, providing new discretizations and a general theory for error estimation.

Contribution

It introduces novel direct discretizations for the non-local terms and a comprehensive framework for error analysis of integro-PDE approximations.

Findings

Schemes handle singular Levy measures effectively

Error estimates are established for the proposed schemes

Applicable to fully nonlinear, degenerate parabolic integro-PDEs

Abstract

We derive and analyze monotone difference-quadrature schemes for Bellman equations of controlled Levy (jump-diffusion) processes. These equations are fully non-linear, degenerate parabolic integro-PDEs interpreted in the sense of viscosity solutions. We propose new ``direct'' discretizations of the non-local part of the equation that give rise to monotone schemes capable of handling singular Levy measures. Furthermore, we develop a new general theory for deriving error estimates for approximate solutions of integro-PDEs, which thereafter is applied to the proposed difference-quadrature schemes.