Difference-Quadrature Schemes for Nonlinear Degenerate Parabolic Integro-PDE

We derive and analyze monotone difference-quadrature schemes for Bellman equations of controlled Lévy (jump-diffusion) processes. These equations are fully nonlinear, degenerate parabolic integro-PDEs interpreted in the sense of viscosity solutions. We propose new “direct” discretizations of the nonlocal part of the equation that give rise to monotone schemes capable of handling singular Lévy 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.