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 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 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.