On finite difference schemes for partial integro-differential equations of Lévy type

In this article we introduce a finite difference approximation for integro-differential operators of Lévy type. We approximate solutions of possibly degenerate integro-differential equations by treating the nonlocal operator as a second-order operator on the whole unit ball, eliminating the need for truncation of the Lévy measure which is present in the existing literature. This yields an approximation scheme with significantly reduced computational cost, especially for Lévy measures corresponding to processes with jumps of infinite variation.