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

Abstract

In this article we introduce a finite difference approximation for integro-differential operators of Lévy type. We approximate solutions of integro-differential equations, where the second order operator is allowed to degenerate. In the existing literature, the Lévy operator is treated as a zero/first order operator outside of a centered ball of radius $\delta$, leading to error estimates of order $\xi (\delta )+N(\delta )(h+\sqrt{\tau })$, where $h$ and $\tau$ are the spatial and temporal discretization parameters respectively. In these estimates $\xi (\delta )\downarrow 0$, but $N(\delta )\uparrow \infty$ as $\delta \downarrow 0$. In contrast, we treat the integro-differential operator as a second order operator on the whole unit ball. By this method we obtain error estimates of order $(h+{\tau }^k)$ for $k\in \{1/2,1\}$, eliminating the additional errors and the blowing up constants. Moreover, we do not pose any conditions on the Lévy measure.

arXiv comment16 pages