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 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 $δ$, leading to error estimates of order $ξ(δ)+N(δ)(h+\sqrtτ)$, where $h$ and $τ$ are the spatial and temporal discretization parameters respectively. In these estimates $ξ(δ) \downarrow 0$, but $N(δ)\uparrow \infty$ as $δ\downarrow 0$. 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 $(h+τ^k)$ 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.