Konstantinos Dareiotis

Konstantinos Dareiotis, Chaman Kumar, Sotirios Sabanis

We extend the taming techniques for explicit Euler approximations of stochastic differential equations (SDEs) driven by Lévy noise with super-linearly growing drift coefficients. Strong convergence results are presented for the case of locally Lipschitz coefficients. Moreover, rate of convergence results are obtained in agreement with classical literature when the local Lipschitz continuity assumptions are replaced by global and, in addition, the drift coefficients satisfy polynomial Lipschitz continuity. Finally, we further extend these techniques to the case of delay equations.

Konstantinos Dareiotis, James-Michael Leahy

We study the rate of convergence of an explicit and an implicit-explicit finite difference scheme for linear stochastic integro-differential equations of parabolic type arising in non-linear filtering of jump-diffusion processes. We show that the rate is of order one in space and order one-half in time.

Konstantinos Dareiotis

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.