Stability of the semi-tamed and tamed Euler schemes for stochastic differential equations with jumps under non-global Lipschitz condition

TL;DR

This paper investigates the stability properties of semi-tamed and tamed Euler schemes for stochastic differential equations with jumps under non-global Lipschitz conditions, providing theoretical analysis and numerical validation.

Contribution

It extends the analysis of tamed Euler schemes to include nonlinear and linear stability for SDEs with jumps under non-global Lipschitz conditions.

Findings

The schemes are shown to be stable under specified conditions.

Numerical simulations confirm the theoretical stability results.

Abstract

Under non-global Lipschitz condition, Euler Explicit method fails to converge strongly to the exact solution, while Euler implicit method converges but requires much computational efforts. Tamed scheme was first introduced in [2] to overcome this failure of the standard explicit method. This technique is extended to SDEs driven by Poisson jump in [3] where several schemes were analyzed. In this work, we investigate their nonlinear stability under non-global Lipschitz and their linear stability. Numerical simulations to sustain the theoretical results are also provided.