On Explicit Approximations for L\'evy Driven SDEs with Super-linear Diffusion Coefficients

TL;DR

This paper develops explicit Euler-type numerical schemes for Lévy-driven SDEs with super-linear coefficients, proving strong convergence at classical rates, including for delay equations, thus advancing numerical methods for complex stochastic systems.

Contribution

It introduces explicit schemes for Lévy-driven SDEs with super-linear growth, extending to delay equations, and establishes their strong convergence with optimal rates.

Findings

Strong convergence of the schemes is proven.

Convergence rate matches classical Euler scheme.

Applicable to delay SDEs driven by Lévy noise.

Abstract

Motivated by the results of \cite{sabanis2015}, we propose explicit Euler-type schemes for SDEs with random coefficients driven by L\'evy noise when the drift and diffusion coefficients can grow super-linearly. As an application of our results, one can construct explicit Euler-type schemes for SDEs with delays (SDDEs) which are driven by L\'evy noise and have super-linear coefficients. Strong convergence results are established and their rate of convergence is shown to be equal to that of the classical Euler scheme. It is proved that the optimal rate of convergence is achieved for $\mathcal{L}^2$-convergence which is consistent with the corresponding results available in the literature.