An Explicit Euler-type Scheme for L\'evy-driven SDEs with Superlinear and Time-Irregular Coefficients

TL;DR

This paper develops a randomized tamed Euler scheme for Le9vy-driven SDEs with superlinear and time-irregular coefficients, achieving near-optimal strong convergence and addressing previously unhandled superlinear cases.

Contribution

It introduces a novel randomized tamed Euler scheme that handles superlinear, time-irregular coefficients in Le9vy-driven SDEs, including stochastic delay differential equations with switching.

Findings

Achieves strong -convergence rate close to 0.5.

First to analyze superlinear coefficients in Carathe9odory-type SDEs.

Applicable to stochastic delay differential equations with Markovian switching.

Abstract

This paper introduces a randomized tamed Euler scheme tailored for L\'evy-driven stochastic differential equations (SDEs) with superlinear random coefficients and Carath\'eodory-type drift. Under assumptions that allow for time-irregular drifts while ensuring appropriate time-regularity of the diffusion and jump coefficients, the proposed scheme is shown to achieve the optimal strong $\mathcal{L}^2$-convergence rate, arbitrarily close to $0.5$. A crucial component of our methodology is the incorporation of drift randomization, which overcomes challenges due to low time-regularity, along with a taming technique to handle the superlinear state dependence. Our analysis moreover covers settings where the coefficients are random, providing for instance strong convergence of randomized tamed Euler schemes for L\'evy-driven stochastic delay differential equations (SDDEs) with Markovian switching. To our knowledge, this is the first {work} that addresses the case of superlinear coefficients in the numerical analysis of Carath\'eodory-type SDEs and even for ordinary differential equations.