What question did this study set out to answer?

The study aims to explore the strong convergence of the semi-implicit Euler–Maruyama method for nonlinear stochastic differential equations driven by Lévy processes.

February 5, 2026

The semi-implicit euler–maruyama method for nonlinear non-autonomous stochastic differential equations driven by a class of lévy processes

Key Points

The study aims to explore the strong convergence of the semi-implicit Euler–Maruyama method for nonlinear stochastic differential equations driven by Lévy processes.
Analyzed the semi-implicit Euler–Maruyama method for nonlinear stochastic differential equations.
Investigated the convergence order related to Lévy process parameters.
Proved the existence and uniqueness of the invariant measure for the numerical method.
Provided numerical examples to validate theoretical findings.
Demonstrated the strong convergence of the semi-implicit EM method under specific conditions.
Identified a novel dependence of convergence order on Lévy process parameters.
Established convergence of the numerical invariant measure to the underlying invariant measure.

Abstract

The strong convergence of the semi-implicit Euler–Maruyama (EM) method for stochastic differential equations with nonlinear coefficients driven by a class of Lévy processes is investigated. The dependence of the convergence order of the numerical scheme on the parameters of the class of Lévy processes is discovered, which is different from existing results. In addition, the existence and uniqueness of the numerical invariant measure for the semi-implicit EM method is studied, and its convergence to the underlying invariant measure is also proved. Numerical examples are provided to confirm our theoretical results.

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