The truncated EM method for stochastic differential equations with Poisson jumps

TL;DR

This paper analyzes the truncated Euler-Maruyama method for stochastic differential equations with Poisson jumps, establishing convergence rates and stability properties under super-linear growth conditions, with implications for numerical simulations.

Contribution

It provides new convergence rate results and stability analysis for the truncated EM method applied to SDEs with jumps under super-linear growth.

Findings

Optimal $\\mathcal{L}^r$-convergence rate close to $1/(1+\gamma)$.

Strong convergence rate not exceeding $1/4$ when all coefficients grow super-linearly.

Truncated EM preserves mean square exponential stability and boundedness.

Abstract

In this paper, we use the truncated EM method to study the finite time strong convergence for the SDEs with Poisson jumps under the Khasminskii-type condition. We establish the finite time $ \mathcal L ^r (r \ge 2) $ convergence rate when the drift and diffusion coefficients satisfy super-linear condition and the jump coefficient satisfies the linear growth condition. The result shows that the optimal $\mathcal L ^r$-convergence rate is close to $ 1/ (1 + \gamma)$, where $\gamma$ is the super-linear growth constant. This is significantly different from the result on SDEs without jumps. When all the three coefficients of SDEs are allowing to grow super-linearly, the $ \mathcal L^r (0<r<2)$ strong convergence results are also investigated and the optimal strong convergence rate is shown to be not greater than $1/4$. Moreover, we prove that the truncated EM method preserve nicely the mean square exponentially stability and asymptotic boundedness of the underlying SDEs with Piosson jumps. Several examples are given to illustrate our results.