Rate of convergence for numerical solutions to SFDEs with jumps

TL;DR

This paper analyzes the convergence rate of Euler-Maruyama numerical solutions for stochastic functional differential equations with jumps, showing a convergence order of 1/p in the pth moment, which differs from the non-jump case.

Contribution

It establishes the convergence order of EM solutions for SFDEs with jumps under global and local Lipschitz conditions, highlighting the impact of jumps on convergence rates.

Findings

pth moment convergence order is 1/p for SFDEs with jumps

Mean-square convergence order is close to 1/2 under local Lipschitz conditions

Convergence behavior differs significantly from SFDEs without jumps

Abstract

In this paper, we are interested in the numerical solutions of stochastic functional differential equations (SFDEs) with {\it jumps}. Under the global Lipschitz condition, we show that the $p$th moment convergence of the Euler-Maruyama (EM) numerical solutions to SFDEs with jumps has order $1/p$ for any $p\ge 2$. This is significantly different from the case of SFDEs without jumps where the order is 1/2 for any $p\ge 2$. It is therefore best to use the mean-square convergence for SFDEs with jumps. Consequently, under the local Lipschitz condition, we reveal that the order of the mean-square convergence is close to 1/2, provided that the local Lipschitz constants, valid on balls of radius $j$, do not grow faster than $\log j$.