Rate of convergence for numerical solutions to SFDEs with jumps
Provides theoretical convergence rates for numerical solutions of SFDEs with jumps, highlighting a key difference from the jump-free case and guiding practical use of mean-square convergence.
The paper proves that for SFDEs with jumps under global Lipschitz conditions, the p-th moment convergence of Euler-Maruyama solutions has order 1/p (p≥2), unlike the 1/2 order for SFDEs without jumps. Under local Lipschitz conditions with slow-growing constants, mean-square convergence order approaches 1/2.
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 $\log j$.