The Truncated EM Method for Jump-Diffusion SDDEs with Super-Linearly Growing Diffusion and Jump Coefficients (Journal of Computational Mathematics, 2023)
This work is concerned with the convergence and stability of the truncated Euler-Maruyama (EM) method for super-linear stochastic differential delay equations (SDDEs)with time-variable delay and Poisson jumps. By constructing appropriate truncated functions to control the super-linear growth of the original coefficients, we present two types ofthe truncated EM method for such jump-diffusion SDDEs with time-variable delay, whichis proposed to be approximated by the value taken at the nearest grid points on the leftof the delayed argument. The first type is proved to have a strong convergence orderwhich is arbitrarily close to 1/2 in mean-square sense, under the Khasminskii-type, globalmonotonicity with $U$ function and polynomial growth conditions. The second type is convergent in $q$-th $(q < 2)$ moment under the local Lipschitz plus generalized Khasminskii-typeconditions. In addition, we show that the partially truncated EM method preserves themean-square and $H_∞$ stabilities of the true solutions. Lastly, we carry out some numericalexperiments to support the theoretical results.