Randomized Milstein algorithm for approximation of solutions of jump-diffusion SDEs

Abstract

We investigate the error of the randomized Milstein algorithm for solving scalar jump-diffusion stochastic differential equations. We provide a complete error analysis under substantially weaker assumptions than known in the literature. In case the jump-commutativity condition is satisfied, we prove optimality of the randomized Milstein algorithm by proving a matching lower bound. Moreover, we give some insight into the multidimensional case by investigating the optimal convergence rate for the approximation of jump-diffusion type L\'evys' areas. Finally, we report numerical experiments that support our theoretical findings.