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 those known in the literature. In case the jump-commutativity condition is satisfied, we prove optimality of the randomized Milstein algorithm by establishing matching lower bounds. Moreover, we give some insight into the multidimensional case by investigating the optimal convergence rate for the approximation of jump–diffusion type Lévys’ areas. Finally, we report numerical experiments that support our theoretical findings.