Optimal approximation of jump-diffusion SDEs using randomized schemes

We prove the optimality of the randomized Milstein algorithm for solving scalar jump-diffusion stochastic differential equations. For this, we first provide a complete error analysis of the randomized Milstein algorithm for approximating scalar jump-diffusion stochastic differential equations under substantially weaker assumptions than those known in the literature. Then we provide lower bounds for randomized schemes assuming the jump-commutativity condition and conclude optimality. Moreover, we give some insight into the multidimensional case and report numerical experiments that support our theoretical findings.

This is joint work with Paweł Przybyłowicz and Michaela Szölgyenyi.