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

TL;DR

This paper analyzes the error of the randomized Milstein algorithm for jump-diffusion SDEs, establishing optimality under certain conditions and providing insights into multidimensional cases with numerical validation.

Contribution

It offers a comprehensive error analysis under weaker assumptions and proves the optimality of the randomized Milstein algorithm for jump-diffusion SDEs.

Findings

Error bounds under weaker assumptions

Optimality of the randomized Milstein algorithm

Numerical experiments supporting theoretical results

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.