  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.
