Approximation and optimality for jump-diffusion SDEs with discontinuo...

In this talk we consider the approximation of jump-diffusion stochastic differential equations with discontinuous drift, possibly degenerate diffusion coefficient, and Lipschitz continuous jump coefficient. We present a jump-adapted higher-order scheme, the so-called transformation-based jump-adapted quasi-Milstein scheme. For this scheme, we provide a complete error analysis: We prove convergence order $3/4$ in $L^p$ for $p\in[1,\infty)$. Further, we provide lower error bounds for non-adaptive and jump-adapted approximation schemes of order $3/4$ in $L^1$. This result is based on a universal Skorokhod measurable functional representation of solutions to semimartingale SDEs and yields optimality of the transformation-based jump-adapted quasi-Milstein scheme.

This is based on joint work with Paweł Przybyłowicz, Alexander Steinicke and Michaela Szölgyenyi.