Stong order 1 adaptive approximation of jump-diffusion SDEs with discontinuous drift
Verena Schwarz
TL;DR
This work tackles the numerical approximation of jump-diffusion SDEs with discontinuous drift by introducing a transformation-based, doubly-adaptive quasi-Milstein scheme that is jump-adapted and drift-discontinuity-aware. The authors prove a strong convergence rate of $1$ in $L^p$ for $p\in[1,\infty)$ and provide explicit cost bounds, using a two-step approach: first establishing rate $1$ for a doubly-adaptive scheme under stronger assumptions and then applying a transformation to handle the discontinuities. The combination of occupation-time estimates, moment bounds, and a meticulous cost analysis leads to a practically implementable method with near-optimal computational efficiency. The transformation back to the original variable yields the same rate for $X$, making the method broadly applicable to jump-diffusion models with piecewise Lipschitz drift in applications such as energy markets.
Abstract
We present an adaptive approximation scheme for jump-diffusion SDEs with discontinuous drift and (possibly) degenerate diffusion. This transformation-based doubly-adaptive quasi-Milstein scheme is the first scheme that has strong convergence rate $1$ in $L^p$ for $p\in[1,\infty)$ with respect to the average computational cost for these SDEs. To obtain our result, we prove that under slightly stronger assumptions which are still weaker than those in existing literature, a related doubly-adaptive quasi-Milstein scheme has convergence order $1$. This scheme is doubly-adaptive in the sense that it is jump-adapted, i.e.~all jump times of the Poisson noise are grid points, and it includes an adaptive stepsize strategy to account for the discontinuities of the drift.