Optimal sampling design for global approximation of jump diffusion SDEs
For researchers in numerical analysis of SDEs, this provides optimal sampling strategies and exact error rates for a class of jump diffusion processes, though the jump commutativity condition limits applicability.
This paper establishes the exact convergence rate of minimal errors for strong global approximation of jump diffusion SDEs with jump commutativity, and constructs optimal Milstein-based methods that asymptotically achieve these rates, showing nonequidistant sampling outperforms equidistant sampling.
The paper deals with strong global approximation of SDEs driven by two independent processes: a nonhomogeneous Poisson process and a Wiener process. We assume that the jump and diffusion coefficients of the underlying SDE satisfy jump commutativity condition. We establish the exact convergence rate of minimal errors that can be achieved by arbitrary algorithms based on a finite number of observations of the Poisson and Wiener processes. We consider classes of methods that use equidistant or nonequidistant sampling of the Poisson and Wiener processes. We provide a construction of optimal methods, based on the classical Milstein scheme, which asymptotically attain the established minimal errors. The analysis implies that methods based on nonequidistant mesh are more efficient than those based on the equidistant mesh.