# Optimal sampling design for global approximation of jump diffusion SDEs

## TL;DR

This paper analyzes the optimal sampling strategies for accurately approximating jump diffusion SDEs driven by Poisson and Wiener processes, establishing convergence rates and constructing asymptotically optimal methods.

## Contribution

It determines the exact convergence rate of minimal errors for approximating jump diffusion SDEs and constructs optimal Milstein-based methods using various sampling schemes.

## Key findings

- Nonequidistant sampling is more efficient than equidistant sampling.
- Optimal methods asymptotically attain the minimal possible errors.
- The convergence rate of approximation errors is precisely characterized.

## Abstract

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.