Asymptotic equivalence of nonparametric diffusion and Euler scheme experiments

TL;DR

This paper establishes that, under certain conditions, observing a nonparametric diffusion process is asymptotically equivalent to observing its Euler scheme, validating the use of the simpler Euler scheme for statistical inference.

Contribution

The paper proves a global asymptotic equivalence between nonparametric diffusion experiments and their Euler scheme approximations, using explicit mappings based on random time changes.

Findings

Asymptotic equivalence holds for diffusions with known diffusion coefficient.

Euler scheme can be used as a valid approximation for inference.

Explicit mappings facilitate practical implementation of the equivalence.

Abstract

We prove a global asymptotic equivalence of experiments in the sense of Le Cam's theory. The experiments are a continuously observed diffusion with nonparametric drift and its Euler scheme. We focus on diffusions with nonconstant-known diffusion coefficient. The asymptotic equivalence is proved by constructing explicit equivalence mappings based on random time changes. The equivalence of the discretized observation of the diffusion and the corresponding Euler scheme experiment is then derived. The impact of these equivalence results is that it justifies the use of the Euler scheme instead of the discretized diffusion process for inference purposes.