Asymptotic equivalence of discretely observed diffusion processes and their Euler scheme: small variance case

TL;DR

This paper proves that scalar diffusion processes with small variance are asymptotically equivalent to nonparametric autoregressive models, facilitating transfer of statistical methods between these models in fixed time horizons.

Contribution

It establishes the first explicit asymptotic equivalence between small-variance diffusion processes and autoregressive models, including cases with non-constant diffusion coefficients.

Findings

Asymptotic equivalence in Le Cam $ riangle$-distance is established.

Explicit mappings for equivalence are constructed.

Results hold for both discrete and continuous observations.

Abstract

This paper establishes the global asymptotic equivalence, in the sense of the Le Cam $\Delta$-distance, between scalar diffusion models with unknown drift function and small variance on the one side, and nonparametric autoregressive models on the other side. The time horizon $T$ is kept fixed and both the cases of discrete and continuous observation of the path are treated. We allow non constant diffusion coefficient, bounded but possibly tending to zero. The asymptotic equivalences are established by constructing explicit equivalence mappings.