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

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