We consider a diffusion model of small variance type with positive drift density varying in a nonparametric set. We investigate Gaussian and Poisson approximations to this model in the sense of asymptotic equivalence of experiments. It is shown that observation of the diffusion process until its first hitting time of level one is a natural model for the purpose of inference on the drift density. The diffusion model can be discretized by the collection of level crossing times for a uniform grid of levels. The random time increments are asymptotically sufficient and obey a nonparametric regression model with independent data. This decoupling is then used to establish asymptotic equivalence to Gaussian signal-in-white-noise and Poisson intensity models on the unit interval, and also to an i.i.d. model when the diffusion drift function $f$ is a probability density. As an application, we find the exact asymptotic minimax constant for estimating the diffusion drift density with sup-norm loss.
"Asymptotic equivalence of estimating a Poisson intensity and a positive diffusion drift." Ann. Statist. 30 (3) 731 - 753, June 2002. https://doi.org/10.1214/aos/1028674840