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

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.

Appendix: Background on Le Cam’s theory

1.1 Asymptotic equivalence in the sense of Le Cam

A statistical model is a triplet $\fancyscript{P}_j=(\fancyscript{X}_j,\fancyscript{A}_j,\{P_{j,\theta }; \theta \in \Theta \})$ where $\{P_{j,\theta }; \theta \in \Theta \}$ is a family of probability distributions all defined on the same $\sigma$-field $\fancyscript{A}_j$ over the sample space $\fancyscript{X}_j$ and $\Theta$ is the parameter space. The deficiency $\delta (\fancyscript{P}_1,\fancyscript{P}_2)$ of $\fancyscript{P}_1$ with respect to $\fancyscript{P}_2$ quantifies “how much information we lose” by using $\fancyscript{P}_1$ instead of $\fancyscript{P}_2$ and is defined as $\delta (\fancyscript{P}_1,\fancyscript{P}_2)=\inf _K\sup _{\theta \in \Theta }||KP_{1,\theta }-P_{2,\theta }||_{TV},$ where TV stands for “total variation” and the infimum is taken over all “transitions” $K$ (see Le Cam (1986), page 18). In our setting, however, the general notion of “transitions” can be replaced with the notion of Markov kernels. Indeed, when the model $\fancyscript{P}_1$ is dominated and the sample space $(\fancyscript{X}_2,\fancyscript{A}_2)$ of the experiment $\fancyscript{P}_2$ is a Polish space, the infimum appearing on the definition of the deficiency $\delta$ can be taken over all Markov kernels $K$ on $\fancyscript{X}_1\times \fancyscript{A}_2$ (see Nussbaum (1996), Proposition 10.2), i.e.

$$\begin{aligned} \delta (\fancyscript{P}_1,\fancyscript{P}_2)=\inf _K\sup _{\theta \in \Theta }\sup _{A\in \fancyscript{A}_2}\bigg |\int _{\fancyscript{X}_1}K(x,A)P_{1,\theta }(dx)-P_{2,\theta }(A)\bigg |. \end{aligned}$$

(18)

The experiment $KP_{1,\theta }=(\fancyscript{X}_1,\fancyscript{A}_1,\{KP_{1,\theta }\}_{\theta \in \Theta })$ is called a randomization of $\fancyscript{P}_1$ by the kernel $K$. If the kernel is deterministic, i.e. for $T:(\fancyscript{X}_1,\fancyscript{A}_1)\rightarrow (\fancyscript{X}_2,\fancyscript{A}_2)$ a random variable, , the experiment $T\fancyscript{P}_1$ is called the image experiment by the random variable $T$. Closely associated with the notion of deficiency is the so called $\Delta$-distance, i.e. the pseudo metric defined by:

$$\begin{aligned} \Delta (\fancyscript{P}_1,\fancyscript{P}_2):=\max (\delta (\fancyscript{P}_1,\fancyscript{P}_2),\delta (\fancyscript{P}_2,\fancyscript{P}_1)). \end{aligned}$$

The sufficiency of a statistic can be expressed in terms of the $\Delta$-distance. More precisely, the following holds (see Genon-Catalot and Laredo (2014), Proposition 8.1, page 23). Let $T:(\fancyscript{X}_1,\fancyscript{A}_1)\rightarrow (\fancyscript{X}_2,\fancyscript{A}_2)$ be a random variable. The statistic $T$ is sufficient for $\fancyscript{P}_1$ if and only if $\Delta (\fancyscript{P}_1,T\fancyscript{P}_1)=0.$

Also, remark that thanks to (18), if $\fancyscript{P}_1=(\fancyscript{X},\fancyscript{A}_1,\{P_\theta ;\theta \in \Theta \})$ and $\fancyscript{P}_2=(\fancyscript{X},\fancyscript{A}_2,\{P_\theta ;\theta \in \Theta \})$ with $\fancyscript{A}_2\subset \fancyscript{A}_1$, then $\delta (\fancyscript{P}_1,\fancyscript{P}_2)=0$.

Two sequences of statistical models $(\fancyscript{P}_{1}^n)_{n\in \mathbb {N}}$ and $(\fancyscript{P}_{2}^n)_{n\in \mathbb {N}}$ are called asymptotically equivalent if $\Delta (\fancyscript{P}_{1}^n,\fancyscript{P}_{2}^n)$ tends to zero as $n$ goes to infinity. Similarly, the statistic $T^n$ is asymptotically sufficient for $\fancyscript{P}_1^n$ if $\Delta (\fancyscript{P}_{1}^n,T^n\fancyscript{P}_{1}^n)$ tends to zero as $n$ goes to infinity.