Parametric inference for discretely observed non-ergodic diffusions

Abstract

We consider a multidimensional diffusion process $X$ whose drift and diffusion coefficients depend respectively on a parameter $\lambda$ and $\theta$. This process is observed at $n+1$ equally spaced times $0,{\Delta }_n,2{\Delta }_n,\dots ,n{\Delta }_n$, and $T_n=n{\Delta }_n$ denotes the length of the `observation window'. We are interested in estimating $\lambda$ and/or $\theta$. Under suitable smoothness and identifiability conditions, we exhibit estimators ${\hat{\lambda }}_n$ and ${\hat{\theta }}_n$, such that the variables $\sqrt{n}.({\hat{\theta }}_n-\theta )$ and $\sqrt{T_n}({\hat{\lambda }}_n-\lambda )$ are tight for ${\Delta }_n\to 0$ and $T_n\to \mathrm{\infty }$. When $\lambda$ is known, we can even drop the assumption that $T_n\to \mathrm{\infty }$. These results hold without any kind of ergodicity or even recurrence assumption on the diffusion process.