A central limit theorem for normalized functions of the increments of a diffusion process, in the presence of round-off errors

Abstract

Let $X$ be a one-dimensional diffusion process. For each $n\ge 1$ we have a round-off level ${\alpha }_n>0$ and we consider the rounded-off value $X_t^{\left(\alpha \right)}=\alpha [X_t/\alpha ]$. We are interested in the asymptotic behaviour of the processes $U(n,\phi {)}_t=\frac{1}{n}{\sum }_{1\le i\le \left[\mathrm{nt}\right]}\phi (X_{(i-1)/n}^{\left(\alpha \right)},\sqrt{n}(X_{i/n}^{\left(\alpha \right)}-X_{(i-1)/n}^{\left(\alpha \right)}))$ as $n$ goes to $+\mathrm{\infty }$: under suitable assumptions on $\phi$, and when the sequence $\alpha \sqrt{n}$ goes to a limit $\beta \in [0,\mathrm{\infty })$, we prove the convergence of $U(n,\phi )$ to a limiting process in probability (for the local uniform topology), and an associated central limit theorem. This is motivated mainly by statistical problems in which one wishes to estimate a parameter occurring in the diffusion coefficient, when the diffusion process is observed at times $i/n$ and is subject to rounding off at some level ${\alpha }_n$ which is 'small' but not 'very small'.