A central limit theorem for jump diffusion processes in the presence of round-off errors

Abstract Let X be the sum of a diffusion process and a Lévy jump process, and for any integer $n\ge 1$ let $\phi_n$ be a function defined on $\mathbb{R}^2$ and taking values in $\mathbb{R}$ , with adequate properties. We study the convergence of functionals of the type \begin{align*} \Delta_n\sum_{i=1}^{[t/\Delta_n]} \phi_n\!\left(\alpha_n\left[\frac{X_{(i-1)\Delta_n}}{\alpha_n}\right], \:\frac{\alpha_n}{\sqrt{\Delta_n}}\left(\left[\frac{X_{i\Delta_n}}{\alpha_n}\right]-\left[\frac{X_{(i-1)\Delta_n}}{\alpha_n}\right]\right)\right),\end{align*} where [ x ] is the integer part of the real number x and the sequences $(\Delta_n)$ and $(\alpha_n)$ tend to 0 as $n\to +\infty$ . We then prove the law of large numbers and establish, in the case where $\frac{\alpha_n}{\sqrt{\Delta_n}}$ converges to a real number in $[0,+\infty)$] , a new central limit theorem which generalizes that in the case where X is a continuous Itô’s semimartingale.