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A central limit theorem for jump diffusion processes in the presence of round-off errors
- Part of
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- Journal:
- Advances in Applied Probability , First View
- Published online by Cambridge University Press:
- 25 March 2025, pp. 1-39
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- Article
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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 where [x] is the integer part of the real number x and the sequences
\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*}
$(\Delta_n)$ 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.
