Joint estimation for volatility and drift parameters of ergodic jump diffusion processes via contrast function

In this paper we consider an ergodic diffusion process with jumps whose drift coefficient depends on μ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu $$\end{document} and volatility coefficient depends on σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}, two unknown parameters. We suppose that the process is discretely observed at the instants (tin)i=0,…,n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(t^n_i)_{i=0,\ldots ,n}$$\end{document} with Δn=supi=0,…,n-1(ti+1n-tin)→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _n=\sup _{i=0,\ldots ,n-1} (t^n_{i+1}-t^n_i) \rightarrow 0$$\end{document}. We introduce an estimator of θ:=(μ,σ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta :=(\mu , \sigma )$$\end{document}, based on a contrast function, which is asymptotically gaussian without requiring any conditions on the rate at which Δn→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _n \rightarrow 0$$\end{document}, assuming a finite jump activity. This extends earlier results where a condition on the step discretization was needed (see Gloter et al. in Ann Stat 46(4):1445–1480, 2018; Shimizu and Yoshida in Stat Inference Stoch Process 9(3):227–277, 2006) or where only the estimation of the drift parameter was considered (see Amorino and Gloter in Scand J Stat 47:279–346, 2019. https://doi.org/10.1111/sjos.12406). In general situations, our contrast function is not explicit and in practise one has to resort to some approximation. We propose explicit approximations of the contrast function, such that the estimation of θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta $$\end{document} is feasible under the condition that nΔnk→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\Delta _n^k \rightarrow 0$$\end{document} where k>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k>0$$\end{document} can be arbitrarily large. This extends the results obtained by Kessler (Scand J Stat 24(2):211–229, 1997) in the case of continuous processes.