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Joint estimation for volatility and drift parameters of ergodic jump diffusion processes via contrast function. (English) Zbl 1469.62324

Summary: In this paper we consider an ergodic diffusion process with jumps whose drift coefficient depends on \(\mu\) and volatility coefficient depends on \(\sigma \), two unknown parameters. We suppose that the process is discretely observed at the instants \((t^n_i)_{i=0,\dots ,n}\) with \(\Delta_n=\sup_{i=0,\dots ,n-1} (t^n_{i+1}-t^n_i) \rightarrow 0\). We introduce an estimator of \(\theta:=(\mu,\sigma)\), based on a contrast function, which is asymptotically gaussian without requiring any conditions on the rate at which \(\Delta_n \rightarrow 0\), assuming a finite jump activity. This extends earlier results where a condition on the step discretization was needed (see [the second author et al., Ann. Stat. 46, No. 4, 1445–1480 (2018; Zbl 1430.60066); Y. Shimizu and N. Yoshida, Stat. Inference Stoch. Process. 9, No. 3, 227–277 (2006; Zbl 1125.62089)]) or where only the estimation of the drift parameter was considered (see [the authors, Scand. J. Stat. 47, No. 2, 279–346 (2020; Zbl 1450.62108)]). 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 \(\theta\) is feasible under the condition that \(n\Delta_n^k\rightarrow 0\) where \(k> 0\) can be arbitrarily large. This extends the results obtained by M. Kessler [Scand. J. Stat. 24, No. 2, 211–229 (1997; Zbl 0879.60058)] in the case of continuous processes.

MSC:

62M05 Markov processes: estimation; hidden Markov models
60J60 Diffusion processes
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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