Joint estimation for volatility and drift parameters of ergodic jump diffusion processes via contrast function
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Joint estimation for volatility and drift parameters of ergodic jump diffusion processes via contrast function Chiara Amorino1
· Arnaud Gloter1
Received: 24 October 2019 / Accepted: 25 August 2020 © Springer Nature B.V. 2020
Abstract In this paper we consider an ergodic diffusion process with jumps whose drift coefficient depends on μ and volatility coefficient depends on σ , two unknown parameters. We suppose that the process is discretely observed at the instants (tin )i=0,...,n with n = n − t n ) → 0. We introduce an estimator of θ := (μ, σ ), based on a contrast supi=0,...,n−1 (ti+1 i function, which is asymptotically gaussian without requiring any conditions on the rate at which n → 0, 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 is feasible under the condition that nkn → 0 where k > 0 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. Keywords Drift estimation · Volatility estimation · Ergodic properties · High frequency data · Lévy-driven SDE · Thresholding methods
1 Introduction Recently, diffusion processes with jumps are becoming powerful tools to model various stochastic phenomena in many areas, for example, physics, biology, medical sciences, social sciences, economics, and so on. In finance, jump-processes were introduced to model the dynamic of exchange rates (Bates 1996), asset prices (Merton 1976; Kou 2002), or volatility processes (Barndorff-Nielsen and Shephard 2001; Eraker et al. 2003). Utilization of jumpprocesses in neuroscience, instead, can be found for instance in Ditlevsen and Greenwood
B 1
Chiara Amorino [email protected] Laboratoire de Mathématiques et Modélisation d’Evry, CNRS, Univ Evry, Université Paris-Saclay, 91037 Evry, France
123
Statistical Inference for Stochastic Processes
(2013). Therefore, inference problems for such models from various types of data should be studied, in particular, inference from discrete observation should be desired since the actual data may be obtained discretely. In this work, our aim is to estimate jointly the drift and the volatility parameter (μ, σ ) =: θ from a discrete sampling of the process X θ solution to X tθ = X 0θ +
t 0
b(μ, X sθ )ds +
0
t
a(σ, X sθ )dWs +
t 0
R\{0}
θ γ (X s− )z μ(ds, ˜ dz),
where W is a one dimensional Brownian motion and μ˜ a compensated Poisson random measure, with a finite jump activity. We assume that the process is sampled at the times
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