Abstract
In this paper we study the problem of parametric inference for multidimensional diffusions based on observations at random stopping times. We work in the asymptotic framework of high frequency data over a fixed horizon. Previous works on the subject (such as [10, 17, 19, 5] among others) consider only deterministic, strongly predictable or random, independent of the process, observation times, and do not cover our setting. Under mild assumptions we construct a consistent sequence of estimators, for a large class of stopping time observation grids (studied in [20, 23]). Further we carry out the asymptotic analysis of the estimation error and establish a Central Limit Theorem (CLT) with a mixed Gaussian limit. In addition, in the case of a 1-dimensional parameter, for any sequence of estimators verifying CLT conditions without bias, we prove a uniform a.s. lower bound on the asymptotic variance, and show that this bound is sharp.
Highlights
In this work we study the problem of parametric inference for a d-dimensional Brownian semimartingale (St)0≤t≤T of the form t t
Our goal is to estimate ξ using these discrete observations and study the asymptotic properties of the estimator sequence as the number of observations goes to infinity; we work in the highfrequency fixed horizon setting
We provide an asymptotic analysis that allows to directly apply the existing results of [21] on Central Limit Theorem (CLT) for discretization errors and show the convergence in distribution of the renormalized error NTn(ξn − ξ ) to an explicitly defined mixture of normal variables
Summary
To the best of our knowledge this setting has not yet been studied in the literature, except in [30] where a Central Limit Theorem (CLT) for estimating the integrated volatility in dimension 1 is established assuming the convergence in probability of renormalized quarticity and tricity (the authors do not characterize the stopping times for which these convergences hold). In [17] the authors consider the problem of the parametric estimation of a multidimensional diffusion under regular deterministic observation grids They construct consistent sequences of estimators of the unknown parameter based on the minimization of certain c√ontrasts and prove the weak convergence of the error renormalized at the rate n to a mixed Gaussian variable, where n is the number of observations.
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