Parameter Inference for Hypo-Elliptic Diffusions under a Weak Design Condition

We address the problem of parameter estimation for degenerate diffusion processes defined via the solution of Stochastic Differential Equations (SDEs) with diffusion matrix that is not full-rank. For this class of hypo-elliptic diffusions recent works have proposed contrast estimators that are asymptotically normal, provided that the step-size in-between observations and their total number satisfy , , , and additionally . This latter restriction places a requirement for a so-called `rapidly increasing experimental design'. In this paper, we overcome this limitation and develop a general contrast estimator satisfying asymptotic normality under the weaker design condition for general . Such a result has been obtained for elliptic SDEs in the literature, but its derivation in a hypo-elliptic setting is highly non-trivial. We provide numerical results to illustrate the advantages of the developed theory.
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