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

7 December 2023
Yuga Iguchi
A. Beskos
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Abstract

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 Δ=Δn\Delta=\Delta_nΔ=Δn​ and their total number nnn satisfy n→∞n \to \inftyn→∞, nΔn→∞n \Delta_n \to \inftynΔn​→∞, Δn→0\Delta_n \to 0Δn​→0, and additionally Δn=o(n−1/2)\Delta_n = o (n^{-1/2})Δn​=o(n−1/2). 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 Δn=o(n−1/p)\Delta_n = o(n^{-1/p})Δn​=o(n−1/p) for general p≥2p \ge 2p≥2. 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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