Parameter Inference for Hypo-Elliptic Diffusions under a Weak Design Condition
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866929482941595648 |
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| author | Iguchi, Yuga Beskos, Alexandros |
| author_facet | Iguchi, Yuga Beskos, Alexandros |
| contents | 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 $Δ=Δ_n$ and their total number $n$ satisfy $n \to \infty$, $n Δ_n \to \infty$, $Δ_n \to 0$, and additionally $Δ_n = o (n^{-1/2})$. 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 $Δ_n = o(n^{-1/p})$ for general $p \ge 2$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_04444 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Parameter Inference for Hypo-Elliptic Diffusions under a Weak Design Condition Iguchi, Yuga Beskos, Alexandros Statistics Theory Methodology 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 $Δ=Δ_n$ and their total number $n$ satisfy $n \to \infty$, $n Δ_n \to \infty$, $Δ_n \to 0$, and additionally $Δ_n = o (n^{-1/2})$. 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 $Δ_n = o(n^{-1/p})$ for general $p \ge 2$. 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. |
| title | Parameter Inference for Hypo-Elliptic Diffusions under a Weak Design Condition |
| topic | Statistics Theory Methodology |
| url | https://arxiv.org/abs/2312.04444 |