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Pré-Publication, Document De Travail Année : 2020

PROBABILISTIC PROPERTIES AND PARAMETRIC INFERENCE OF SMALL VARIANCE NONLINEAR SELF-STABILIZING STOCHASTIC DIFFERENTIAL EQUATIONS

Résumé

We consider a process (Xt) solution of a one-dimensional nonlinear self-stabilizing stochastic differential equation, with classical drift term V (α, x) depending on an unknown parameter α, self-stabilizing term Φ(β, x) depending on another unknown parameter β and small noise amplitude ε. Self-Stabilization is the effect of including a mean-field interaction in addition to the state-dependent drift. Adding this term leads to a nonlinear or Mac Kean-Vlasov Markov process with transitions depending on the distribution of Xt. We study the probabilistic properties of (Xt) as ε tends to 0 and exhibit a Gaussian approximating process for (Xt). Next, we study the estimation of (α, β) from a continuous observation of (Xt, t ∈ [0, T ]). We build explicit estimators using an approximate log-likelihood function obtained from the exact log-likelihood function of a proxi-model. We prove that, for fixed T , as ε tends to 0, α can be consistently estimated with rate ε −1 but not β. Then, considering n i.i.d. sample paths (X i t , i = 1,. .. , n), we build consistent and asymptotically Gaussian estimators of (α, β) with rates √ nε −1 for α and √ n for β. Finally, we prove that the statistical experiments generated by (Xt) and the proxi-model are asymptotically equivalent in the sense of the Le Cam ∆-distance both for the continuous observation of one path and for n i.i.d. paths under the condition √ nε → 0, which justifies our statistical method.
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Dates et versions

hal-02955171 , version 1 (01-10-2020)
hal-02955171 , version 2 (05-01-2021)

Identifiants

  • HAL Id : hal-02955171 , version 1

Citer

Valentine Genon-Catalot, Catherine Laredo. PROBABILISTIC PROPERTIES AND PARAMETRIC INFERENCE OF SMALL VARIANCE NONLINEAR SELF-STABILIZING STOCHASTIC DIFFERENTIAL EQUATIONS. 2020. ⟨hal-02955171v1⟩
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