PARAMETRIC INFERENCE FOR ERGODIC MCKEAN-VLASOV STOCHASTIC DIFFERENTIAL EQUATIONS
Résumé
We consider a one-dimensional McKean-Vlasov stochastic differential equation with potential and interaction terms depending on unknown parameters. The sample path is continuously observed on a time interval [0, 2T ]. We assume that the process is in stationary regime. As this distribution is not explicit, the exact likelihood does not lead to computable estimators. To overcome this difficulty, we consider a standard kernel estimator of the invariant density based on the sample path on [0, T ] and obtain original properties of this estimator. Then, we derive an explicit approximate likelihood using the sample path on [T, 2T ], including the kernel estimator of the invariant density and study the associated estimators of the unknown parameters. We prove their consistency and asymptotic normality with rate √ T as T grows to infinity. Several classes of models illustrate the theory.
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