Laboratoire de Mathématiques Raphaël Salem

UMR 6085 CNRS - Université de Rouen Normandie

We consider the solution $X = (X_t)_{t\geq 0}$ of a multivariate stochastic differential equation with Levy-type jumps and with unique invariant probability measure with density $\pi$. We assume that a continuous record of observations $X_T=(X_t )_{0\leq t\leq T}$ is available. In the case without jumps, Dalalyan and Reiss [1] and Strauch [2] have found convergence rates of invariant density estimators, under respectively isotropic and anisotropic Hölder smoothness constraints, which are considerably faster than those known from standard multivariate density estimation. We extend the previous works by obtaining, in presence of jumps, some estimators which achieve  convergence rates faster than the ones found by Strauch [2] for $d\geq 3$ and a rate which depends on the degree of the jumps in the one-dimensional setting. Moreover, we obtain a minimax lower bound on the $L²$-risk for pointwise estimation, with the same rate up to a $\log(T)$ term. It implies that, on a class of diffusions whose invariant density belongs to the anisotropic Holder class we are considering, it is impossible to find an estimator with a rate of estimation faster than the one we propose.

References
[1] Dalalyan, A. and Reiss, M. (2007). Asymptotic statistical equivalence for ergodic diffusions: the multidimensional case. Probab. Theory Relat. Fields, 137(1), 25–47.
[2] Strauch, C. (2018). Adaptive invariant density estimation for ergodic diffusions over anisotropic classes. The Annals of Statistics, 46(6B), 3451-3480.