We study the problem of the nonparametric estimation for the density $\pi$ of the stationary distribution of a $d$-dimensional stochastic differential equation $(X_t)_{ t \in [0, T]}$. From the continuous observation of the sampling path on $[0, T]$, we study the estimation of $\pi(x)$ as $T$ goes to infinity. For $d\ge2$, we characterize the minimax rate for the $\mathbf{ L}^2$-risk in pointwise estimation over a class of anisotropic H\"older functions $\pi$ with regularity $\beta = (\beta_1, ... , \beta_d)$. For $d \ge 3$, our finding is that, having ordered the smoothness such that $\beta_1 \le ... \le \beta_d$, the minimax rate depends on whether $\beta_2 < \beta_3$ or $\beta_2 = \beta_3$. In the first case, this rate is $(\frac{ \log T}{ T})^\gamma$, and in the second case, it is $(\frac{ 1}{ T})^\gamma$, where $\gamma$ is an explicit exponent dependent on the dimension and $\bar{ \beta}_3$, the harmonic mean of smoothness over the $d$ directions after excluding $\beta_1$ and $\beta_2$, the smallest ones. We also demonstrate that kernel-based estimators achieve the optimal minimax rate. Furthermore, we propose an adaptive procedure for both $L^2$ integrated and pointwise risk. In the two-dimensional case, we show that kernel density estimators achieve the rate $\frac{ \log T}{ T}$, which is optimal in the minimax sense. Finally we illustrate the validity of our theoretical findings by proposing numerical results.
Categories: math.ST stat.TH