| Abstract In this article, our aim is to estimate the successive derivatives of the stationary density $f$ of a strictly stationary and $ \beta$-mixing process$ \left(X_{t}\right)_{t\geq0}$. This process is observed at discrete times $t=0,\Delta,\ldots,n\Delta$. The sampling interval $\Delta$ can be fixed or small. We use a penalized least-square approach to compute adaptive estimators. If the derivative $f^{(j)}$ belongs to the Besov space $\mathcal{B}_{2,\infty}^{\alpha}$, then our estimator converges at rate $\left(n\Delta\right)^{-\alpha/(2\alpha+2j+1)}$. Then we consider a diffusion with known diffusion coefficient. We use the particular form of the stationary density to compute an adaptive estimator of its first derivative $f'$. When the sampling interval $\Delta$ tends to 0, and when the diffusion coefficient is known, the convergence rate of our estimator is $\left(n\Delta\right)^{-\alpha/(2\alpha+1)}$. When the diffusion coefficient is known, we also construct a quotient estimator of the drift for low-frequency data. |
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