## Sharp adaptive estimation of the drift function for ergodic diffusions.(English)Zbl 1084.62079

From the paper: We consider the statistical problem of estimating the drift function of a diffusion process $$X$$, given as the solution of the stochastic differential equation $dX_t= S(X_t)dt+\sigma(X_t) dW_t,\quad X_0=\xi,\;t\geq 0,$ where $$W$$ is a standard Brownian motion and the initial value $$\xi$$ is a random variable independent of $$W$$. We assume that a continuous record of observations $$X^T=(X_t,0\leq t\leq T)$$ is available. The goal is to estimate the function $$S(\cdot)$$, which is commonly referred to as the drift function and is interpreted as the instantaneous mean of the process $$X$$.
The global estimation problem of the drift function is considered for ergodic diffusion processes. The unknown drift $$S(\cdot)$$ is supposed to belong to a nonparametric class of smooth functions of order $$k\geq 1$$, but the value of $$k$$ is not known to the statistician. A fully data-driven procedure of estimating the drift function is proposed, using the estimated risk minimization method. The sharp adaptivity of this procedure is proven up to an optimal constant, when the quality of the estimation is measured by the integrated squared error weighted by the square of the invariant density.

### MSC:

 62M05 Markov processes: estimation; hidden Markov models 62G07 Density estimation 62G20 Asymptotic properties of nonparametric inference
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