Nonparametric adaptive estimation for integrated diffusions

Let (Vt)(Vt) be a stationary and ββ-mixing diffusion with unknown drift and diffusion coefficient. The integrated process Xt=∫0tVsds is observed at discrete times with regular sampling interval Δ. For both the drift function and the diffusion coefficient of the unobserved diffusion (Vt)(Vt), we build nonparametric adaptive estimators based on a penalized least square approach. We derive risk bounds for the estimators. Interpreting these bounds through the asymptotic framework of high frequency data, we show that our estimators reach the minimax optimal rates of convergence, under some constraints on the sampling interval. The algorithms of estimation are implemented for several examples of diffusion models.