Nonparametric Estimation for I.I.D. Paths of a Martingale Driven Model with Application to Non-Autonomous Fractional SDE

This paper deals with a projection least square estimator of the function $J_0$ computed from multiple independent observations on $[0,T]$ of the process $Z$ defined by $dZ_t = J_0(t)d\langle M\rangle_t + dM_t$, where $M$ is a centered, continuous and square integrable martingale vanishing at $0$. Risk bounds are established on this estimator and on an associated adaptive estimator. An appropriate transformation allows to rewrite the differential equation $dX_t = V(X_t)(b_0(t)dt +\sigma(t)dB_t)$, where $B$ is a fractional Brownian motion of Hurst parameter $H\in (1/2,1)$, as a model of the previous type. So, the second part of the paper deals with risk bounds on a nonparametric estimator of $b_0$ derived from the results on the projection least square estimator of $J_0$. In particular, our results apply to the estimation of the drift function in a non-autonomous extension of the fractional Black-Scholes model introduced in Hu et al. (2003).