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).

翻译：本文涉及一个对函数的预测最小平方估计值 $0 J_ 0美元,该函数的预测最小估计值来自对 $[0,T]$的多重独立观察,该函数由美元=J_0(t)d\langle Mrcle_t+dM_t_t$t美元定义,美元为美元是一个中心、连续和可平方分辨的马丁格列以0美元消失的模型。该估计值和相关的适应估计值设定了风险界限。适当的转换允许重写对 $X_t=V(X_t)(b_0)(t)t = ⁇ sgmam(t)dB_t美元定义的差方程方程方程,其中美元是赫斯特参数($H) / in (1.2,1美元) 的分数的布朗运动。因此,该文件的第二部分涉及对一个非准数估计值为$b_0美元的风险约束值,这是根据对 $_0美元(X_t) (b_0) (t) a dd) t) t = {s\ disgraphalisalisimal 函数的预测值的最小估测算结果。具体应用了我们推算结果。