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, on an associated adaptive estimator and on an associated discrete time version used in practice. 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 Black-Scholes model and to nonparametric estimation in a non-autonomous fractional stochastic volatility model.
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 。 适当的转换允许重写由美元= J_0 (t) d\langle M\rangle_t + dM_t$, 美元是核心的、连续的和可平方的martingale, 以美元消失。 此估计值、 相关适应估计器和实践中使用的相关离散时间版本设定了风险界限。 适当的转换允许重写 $X_ t = V(X_ t) 美元 定义的差方方方 $Z$ (b_ 0 (t) t) ⁇ s sgmam (t) d_t 美元 美元, 其中$B$是 Hurst 参数的分数布朗运动 $H\ in [2/2, 1] 。 因此, 本文的第二部分涉及风险界限是用一个非参数估测算 $0 的 美元 模型, 从预测结果中推算为 最低平方平方 的模型, 。