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This paper deals with a projection least squares 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 acontinuous and square integrable martingale vanishing at $0$. Risk bounds areestablished on this estimator, on an associated adaptive estimator and on anassociated discrete-time version used in practice. An appropriatetransformation 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 ofHurst parameter $H\in [1/2,1)$, as a model of the previous type. So, the secondpart of the paper deals with risk bounds on a nonparametric estimator of $b_0$derived from the results on the projection least squares estimator of $J_0$. Inparticular, our results apply to the estimation of the drift function in anon-autonomous Black-Scholes model and to nonparametric estimation in anon-autonomous fractional stochastic volatility model.

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