Nonparametric estimation for i.i.d. paths of a martingale-driven model with application to non-autonomous financial models

This paper deals with a projection least squares estimator of the function J 0 $J_{0}$ computed from multiple independent observations on [ 0 , T ] $[0,T]$ of the process Z $Z$ defined by d Z t = J 0 ( t ) d 〈 M 〉 t + d M t $dZ_{t} = J_{0}(t)d\langle M\rangle _{t} + dM_{t}$ , where M $M$ is a continuous square-integrable martingale vanishing at 0. Risk bounds are established for this estimator, an associated adaptive estimator and an associated discrete-time version used in practice. An appropriate transformation allows us to rewrite the differential equation d X t = V ( X t ) ( b 0 ( t ) d t + σ ( t ) d B t ) $dX_{t} = V(X_{t})(b_{0}(t)dt +\sigma (t)dB_{t})$ , where B $B$ is a fractional Brownian motion with Hurst parameter H ∈ [ 1 / 2 , 1 ) $H\in [1/2,1)$ , as a model of the previous type. The second part of the paper deals with risk bounds for a nonparametric estimator of b 0 $b_{0}$ derived from the results on the projection least squares estimator of J 0 $J_{0}$ . 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.