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 \(J_{0}\) 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 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 \(dX_{t} = V(X_{t})(b_{0}(t)dt +\sigma (t)dB_{t})\), where \(B\) is a fractional Brownian motion with Hurst parameter \(H\in [1/2,1)\), as a model of the previous type. The second part of the paper deals with risk bounds for a nonparametric estimator of \(b_{0}\) derived from the results on the projection least squares 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.