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

Abstract

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.