2021/10/29 by Nicolas Marie, Marie, Nicolas · 1 citation
Decision Sciences · Economics, Econometrics and Finance · #Auction Theory and Applications #Complex Systems and Time Series Analysis #FOS: Mathematics #Statistics Theory (math.ST) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2110.15637
openalex publication_date 2021/10/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper deals with a projection least squares estimator of the function J0 computed from multiple independent observations on [0,T] of the process Z defined by dZt = J0(t)d⟨ M⟩t + dMt, where M is a 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 dXt = V(Xt)(b0(t)dt +σ(t)dBt), where B is a fractional Brownian motion of Hurst parameter H∈ [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 b0 derived from the results on the projection least squares estimator of J0. 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.