Nonparametric Estimation for I.I.D. Paths of a Martingale Driven Model with Application to Non-Autonomous Fractional SDE
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This paper deals with a projection least square 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⟨ M⟩_t + dM_t, where M is a centered, continuous and square integrable martingale vanishing at 0. Risk bounds are established on this estimator and on an associated adaptive estimator. An appropriate transformation allows to rewrite the differential equation dX_t = V(X_t)(b_0(t)dt +σ(t)dB_t), 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 b_0 derived from the results on the projection least square estimator of J_0. In particular, our results apply to the estimation of the drift function in a non-autonomous extension of the fractional Black-Scholes model introduced in Hu et al. (2003).
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