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On a Computable Skorokhod's Integral Based Estimator of the Drift Parameter in Fractional SDE

Abstract

This paper deals with a Skorokhod's integral based least squares type estimator θ^N\widehat\theta_N of the drift parameter θ0\theta_0 computed from NNN\in\mathbb N^* copies X1,,XNX^1,\dots,X^N of the solution XX to dXt=θ0b(Xt)dt+σdBtdX_t =\theta_0b(X_t)dt +\sigma dB_t, where BB is a fractional Brownian motion of Hurst index H[1/2,1)H\in [1/2,1). On the one hand, a risk bound is established on θ^N\widehat\theta_N when H=1/2H = 1/2 and X1,,XNX^1,\dots,X^N are dependent copies of XX. On the other hand, when H>1/2H > 1/2, Skorokhod's integral based estimators as θ^N\widehat\theta_N cannot be computed directly from data, but in this paper some convergence results are established on a computable approximation of θ^N\widehat\theta_N when X1,,XNX^1,\dots,X^N are independent.

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