On estimation of the Hurst index of solutions of stochastic integral equations

Abstract

Let X be a solution of a stochasti Let X be a solution of a stochastic integral equation driven by a fractional Brownian motion B_H and let V_n(X, 2) = \sumⁿ _k=1(\Delta_kX)², where \Delta_kX = X( k+1/n ) - X(k/n ). We study the

ditions n^2H-1V_n(X, 2) convergence almost surely as n → ∞ holds. This fact is used to obtain a strongly consistent estimator of the Hurst index H, 1/2 < H < 1.