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A formula for the likelihood ratio of measures generated by solutions of a stochastic differential equation with a fractional Brownian motion is established in the paper. We find sufficient conditions that the family of measures generated by solutions of such an equation is locally asymptotically normal. Introduction We consider the stochastic differential equation (1) Xt = x0 + ∫ t 0 S(θ, u, Xu) du + εBt, t ∈ [0, T ], where x0 ∈ R, ε ∈ (0, 1); S(θ, t, x) : R × [0, T ]×R → R is a nonrandom function of drift; θ ∈ Θ ⊂ R is an unknown parameter of the system; Bt = B t is a fractional Brownian motion with the Hurst parameter H ∈ ( 1 2 , 1). Along with equation (1) we consider the deterministic equation (2) xt = x0 + ∫ t 0 S(θ, u, xu) du, t ∈ [0, T ], whose solution is x = x(θ). Equation (1) describes the evolution of a dynamic system with a small noise being a fractional Brownian motion. The problem of the statistical estimation is well studied for systems with a small noise being a standard Brownian process (see [1]). In particular, the consistency and asymptotic normality of the maximum likelihood estimator of the parameter θ is proved under certain assumptions for systems with Brownian noise. As shown in the monograph [2, Chapter II], several important properties of statistical estimators follow from the local asymptotic normality of a system of measures generated by the random element X θ . Thus the proof of the local asymptotic normality is a necessary step to obtain results similar to the Kutoyants results [1] in the case of a fractional Brownian motion. In this paper, we obtain some conditions under which the family of probability measures {P (ε) θ , θ ∈ Θ} generated by solutions of equation (1) that correspond to different parameters θ in the measurable space ( C[0, T ],BT ) is locally asymptotically normal as ε → 0. 2000 Mathematics Subject Classification. Primary 62F12; Secondary 60G15, 60H10.

theory of probability and mathematical statistics

The local asymptotic normality of a family of measures generated by solutions of stochastic differential equations with a small fractional Brownian motion