Optimal estimation for semimartingales

Abstract

This paper extends a result of Godambe's theory of parametric estimation for discrete-time stochastic processes to the continuous-time case. Let P ={P} be a family of probability measures such that (Ω, F, P) is complete, (Ft, t≧0) is a standard filtration, and X = (Xt Ft, t ≧ 0) is a semimartingale for every P ∈ P. For a parameter θ (Ρ), suppose Xt = Vt,θ + Ht,θ where the Vθ process is predictable and locally of bounded variation and the Hθ process is a local martingale. Consider estimating equations for θ of the form process is predictable. Under regularity conditions, an optimal form for α θ in the sense of Godambe (1960) is determined. When Vt,θ is linear in θ the optimal , corresponds to certain maximum likelihood or least squares estimates derived previously in special cases. Asymptotic properties of , are discussed.