Deterministic Estimation and Asymptotic Stochastic Estimation

Abstract

Let t → x(t) be a state process and consider observations t → y(t) of a signal t → h(x(t)) in the presence of additive white noise ẏ = h(x) + v. The stochastic filter is the map that associates to each observation record y(τ), 0 ≤ τ ≤ t, the conditional mean E( ∅ (x(t)) ∣ y(τ), 0 ≤ τ ≤ t). In this paper it is shown that the output of the stochastic filter converges to the output of what is known as the deterministic filter as the variances of the impinging noises go to zero, exactly in analogy with the physical fact that quantum mechanics converges to classical mechanics as Plank’s constant goes to zero.

The contents of this paper appear in the author’s Ph.D. thesis writtem under the direction of A. J. Krener.