New explicit filters and smoothers for diffusions with nonlinear drift and measurements

Abstract

The optimal least-squares filtering of a diffusion x(t) from its noisy measurements {y(τ); 0 ≤ τ ≤ t} is given by the conditional mean E[x(t)\y(τ); 0 ≤ τ ≤ t]. When x(t) satisfies the stochastic diffusion equation dx(t) = f(x(t))dt + dw(t) and y(t) = ∫t0x(s)ds + b(t), where f(·) is a global solution of the Riccati equation ∂/∂xf(x) + f(x)2 = αx2 + βx + γ, for some (α,β, γ) ∈ ℛ3, and w(·), b(·) are independent Brownian motions, Benes gave an explicit formula for computing the conditional mean. This paper extends Benes results to measurements y(t) = ∫t0x(s)ds + ∫t0 dx(s) + b(t) (and its multidimensional version) without imposing additional conditions on f(·). Analogous results are also derived for the optimal least-squares smoothed estimate E[x(s)|y(τ); 0 ≤ τ ≤ t], s < t. The methodology relies on Girsanov's measure transformations, gauge transformations, function space integrations, Lie algebras, and the Duncan-Mortensen-Zakai equation. © 1998 Elsevier Science B.V. All rights reserved.