Therepresentation theorem

Abstract

Representing the solutions of partial differential equations by integrals over function space has been done for various problems. In quantum mechanics, Feynman solved the Schroedinger equation in this way, see [5].Mark Kac used integrals over Brownian paths to represent solutions of a generalized Fokker-Planck equation with particle birth and death, see [7]. Function space integrals with respect to Brownian paths have been consideted by Wiener, [12] and Friedricks, [6].Ito in [8],introduced path descriptions of Markov diffusion processes, stochastic differential equations, with these processes having probability distributions satisfying generalized Fokker-Planck equations. In the early 6O’s, Stratonovich derived a non-linear partial differential driven by stochastic term governing the evolution of the conditional density of the signal given the observations for the nonlinear filtering problem, see [10]. Controversy arose over the form of the stochastic driving term in this equation which hinged on the stochastic calculus used.In [1], Bucy proposed a solution to a version of the Stratonovich partial differential equation valid for the Ito calculus as a function space integral with respect to the signal process paths.This result was known as the representation theorem. Proofs were given for the representation theorem in [2],[3] and [9]. Duncan in his thesis, [4], resolved the statistical testing problem for processes using the representation theorem.This theorem was used to synthesize nonlinear filters with digital computers, see [2].In this paper, we will derive the discrete time version of the representation theorem. It is interesting to do this, as the details are less technical than in the continuous time case.Further some interesting connections with Statistical Mechanics are apparent when this is done. Integrability conditions for systems of partial differential equations are used to characterize the solution of the nonlinear filtering problem. In the special case of the linear gaussian filtering problem, this characterization coincides with the Krein-Bellman equation.