Stochastic PDE Projection on Manifolds: Assumed-Density and Galerkin Filters

We review the manifold projection method for stochastic nonlinear filtering in a more general setting than in our previous paper in Geometric Science of Information 2013. We still use a Hilbert space structure on a space of probability densities to project the infinite dimensional stochastic partial differential equation for the optimal filter onto a finite dimensional exponential or mixture family, respectively, with two different metrics, the Hellinger distance and the L direct metric. This reduces the problem to finite dimensional stochastic differential equations. In this paper we summarize a previous equivalence result between Assumed Density Filters (ADF) and Hellinger/Exponential projection filters, and introduce a new equivalence between Galerkin method based filters and Direct metric/Mixture projection filters. This result allows us to give a rigorous geometric interpretation to ADF and Galerkin filters. We also discuss the different finite-dimensional filters obtained when projecting the stochastic partial differential equation for either the normalized (Kushner-Stratonovich) or a specific unnormalized (Zakai) density of the optimal filter. 1 The filtering problem in continuous time The state of a system X evolves over time according to some stochastic process driven by a noise W . We cannot observe the state directly but we make an imperfect measurement Y which is also perturbed stochastically by random noise V . In a diffusion setting this problem is formulated as dXt = ft(Xt) dt+ σt(Xt) dWt, X0, dYt = bt(Xt) dt+ dVt, Y0 = 0 . (1) In these equations the unobserved state process {Xt, t ≥ 0} takes values in R, the observation {Yt, t ≥ 0} takes values in R and the noise processes {Wt, t ≥ 0} and {Vt, t ≥ 0} are two Brownian motions. The nonlinear filtering problem consists in finding the conditional probability distribution πt of the state Xt given the observations up to time t and the prior distribution π0 for X0. Let us assume that X0, and the two Brownian motions are independent. Let us also assume that the covariance matrix for Vt is invertible. We can then assume without any further loss of generality that its covariance matrix is the identity. We introduce a variable at defined by at = σtσ T t . With these preliminaries, and a number of rather more technical conditions for which we refer to [9], one can show that πt satisfies the Kushner–Stratonovich equation. We further suppose that the measure πt is determined by a probability density pt. A formal calculation then gives the following Stratonovich calculus version of the optimal filter stochastic PDE (SPDE) for the evolution of p: dpt = Lt pt dt− 1 2 pt [|bt| − Ept{|bt|}] dt+ d ∑ k=1 pt [b k t − Ept{bt }] ◦ dY k t . (2) We use Stratonovich calculus because we need the formal chain rule to hold when identifying the projected evolution from the projected right hand side of the equation, as we hint below after Equation (7). Here L∗ is the formal adjoint of L – the so-called forward diffusion operator for X, where the backward diffusion operator is defined by: