Nongaussian linear filtering, identification of linear systems, and the symplectic group

Abstract

Consider stochastic linear dynamical systems, dx=Axdl+Bdw,dy=Cxdt+d,,y(O)=O,x(O) a given initialrandom variable independent of the standard independent Wiener noise processes w, v. The matrices A, 8, C are supposed to be constant. In this paper I consider two problems. For the first one A, 8 and C are supposed known and the question is how to calculate the conditional probability density of x at time t given the observations y(s),O<s<I in the case that x(O) is not necessarily gaussian. (In the gaussian case the answer is given by the Kalman-Bucy filter). The second problem concerns identification, i.e. the A. 8, C are unknown (but assumed constant so that dA=O,d8=0,dC=O), and one wants to calculate the joint conditional probability density at time t of (x,A,8,C), again given the obselvations y(s),O..;s,.;;t. The methods used rely on Wei-Norman theory, the Duncan-Mortensen-Zakai equation and a "real form" of the SegalShale-Weil representation of the symplectic group Spn(R).

Finite dimensional solutions to SPDEs and the geometry of infinite jet bundles

“…When the filtering problem can be reduced to a finite dimensional process we speak of finite dimensional filters. Examples of such filters are the Kalman filter, the Benes filter and the related ones (see [1,5,34]). In many cases it is possible to reduce the problem of finding and studying finite dimensional filters to the problem of calculating finite dimensional solutions to particular SPDEs.…”

“…stochastic filters described by a finite number of parameters such as the Kalman filter or the Benes filter (see [5]). This problem is equivalent to studying the finite dimensional solutions to a special linear SPDE called Zakai equation (see the classical works [1,7,9,34,59] and the papers of Cohen De Lara [11,12]). A second application of the problem of finding finite dimensional solutions to SPDEs is the Heath-Jarrow-Morton (HJM) model used in mathematical finance for describing the interest rate (see [35]).…”