On partially observed jump diffusions I. The filtering equations

Abstract

This paper is the first part of a series of papers on filtering for partially observed jump diffusions satisfying a stochastic differential equation driven by Wiener processes and Poisson martingale measures. The coefficients of the equation only satisfy appropriate growth conditions. Some results in filtering theory of diffusion processes are extended to jump diffusions and equations for the time evolution of the conditional distribution and the unnormalised conditional distribution of the unobserved process at time t, given the observations until t, are presented.

On partially observed jump diffusions II. The filtering density

“…Remark 3.2. We also recall from [4] that there exists a cadlag F Y t -adapted positive process p o γ t q tPr0,T s , the optional projection of pγ t q tPr0,T s under P with respect to pF Y t q tPr0,T s , such that for every F Y t -stopping time τ ď T we have…”

“…As in [4] we consider a partially observed jump diffusion Z " pX t , Y t q tPr0,T s satisfying the system of stochastic differential equations dX t " bpt, Z t qdt `σpt, Z t qdW t `ρpt, Z t qdV t `żZ 0 ηpt, Z t´, zq Ñ0 pdz, dtq `żZ 1 ξpt, Z t´, zq Ñ1 pdz, dtq, dY t " Bpt, Z t qdt `dV t `żZ 1 z Ñ1 pdz, dtq,…”

“…In [4] we were interested in the equations for the evolution of the conditional distribution P t pdxq " P pX t P dx|Y s , s ď tq of the unobserved component X t given the observations pY s q sPr0,T s . Our aim in the present paper is to show, under fairly general conditions, that if the conditional distribution of X 0 given Y 0 has a density π 0 , such that it is almost surely in L p for some p ě 2, then X t for every t has a conditional density π t given pY s q tPr0,ts , which belongs also to L p , almost surely for all t. In a subsequent paper we investigate the regularity properties of the conditional density.…”

“…For a brief account on the filtering problem and on the history of the presentation of the filtering equations for partially observed diffusion processes we refer to [9]. Concerning the filtering equations associated to (1.1) we refer the reader to [4] and the references therein.…”

“…In section 2 we formulate our main result. In section 3 we recall important results from [4] together with the filtering equations obtained therein. In section 4 we prove L p estimates needed for a priori bounds for the smoothed conditional distribution.…”