The Zakai Equation of Nonlinear Filtering for Jump

Authors: Claudia Ceci Katia Colaneri

Publish Date: 2013/09/13

Volume: 69, Issue: 1, Pages: 47-82

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Abstract

In this paper we study a nonlinear filtering problem for a general Markovian partially observed system XY whose dynamics is modeled by correlated jumpdiffusions having common jump times At any time t∈0T the σalgebra mathcalFY t= sigma Y s sleq t provides all the available information about the signal X t The central goal of stochastic filtering is to characterize the filter π t which is the conditional distribution of X t given the observed data mathcalFY t In Ceci and Colaneri Adv Appl Probab 443678–701 2012 it is proved that π is the unique probability measurevalued process satisfying a nonlinear stochastic equation the socalled KushnerStratonovich equation in short KS equation In this paper the aim is to improve the hypotheses to obtain the KS equation and describe the filter π in terms of the unnormalized filter ϱ which is solution of a linear stochastic differential equation the socalled Zakai equation We prove the equivalence between strong uniqueness of the solution of the KS equation and strong uniqueness of the solution of the Zakai one and as a consequence we deduce pathwise uniqueness of the solution of the Zakai equation by applying the Filtered Martingale Problem approach Kurtz and Ocone in Ann Probab 1680–107 1988 Ceci and Colaneri in Adv Appl Probab 443678–701 2012 To conclude we discuss some particular modelsLet θ be the process defined in 49 where the process μ is a strong solution of the KushnerStratonovich equation such that mu tlambdaphimathrmdz is equivalent to the measure pi tlambdaphimathrmdz for every t∈0T and assume the hypotheses of Theorem 36 Then sup t∈0T θ t ∞ P 0asObserve that the RadonNikodym derivative fracmathrmdmu tlambdaphimathrmdeta tz is well defined in fact the measures mu tlambdaphimathrmdz and etatY t mathrmdz are equivalent since they are both equivalent to pi tlambdaphimathrmdz

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