# The Zakai Equation of Nonlinear Filtering for Jump-Diffusion Observations: Existence and Uniqueness

The Zakai Equation of Nonlinear Filtering for Jump-Diffusion Observations: Existence and Uniqueness In this paper we study a nonlinear filtering problem for a general Markovian partially observed system ( X , Y ), whose dynamics is modeled by correlated jump-diffusions having common jump times. At any time t ∈(0, T ), the σ -algebra $\mathcal{F}^{Y}_{t}:= \sigma\{ Y_{s}: s\leq t\}$ 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 $\mathcal{F}^{Y}_{t}$ . In Ceci and Colaneri (Adv. Appl. Probab. 44(3):678–701, 2012 ) it is proved that π is the unique probability measure-valued process satisfying a nonlinear stochastic equation, the so-called Kushner-Stratonovich 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 so-called 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. 16:80–107, 1988 ; Ceci and Colaneri in Adv. Appl. Probab. 44(3):678–701, 2012 ). To conclude, we discuss some particular models. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

# The Zakai Equation of Nonlinear Filtering for Jump-Diffusion Observations: Existence and Uniqueness

, Volume 69 (1) – Feb 1, 2014
36 pages

/lp/springer_journal/the-zakai-equation-of-nonlinear-filtering-for-jump-diffusion-uaL0jRrGo3
Publisher
Springer US
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-013-9217-1
Publisher site
See Article on Publisher Site

### Abstract

In this paper we study a nonlinear filtering problem for a general Markovian partially observed system ( X , Y ), whose dynamics is modeled by correlated jump-diffusions having common jump times. At any time t ∈(0, T ), the σ -algebra $\mathcal{F}^{Y}_{t}:= \sigma\{ Y_{s}: s\leq t\}$ 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 $\mathcal{F}^{Y}_{t}$ . In Ceci and Colaneri (Adv. Appl. Probab. 44(3):678–701, 2012 ) it is proved that π is the unique probability measure-valued process satisfying a nonlinear stochastic equation, the so-called Kushner-Stratonovich 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 so-called 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. 16:80–107, 1988 ; Ceci and Colaneri in Adv. Appl. Probab. 44(3):678–701, 2012 ). To conclude, we discuss some particular models.

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Feb 1, 2014

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