Open Access
May 2018 Nonlinear filtering with correlated Lévy noise characterized by copulas
B. P. W. Fernando, E. Hausenblas
Braz. J. Probab. Stat. 32(2): 374-421 (May 2018). DOI: 10.1214/16-BJPS347


The objective in stochastic filtering is to reconstruct the information about an unobserved (random) process, called the signal process, given the current available observations of a certain noisy transformation of that process.

Usually $X$ and $Y$ are modeled by stochastic differential equations driven by a Brownian motion or a jump (or Lévy) process. We are interested in the situation where both the state process $X$ and the observation process $Y$ are perturbed by coupled Lévy processes. More precisely, $L=(L_{1},L_{2})$ is a $2$-dimensional Lévy process in which the structure of dependence is described by a Lévy copula. We derive the associated Zakai equation for the density process and establish sufficient conditions depending on the copula and $L$ for the solvability of the corresponding solution to the Zakai equation. In particular, we give conditions of existence and uniqueness of the density process, if one is interested to estimate quantities like $\mathbb{P}(X(t)>a)$, where $a$ is a threshold.


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B. P. W. Fernando. E. Hausenblas. "Nonlinear filtering with correlated Lévy noise characterized by copulas." Braz. J. Probab. Stat. 32 (2) 374 - 421, May 2018.


Received: 1 June 2015; Accepted: 1 November 2016; Published: May 2018
First available in Project Euclid: 17 April 2018

zbMATH: 06914680
MathSciNet: MR3787759
Digital Object Identifier: 10.1214/16-BJPS347

Keywords: Lévy copula , Lévy processes , Nonlinear filtering

Rights: Copyright © 2018 Brazilian Statistical Association

Vol.32 • No. 2 • May 2018
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