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The objective in stochastic filtering is to reconstruct information about anunobserved (random) process, called the signal process, given the currentavailable observations of a certain noisy transformation of that process.  Usually X and Y are modeled by stochastic differential equations driven by aBrownian motion or a jump (or Levy) process.  We are interested in the situation where both the state process X and theobservation process Y are perturbed by coupled Levy processes. More precisely,L=(L_1,L_2) is a 2--dimensional Levy process in which the structure ofdependence is described by a Levy copula. We derive the associated Zakaiequation for the density process and establish sufficient conditions dependingon the copula and $L$ for the solvability of the corresponding solution to theZakai equation. In particular, we give conditions of existence and uniquenessof the density process, if one is interested to estimate quantities like P(X(t)>a), where a is a threshold.

Nonlinear filtering with correlated L\'evy noise characterized by copulas