Nonlinear filtering with correlated Lévy noise characterized by copulas

Abstract

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.