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

TL;DR

This paper develops a nonlinear filtering framework for systems driven by coupled Levy processes with dependence characterized by copulas, deriving the Zakai equation and establishing conditions for solution existence and uniqueness.

Contribution

It introduces a novel filtering approach for Levy-driven systems with dependence modeled by copulas, deriving the Zakai equation and conditions for its solvability.

Findings

Derived the Zakai equation for Levy processes with copula dependence.

Established conditions for existence and uniqueness of the filtering density.

Provided a framework for estimating probabilities like P(X(t)>a).

Abstract

The objective in stochastic filtering is to reconstruct 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 Levy) process. We are interested in the situation where both the state process X and the observation 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 of dependence is described by a Levy 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 P( X(t)>a), where a is a threshold.