# Nonlinear filtering of stochastic differential equations driven by correlated L\'evy noises

## TL;DR

This paper develops nonlinear filtering equations for stochastic differential equations driven by correlated Lévy noises, establishing existence, uniqueness, and representation results for the associated filtering equations.

## Contribution

It introduces new filtering equations for systems with correlated Lévy noises and proves their well-posedness, extending classical results to more complex noise structures.

## Key findings

- Established Kushner-Stratonovich and Zakai equations for correlated Lévy noises.
- Proved pathwise uniqueness and joint law uniqueness of solutions.
- Extended filtering theory to systems with correlated Lévy noise.

## Abstract

The work concerns nonlinear filtering problems of stochastic differential equations with correlated L\'evy noises. First, we establish the Kushner-Stratonovich and Zakai equations through martingale representation theorems and the Kallianpur-Striebel formula. Second, we show the pathwise uniqueness and uniqueness in joint law of weak solutions for the Zakai equation. Finally, we investigate the uniqueness in joint law of weak solutions to the Kushner-Stratonovich equation.