Fractional nonlinear filtering problems and their associated fractional Zakai equations

TL;DR

This paper introduces fractional generalizations of filtering problems involving time-changed stochastic processes, leading to Zakai equations with Riemann-Liouville fractional derivatives, expanding the mathematical framework for stochastic filtering.

Contribution

It presents a novel fractional filtering framework with associated fractional Zakai equations driven by time-changed processes, extending classical filtering theory.

Findings

Derivation of fractional Zakai equations with Riemann-Liouville derivatives

Extension of filtering problems to time-changed Brownian motion and Lévy processes

Mathematical formulation of fractional filtering models

Abstract

In this paper we discuss fractional generalizations of the filtering problem. The "fractional" nature comes from time-changed state or observation processes, basic ingredients of the filtering problem. The mathematical feature of the fractional filtering problem emerges as the Riemann-Liouville fractional derivative in the associated Zakai equation. We discuss fractional generalizations of the filtering problem whose state and observation processes are driven by time-changed Brownian motion or/and L\'evy process.