Filtering and Statistical Properties of Unimodal Maps Perturbed by Heteroscedastic Noises

TL;DR

This paper develops a theory for filtering unimodal maps affected by heteroscedastic noise, providing limit theorems and applying to systemic risk models in finance.

Contribution

It introduces a new framework for analyzing the filtering of unimodal maps under heteroscedastic noise, including convergence and limit theorems.

Findings

Filtering leads to a unique distribution regardless of initial guess.

Established concentration inequalities and limit theorems for the model.

Application to systemic risk models in finance.

Abstract

We propose a theory of unimodal maps perturbed by an heteroscedastic Markov chain noise and experiencing another heteroscedastic noise due to uncertain observation. We address and treat the filtering problem showing that by collecting more and more observations, one would predict the same distribution for the state of the underlying Markov chain no matter one's initial guess. Moreover we give other limit theorems, emphasizing in particular concentration inequalities and extreme value and Poisson distributions. Our results apply to a family of maps arising from a model of systemic risk in finance.