Filtering and statistical properties of unimodal maps perturbed by heteroscedastic noise

We propose a theory of unimodal maps perturbed by a 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.