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We investigate and prove the mathematical properties of a general class ofone-dimensional unimodal smooth maps perturbed with a heteroscedastic noise.Specifically, we investigate the stability of the associated Markov chain, showthe weak convergence of the unique stationary measure to the invariant measureof the map, and show that the average Lyapunov exponent depends continuously onthe Markov chain parameters. Representing the Markov chain in terms of randomtransformation enables us to state and prove the Central Limit Theorem, thelarge deviation principle, and the Berry-Ess\`een inequality. We perform amultifractal analysis for the invariant and the stationary measures, and weprove Gumbel's law for the Markov chain with an extreme index equal to 1. Inaddition, we present an example linked to the financial concept of systemicrisk and leverage cycle, and we use the model to investigate the finite sampleproperties of our asymptotic results.

Unimodal maps perturbed by heteroscedastic noise: an application to a financial systems