On the growth rate of a linear stochastic recursion with Markovian dependence

TL;DR

This paper analyzes the growth rates of a linear stochastic recursion with Markovian dependence, revealing phase transitions in Lyapunov exponents through large deviations theory and numerical analysis.

Contribution

It introduces a novel analysis of Lyapunov exponents for Markov-dependent stochastic recursions, identifying phase transitions and critical behavior.

Findings

Lyapunov exponents exist under certain conditions

Phase transitions occur in the growth rate behavior

Analytic and numerical methods characterize critical exponents

Abstract

We consider the linear stochastic recursion $x_{i+1} = a_{i}x_{i}+b_{i}$ where the multipliers $a_i$ are random and have Markovian dependence given by the exponential of a standard Brownian motion and $b_{i}$ are i.i.d. positive random noise independent of $a_{i}$. Using large deviations theory we study the growth rates (Lyapunov exponents) of the positive integer moments $\lambda_q = \lim_{n\to \infty} \frac{1}{n} \log\mathbb{E}[(x_n)^q]$ with $q\in \mathbb{Z}_+$. We show that the Lyapunov exponents $\lambda_q$ exist, under appropriate scaling of the model parameters, and have non-analytic behavior manifested as a phase transition. We study the properties of the phase transition and the critical exponents using both analytic and numerical methods.