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

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.

39 pages, 4 figures

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• On the phase transition curve in a directed exponential random graph model
• Erratum: Propagation Effects on the Breakdown of a Linear Amplifier Model: Complex-Mass Schrodinger Equation Driven by the Square of a Gaussian Field
• Long-run growth rate in a random multiplicative model

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