On the Growth Rate of a Linear Stochastic Recursion with Markovian Dependence

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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 \(\lambda _q = \lim _{n\rightarrow \infty } \frac{1}{n} \log \mathbb {E}[(x_n)^q]\) 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.