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

We consider the linear stochastic recursion $$x_{i+1} = a_{i}x_{i}+b_{i}$$xi+1=aixi+bi where the multipliers $$a_i$$ai are random and have Markovian dependence given by the exponential of a standard Brownian motion and $$b_{i}$$bi are i.i.d. positive random noise independent of $$a_{i}$$ai. Using large deviations theory we study the growth rates (Lyapunov exponents) of the positive integer moments $$\lambda _q = \lim _{n\rightarrow \infty } \frac{1}{n} \log \mathbb {E}[(x_n)^q]$$λq=limn→∞1nlogE[^(xn)q] with $$q\in \mathbb {Z}_+$$q∈Z+. We show that the Lyapunov exponents $$\lambda _q$$λq 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.