  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.
