Growth rate of a stochastic growth process driven by an exponential Ornstein-Uhlenbeck process

We study the stochastic growth process in discrete time $x_{i+1} = (1 + \mu_i) x_i$ with growth rate $\mu_i = \rho e^{Z_i - \frac12 var(Z_i)}$ proportional to the exponential of an Ornstein-Uhlenbeck (O-U) process $dZ_t = - \gamma Z_t dt + \sigma dW_t$ sampled on a grid of uniformly spaced times $\{t_i\}_{i=0}^n$ with time step $\tau$. Using large deviation theory methods we compute the asymptotic growth rate (Lyapunov exponent) $\lambda = \lim_{n\to \infty} \frac{1}{n} \log \mathbb{E}[x_n]$. We show that this limit exists, under appropriate scaling of the O-U parameters, and can be expressed as the solution of a variational problem. The asymptotic growth rate is related to the thermodynamical pressure of a one-dimensional lattice gas with attractive exponential potentials. For $Z_t$ a stationary O-U process the lattice gas coincides with a system considered previously by Kac and Helfand. We derive upper and lower bounds on $\lambda$. In the large mean-reversion limit $\gamma n \tau \gg 1$ the two bounds converge and the growth rate is given by a lattice version of the van der Waals equation of state. The predictions are tested against numerical simulations of the stochastic growth model.