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 + μ_i)x_i with the growth rate $μi=ρeZi−12Var(Zi)$ proportional to the exponential of an Ornstein–Uhlenbeck (O–U) process dZ_t = −γZ_tdt + σdW_t sampled on a grid of uniformly spaced times with time step τ. Using large deviation theory, we compute the asymptotic growth rate (Lyapunov exponent) $λ=limn→∞1nlogE[xn]$⁠. We show that this limit exists, under appropriate scaling of the O–U parameters, and is expressed as the solution of a variational problem. The asymptotic growth rate is equal to the thermodynamical pressure of a one-dimensional lattice gas with attractive exponential potentials. For Z_t, a stationary O–U process of the lattice gas coincides with a model considered previously by Kac and Helfand. We derive upper and lower bounds on λ. In the large mean-reversion limit γnτ ≫ 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.