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

Abstract

We study the stochastic growth process in discrete time $x_{i+1}=(1+{\mu }_i)x_i$ with growth rate ${\mu }_i=\rho e^{Z_i-\frac{1}{2}var(Z_i)}$ proportional to the exponential of an Ornstein-Uhlenbeck (O-U) process $d{Z}_t=-\gamma Z_{t}dt+\sigma d{W}_t$ 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.

24 pages, 3 figures

References in corpus (3)

• On the phase transition curve in a directed exponential random graph model
• On the growth rate of a linear stochastic recursion with Markovian dependence
• Long-run growth rate in a random multiplicative model