README.md

Stochastic growth process with exp-OU growth rates

This repository contains codes for the numerical simulations for the paper Growth rate of a stochastic growth process driven by an exponential Ornstein-Uhlenbeck process (2022) also available on arxiv. A brief summary of the main results of the paper is given below.

Consider a stochastic growth process in discrete time for a quantity $B_i$ with random growth rates $r_i$ $$B_{i+1}=(1+r_i) B_i$$ started at $B_0=1$.

Take the growth rate $r_i$ proportional to the exponential of a stationary Ornstein-Uhlenbeck (exp-OU) process $$r_i = \rho e^{Z_{t_i} - \frac12 var(Z_{t_i})}\qquad (1)$$ where the OU process $Z_t$ is sampled on uniformly spaced times $t_i = i\tau$ with time step $\tau$. The process $Z_t$ satisfies the SDE $$dZ_t = - \gamma Z_t dt + \sigma dW_t$$ and mean-reverts to zero. Recall that for a stationary OU process $Z = N(0,\frac{\sigma^2}{2\gamma})$. Thus the expectation of the growth rate is constant $\mathbb{E}[r_i]=\rho$.

The limiting case of $Z_t = \sigma W_t$ a Brownian motion, corresponds to the growth rates $r_t$ following a geometric Brownian motion in discrete time. This case is more tractable and all positive integer moments $\mathbb{E}[(B_i)^n]$ can be computed exactly. This case is discussed in this paper, and a statistical mechanics analogy to a lattice gas is given here. Chapter 2 of Pirjol (2022) gives a pedagogical discussion with Mathematica codes.

The process (1) describes for example the growth of a bank account accruing interest each period at an interest rate which follows an exp-OU process. This is the interest rates process assumed in the Black-Karasinski model. A similar model can be used to describe the growth of a positive quantity with correlated growth rates, for example a population undergoing growth in a random environment. More precisely the growth rates $r_i$ have Markovian dependence, which is appropriate for example for an environmental variable such as temperature, oxygen supply or food resources.

Mathematically the process for $B_i$ is a stochastic recursion $$B_{i+1}= a(Z_i) B_i$$ with Markov dependence of the multipliers $a(Z) = 1 + \rho e^{Z - \frac12 var(Z)}$.

The process can be generalized such that the growth rates $r_i$ can be both positive and negative, for example by adding deterministic multipliers as $$B_{i+1} = \frac{1}{1+\rho} (1 + \rho e^{Z_i - \frac12 var(Z_i) }) B_i\qquad (2)$$ which corresponds to a growth rate $$r_i = \frac{1}{1+\rho} (\rho e^{Z_i - \frac12 var(Z_i) } - \rho )$$ This process has growth rates averaging to zero $\mathbb{E}[r_i]=0$.

Let us study the expectation $M_t = \mathbb{E}[B_t]$. If the growth rates $r_t$ were uncorrelated, the expectation $M_t$ would have an exponential growth $M_n = \Pi_{i=0}^{n-1}(1 + \mathbb{E}[r_i]) = (1+\rho)^n$. The exponential growth rate is $\lambda_n := \frac{1}{n} \log M_t=\log(1+\rho)$.

However, the growth rates $r_i$ in the process considered here have serial correlation. Let's simulate the process and compute the sample autocorrelation of $r_i$

Simulation

# stationary exp-OU compounding process

n <- 1000 # no time steps
tau <- 0.01 # time step
gamma <- 0.1 # mean reversion
sigma <- 0.2 # volatility

rho <- 0.025

dG <- sigma^2/(2*gamma)*(1-exp(-2*gamma*tau))
varZ <- sigma^2/(2*gamma) # stationary OU process variance
beta <- exp(-gamma*tau)

# Generate random normals N(0,sqrt(dG))
e <- rnorm(n, mean=0, sd=sqrt(dG))

rate = numeric(n)
Z = numeric(n)

Z1 = rnorm(1,0,sqrt(varZ))

Z[1] = Z1
rate[1] = (rho*exp(Z1 - 0.5*varZ)-rho)/(1+rho)

for(i in 2:n) {
  Z[i]= beta*Z[i-1] + e[i]
  rate[i] = (rho*exp(Z[i] - 0.5*varZ)-rho)/(1+rho)
}

plot(rate, type="l", col="blue",xlab="Time",main="Growth rate r(t)")
abline(h=0, col="red")

acf(rate,type="correlation", lag.max=1500, main="ACF(r)")

Sample paths of $r_t$ for $\gamma=0.1,\sigma=0.2$

The autocorrelation plot of the growth rates $r_i$ shows long-range correlation. This will be seen to change the growth pattern of $B_n$ in an unexpected way.

The Mathematica code attached evaluates the expectation $M_n$ by Monte Carlo simulation. The left plot below shows the expectation $M_n$ for $n=100$ and time step $\tau=0.01$ with $\gamma=0.1$ as $\sigma$ increases. The parameter $\rho=0.025$. The horizontal black line is at 1. The expectation increases with $\sigma$, first smoothly and then more erratically.

The growth rate $\lambda_n = \frac{1}{n} \log M_n$ is shown in the right plot. For small $\sigma$, it is close to $\log(1+\rho) \simeq \rho$, but as $\sigma$ increases, it becomes larger and with larger errors.

The theoretical result for $\lambda_n$ is shown as the solid blue curve in the right plot. It has a sharp turn and increases very rapidly. This phenomenon cannot be seen in a MC simulation because the standard deviation of $B_n$ also explodes to very large values. Instead we see an explosion of the MC error, observed as a widening of the error bars.

Main result of the paper

The paper proves the existence of the limit $$\lambda(\rho,\beta,a) = \lim_{n\to \infty} \frac{1}{n} \log M_n$$ taken at fixed $\rho$, $\beta := \frac12\sigma^2\tau n^2$ and $a:= \gamma n\tau$. The limit is mathematically equivalent with the thermodynamical pressure of a one-dimensional lattice gas of particles interacting by attractive exponential potentials $\varepsilon_{ij} \sim e^{-\frac{a}{n}|i-j|}$ at temperature $T=1/\beta$ and chemical potential $T\log \rho$. This system was studied by Kac and Helfand (1963) so we call it here the Kac-Helfand gas. Kac and Helfand showed that in the long-range limit $a\to \infty$, the thermodynamical pressure approaches the van der Waals equation of state. A similar result is recovered here for $\lambda(\rho,\beta,a)$ in the $a\to \infty$ limit.

The function $\lambda(\rho,\beta,a)$ has discontinuous derivatives with respect to $\rho,\beta$ along a critical curve $\beta_c(\rho,a)$. The critical curves for 3 values of $a$ are shown below, in coordinates $(T,\rho)$ with $T=1/\beta$.

This phenomenon is associated with a liquid-gas phase transition in the analog lattice gas.

Analytical upper and lower bounds on $\lambda(\rho,\beta,a)$ are obtained, which constrain it with an error less than 4% over the entire range of parameters. (The lower bound is shown in the right plot above.) For $a=1$ and $\rho=0.025$, the lower and upper bounds on $\lambda$ are shown below. They are so close that the curves are practically indistinguishable, and determine $\lambda$ for all practical purposes. The kink in this curve is the same as the upwards kink in the growth rate $\lambda$ shown as the solid blue curve in the right plot with the MC simulation.

An exact solution for $\lambda(\rho,\beta,a)$ is derived in Chapter 5 of Pirjol (2022).