Optimal growth trajectories with finite carrying capacity

TL;DR

This paper extends the Kelly criterion to stochastic processes with finite carrying capacity, deriving optimal growth strategies for biological, ecological, and financial models with nonlinear constraints.

Contribution

It generalizes the Kelly criterion to processes with finite carrying capacity, providing analytical solutions for specific nonlinear functions and validating them with simulations.

Findings

Optimal control strategies depend on the form of carrying capacity.

Analytical solutions are derived for power-law and logarithmic capacities.

Numerical simulations confirm the theoretical results.

Abstract

We consider the problem of finding optimal strategies that maximize the average growth-rate of multiplicative stochastic processes. For a geometric Brownian motion the problem is solved through the so-called Kelly criterion, according to which the optimal growth rate is achieved by investing a constant given fraction of resources at any step of the dynamics. We generalize these finding to the case of dynamical equations with finite carrying capacity, which can find applications in biology, mathematical ecology, and finance. We formulate the problem in terms of a stochastic process with multiplicative noise and a non-linear drift term that is determined by the specific functional form of carrying capacity. We solve the stochastic equation for two classes of carrying capacity functions (power laws and logarithmic), and in both cases compute optimal trajectories of the control parameter. We further test the validity of our analytical results using numerical simulations.