title: Optimal growth trajectories with finite carrying capacity.
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 nonlinear 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 we compute the optimal trajectories of the control parameter. We further test the validity of our analytical results using numerical simulations.
authors.affiliation: Invenia Labs, 27 Parkside Place, Cambridge CB1 1HQ, United Kingdom.|London Institute of Mathematical Sciences, 35a South Street, London W1K 2XF, United Kingdom.
authors.forename: F
authors.initials: F
authors.lastname: Caravelli
authors.affiliation: Invenia Labs, 27 Parkside Place, Cambridge CB1 1HQ, United Kingdom.
authors.forename: L
authors.initials: L
authors.lastname: Sindoni
authors.affiliation: Department of Computer Science, University College London, Gower Street, London WC1E 6BT, United Kingdom.|Systemic Risk Centre, London School of Economics and Political Sciences, London, United Kingdom.
authors.forename: F
authors.initials: F
authors.lastname: Caccioli
authors.affiliation: Invenia Labs, 27 Parkside Place, Cambridge CB1 1HQ, United Kingdom.
authors.forename: C
authors.initials: C
authors.lastname: Ududec