We consider the problem of finding optimal strategies that maximize theaverage growth-rate of multiplicative stochastic processes. For a geometricBrownian motion the problem is solved through the so-called Kelly criterion,according to which the optimal growth rate is achieved by investing a constantgiven fraction of resources at any step of the dynamics. We generalize thesefinding to the case of dynamical equations with finite carrying capacity, whichcan find applications in biology, mathematical ecology, and finance. Weformulate the problem in terms of a stochastic process with multiplicativenoise and a non-linear drift term that is determined by the specific functionalform of carrying capacity. We solve the stochastic equation for two classes ofcarrying capacity functions (power laws and logarithmic), and in both casescompute optimal trajectories of the control parameter. We further test thevalidity of our analytical results using numerical simulations.
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