Gompertz and logistic stochastic dynamics: Advances in an ongoing quest

TL;DR

This paper reviews methods for solving stochastic differential equations related to Gompertz and logistic models, providing explicit solutions for Gompertz and discussing challenges and prospects for logistic dynamics, with potential links to quantum stochastic mechanics.

Contribution

It offers explicit solutions for Gompertz SDEs and discusses the complexities of logistic SDEs, proposing new avenues including quantum mechanics analogies for stochastic control.

Findings

Gompertz SDE solutions are fully known.

Logistic SDE solutions are explicit but lack closed-form distributions.

Potential connections to quantum stochastic mechanics are suggested.

Abstract

In this report we summarize a few methods for solving the stochastic differential equations (SDE) and the corresponding Fokker-Planck equations describing the Gompertz and logistic random dynamics. It is shown that the solutions of the Gompertz SDE are completely known, while for the logistic SDE's we provide the solution as an explicit process, but we can not yet write down its distributions in closed form. Many details of possible ways out of this maze are listed in the paper and its appendices. We also briefly discuss the prospects of performing a suitable averaging, or a deterministic limit. The possibility is also suggested of associating these equations to the stochastic mechanics of a quantum harmonic oscillator adopted as a tool serviceable also in the field of stochastic control: in particular we propose to investigate the equations associated to the quantum stationary states