Optimal growth trajectories with finite carrying capacity

We consider the harvesting of a population in a stochastic environment whose dynamics in the absence of harvesting is described by a one dimensional diffusion. Using ergodic optimal control, we find the optimal harvesting strategy which maximizes the asymptotic yield of harvested individuals. To our knowledge, ergodic optimal control has not been used before to study harvesting strategies. However, it is a natural framework because the optimal harvesting strategy will never b...

We study a stochastic, continuous time model on a finite horizon for a firm that produces a single good. We model the production capacity as an Ito diffusion controlled by a nondecreasing process representing the cumulative investment. The firm aims to maximize its expected total net profit by choosing the optimal investment process. That is a singular stochastic control problem. We derive some first order conditions for optimality and we characterize the optimal solution in ...

Aiming for more realistic optimal dividend policies, we consider a stochastic control problem with linearly bounded control rates using a performance function given by the expected present value of dividend payments made up to ruin. In a Brownian model, we prove the optimality of a member of a new family of control strategies called delayed linear control strategies, for which the controlled process is a refracted diffusion process. For some parameters specifications, we retr...

In this paper, we study the existence of an optimal strategy for the stochastic control of diffusion in general case and a saddle-point for zero-sum stochastic differential games. The problem is formulated as an extended BSDE with logarithmic growth in the $z$-variable and terminal value in some $L^p$ space. We also show the existence and uniqueness of solution of this BSDE.

We investigate the growth optimal strategy over a finite time horizon for a stock and bond portfolio in an analytically solvable multiplicative Markovian market model. We show that the optimal strategy consists in holding the amount of capital invested in stocks within an interval around an ideal optimal investment. The size of the holding interval is determined by the intensity of the transaction costs and the time horizon.

We address a novel approach for stochastic individual-based modelling of a single species population. Individuals are distinguished by their remaining lifetimes, which are regulated by the interplay between the inexorable running of time and the individual's nourishment history. A food-limited environment induces intraspecific competition and henceforth the carrying capacity of the medium may be finite, often emulating the qualitative features of logistic growth. Inherently n...

This work faces the problem of the origin of the logarithmic character of the Gompertzian growth. We show that the macroscopic, deterministic Gompertz equation describes the evolution from the initial state to the final stationary value of the median of a log-normally distributed, stochastic process. Moreover, by exploiting a stochastic variational principle, we account for self-regulating feature of Gompertzian growths provided by self-consistent feedback of relative density...

In this work, we study the optimization problem of a renewable resource in finite time. The resource is assumed to evolve according to a logistic stochastic differential equation. The manager may harvest partially the resource at any time and sell it at a stochastic market price. She may equally decide to renew part of the resource but uniquely at deterministic times. However, we realistically assume that there is a delay in the renewing order. By using the dynamic programmin...

We propose and study a stochastic capital-labour model with logistic growth function. First, we show that the model has a unique positive global solution. Then, using the Lyapunov analysis method, we obtain conditions for the extinction of the total labour force. Furthermore, we also prove sufficient conditions for their persistence in mean. Finally, we illustrate our theoretical results through numerical simulations.

This paper concerns an optimal stopping problem driven by the running maximum of a spectrally negative Levy process X. More precisely, we are interested in capped versions of the American lookback optimal stopping problem, which has its origins in mathematical finance, and provide semi-explicit solutions in terms of scale functions. The optimal stopping boundary is characterised by an ordinary first-order differential equation involving scale functions and, in particular, cha...