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In this paper, we address the problem of optimal management of renewable resources such as agricultural commodities and fishery production. For that purpose, we consider the population associated with such commodities and assume that its size evolves according to a logistic growth model with a predation term given by a Holling type- n functional response. Additionally, we assume that such population is subject to random fluctuations, modeled by a diffusive term driven by a one-dimensional Brownian motion and having a power-type coefficient, thus endowing the model under consideration with the property of having constant elasticity of variance. Since the stochastic differential equation associated with this model does not fit the standard assumptions in the stochastic optimal control literature, namely sublinear growth, we develop an appropriate version of the dynamic programming principle for the problem under consideration herein, proceeding also to provide a characterization of the optimal harvesting strategies and discuss some qualitative properties of the corresponding value function. [ABSTRACT FROM AUTHOR]

For a sequence of dynamic optimization problems, we aim at discussing a notion of consistency over time. This notion can be informally introduced as follows. At the very first time step t, the decision maker formulates an optimization problem that yields optimal decision rules for all the forthcoming time steps t, t,..., T; at the next time step t, he is able to formulate a new optimization problem starting at time t that yields a new sequence of optimal decision rules. This process can be continued until the final time T is reached. A family of optimization problems formulated in this way is said to be dynamically consistent if the optimal strategies obtained when solving the original problem remain optimal for all subsequent problems. The notion of dynamic consistency, well-known in the field of economics, has been recently introduced in the context of risk measures, notably by Artzner et al. (Ann. Oper. Res. 152(1):5-22, ) and studied in the stochastic programming framework by Shapiro (Oper. Res. Lett. 37(3):143-147, ) and for Markov Decision Processes (MDP) by Ruszczynski (Math. Program. 125(2):235-261, ). We here link this notion with the concept of 'state variable' in MDP, and show that a significant class of dynamic optimization problems are dynamically consistent, provided that an adequate state variable is chosen. [ABSTRACT FROM AUTHOR]