Stochastic Mitra–Wan forestry models analyzed as a mean field optimal control problem

Abstract

This paper concerns with a stochastic version of the discrete-time Mitra–Wan forestry model defined as follows. Consider a system composed by a large number of N trees of the same species, classified according to their ages ranging from 1 to s. At each stage, all trees have a common nonnegative probability of dying (known as the mortality rate). Further, there is a central controller who must decide how many trees to harvest in order to maximize a given reward function. Considering the empirical distribution of the trees over the ages, we introduce a suitable stochastic control model ${\mathcal {M}}_{N}$ to analyze the system. However, due N is too large and the complexity involved in defining an optimal steady policy for long-term behavior, as is typically done in deterministic cases, we appeal to the mean field theory. That is we study the limit as $N\rightarrow \infty$ of the model ${\mathcal {M}}_N$. Then, under a suitable law of large numbers we obtain a control model ${\mathcal {M}}$, the mean field control model, that is deterministic and independent of N, and over which we can obtain a stationary optimal control policy $\pi ^{*}$ under the long-run average criterion. It turns out that $\pi ^*$ is one of the so-called normal forest policy, which is completely determined by the mortality rate. Consequently, our goal is to measure the deviation from optimality of $\pi ^*$ when it is used to control the original process in ${\mathcal {M}}_N$.