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

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 $${\mathcal {M}}_{N}$$ 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 $$N\rightarrow \infty $$ N → ∞ of the model $${\mathcal {M}}_N$$ M N . Then, under a suitable law of large numbers we obtain a control model $${\mathcal {M}}$$ M , the mean field control model, that is deterministic and independent of N, and over which we can obtain a stationary optimal control policy $$\pi ^{*}$$ π ∗ under the long-run average criterion. It turns out that $$\pi ^*$$ π ∗ 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 $$\pi ^*$$ π ∗ when it is used to control the original process in $${\mathcal {M}}_N$$ M N .