On Optimal Harvesting Problems in Random Environments

Chao Zhu is an assistant professor at UWMilwaukee. His research interests include applied probability and stochastic processes; actuarial sciences and fi nance; computational linear algebra; and mathematical statistics. We consider the optimal harvesting problem for a single species living in random environments whose growth is given by a regimeswitching diff usion. Harvesting acts as a (stochastic) control on the size of the population. The objective is to fi nd a harvesting strategy which maximizes the expected total discounted income from harvesting up to the time of extinction of the species; the income rate is allowed to be stateand environment-dependent. This is a singular stochastic control problem with both the extinction time and the optimal harvesting policy depending on the initial condition. One aspect of receiving payments up to the random time of extinction is that small changes in the initial population size may signifi cantly alter the extinction time when using the same harvesting policy. Consequently, one no longer obtains continuity of the value function using standard arguments for either regular or singular control problems having a fi xed time horizon. We introduce a new suffi cient condition under which the continuity of the value function for the regime-switching model is established. Further, it is shown that the value function is the unique viscosity solution of a coupled system of quasi-variational inequalities. We also establishes a verifi cation theorem and, based on this theorem, an $\varepsilon$-optimal harvesting strategy is constructed under certain conditions on the model.