Statistics and Data Science Seminar
Chao Zhu
University of Wisconsin-Milwaukee
On Optimal Harvesting Problems in Random Environments
Abstract: We consider the optimal harvesting strategy for a single species living in
random environments whose growth is given by a regime-switching diffusion. Harvesting
acts as a (stochastic) control on the size of the population. The objective is to
find 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 state- and 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 significantly
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 fixed time horizon. We introduce a new
sufficient 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 verification theorem and, based on this theorem,
an $\varepsilon$-optimal harvesting strategy is constructed under certain conditions
on the model. This is a joint work with Qingshuo Song and Richard Stockbridge.
Wednesday March 7, 2012 at 4:00 PM in SEO 636