Linear Programming Techniques and Numerics in Stochastic Optimal Harvesting

A numerical technique for stochastic control problems featuring singular control is derived. In these problems, the control is given by positioning a reflecting boundary in the state space of a given stochastic process. The position of the reflecting boundary has to be adjusted to maximize a given optimality criterion. The presented numerical technique uses a novel discretization approach to a linear programming framework for the occupation measures of the process. However, a non-linear solver has to be used to find the optimal position of the boundary. Proofs of the existence and uniqueness of solutions to the underlying linear constraints are outlined. The discretization of these constraints is conducted by a finite-element type approach using finite dimensional subspaces, in order to find computable approximations. A convergence analysis for this approach is given. The technique is illustrated with an example from stochastic optimal harvesting.