An implicit method for the finite time horizon Hamilton-Jacobi-Bellman quasi-variational inequalities

We propose a new numerical method for solving the Hamilton-Jacobi-Bellman quasi-variational inequality associated with the combined impulse and stochastic optimal control problem over a finite time horizon. Our method corresponds to an implicit method in the field of numerical methods for partial differential equations, and thus it is advantageous in the sense that the stability condition is independent of the discretization parameters. We apply our method to the finite time horizon optimal forest harvesting problem, which considers exiting from the forestry business at a finite time. We show that the behavior of the obtained optimal harvesting strategy of the extended problem coincides with our intuition.