A fitted finite volume method for stochastic optimal control problems in finance

Abstract

In this article, we provide a numerical method based on fitted finite volume method to approximate the Hamilton-Jacobi-Bellman (HJB) equation coming from stochastic optimal control problems in one and two dimensional domain. The computational challenge is due to the nature of the HJB equation, which may be a second-order degenerate partial differential equation coupled with optimization. For such problems, standard scheme such as finite difference losses its monotonicity and therefore the convergence toward the viscosity solution may not be guarantee. In the work, we discretize the HJB equation using the fitted finite volume method, which has for main feature to tackle the degeneracy of the equation. The time discretisation is performed using the Implicit Euler method, which is unconditionally stable. We show that matrices resulting from spatial discretization and temporal discretization are M-matrices. The optimization problem is solved at every time step using iterative method. Numerical results are presented to show the robustness of the fitted finite volume numerical method comparing to the standard finite difference method.