A novel high dimensional fitted scheme for stochastic optimal control problems

TL;DR

This paper introduces a new fitted finite volume method for solving high-dimensional degenerated Hamilton-Jacobi-Bellman equations in stochastic optimal control, ensuring convergence and accuracy in complex financial models.

Contribution

The paper presents a novel fitted finite volume scheme combined with Implicit Euler time discretization for high-dimensional degenerated HJB equations, addressing limitations of standard methods.

Findings

The method effectively solves high-dimensional degenerated HJB equations.

Numerical results show improved accuracy over finite difference methods.

Matrices from discretization are M-matrices, ensuring stability.

Abstract

Stochastic optimal principle leads to the resolution of a partial differential equation (PDE), namely the Hamilton-Jacobi-Bellman (HJB) equation. In general, this equation cannot be solved analytically, thus numerical algorithms are the only tools to provide accurate approximations. The aims of this paper is to introduce a novel fitted finite volume method to solve high dimensional degenerated HJB equation from stochastic optimal control problems in high dimension ($ n\geq 3$). The challenge here is due to the nature of our HJB equation which is a degenerated second-order partial differential equation coupled with an optimization problem. For such problems, standard scheme such as finite difference method losses its monotonicity and therefore the convergence toward the viscosity solution may not be guarantee. We discretize the HJB equation using the fitted finite volume method, well known to tackle degenerated PDEs, while the time discretisation is performed using the Implicit Euler scheme. We show that matrices resulting from spatial discretization and temporal discretization are M--matrices. Numerical results in finance demonstrating the accuracy of the proposed numerical method comparing to the standard finite difference method are provided.