  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.
