A fast algorithm for the two dimensional HJB equation of stochastic control

Abstract : This paper analyses the implementation of the generalized finite differences method for the HJB equation of stochastic control, introduced by two of the authors in [Bonnans and Zidani, SIAM J. Numer. Anal. 41 (2003) 1008-1021]. The computation of coefficients needs to solve at each point of the grid (and for each control) a linear programming problem. We show here that, for two dimensional problems, this linear programming problem can be solved in O(p max) operations, where p max is the size of the stencil. The method is based on a walk on the Stern-Brocot tree, and on the related filling of the set of positive semidefinite matrices of size two.