Gauss–Galerkin approximation of diffusion processes with boundary conditions

The Gauss–Galerkin approximation of the laws of some diffusion processes with boundary conditions is considered. The Gauss–Galerkin approximation was originally proposed by Dawson. We obtain a sequence of discrete measures which converges weakly to the law of the process. The Gauss–Galerkin approximation is obtained through a basic differential equation describing the evolution of the expected values of a certain functional of the process. Dawson and HajJafar derived this basic differential equation through the Fokker–Planck equation. They then obtained the Gauss–Galerkin approximation with polynomial basis functions. The approach considered here covers diffusion processes for which the Fokker–Planck equation may not be satisfied or situations where the polynomial basis functions are inappropriate. Conditions are specified under which the Gauss–Galerkin approximation of order n converge weakly to the true distribution as n.