| This dissertation concerns the convergence of the Gauss-Galerkin method for some Markov processes. This method provides approximations to p(t,x), the solution of the Fokker-Planck equation (PAR-DIFF)p/(PAR-DIFF)t = -((PAR-DIFF)/(PAR-DIFF)x)(ap) + (, 1/2) ((PAR-DIFF)('2)/(PAR-DIFF)x('2))(b('2)p) corresponding to the stochastic differential equation dX(t) = a(X(t) ,t)dt + b(X(t) ,t)dW(t) defined on (r(,1),r(,2)) with 0 (LESSTHEQ) r(,1) < r(,2) (LESSTHEQ) (INFIN). The approximation is in the sense of approximation of probability measures.; We first show that when the coefficients of the stochastic differential equation are polynomials, the resulting Gauss-Galerkin system is equivalent to the Hankel system of moments which is closed in an appropriate manner. We then compare the Gauss-Christoffel approximations to p(t,x) with the Gauss-Galerkin approximations and derive upper bounds for the inherent errors.; The Gauss-Galerkin measures, which are atomic in nature, form the basis for numerical integration quadrature formulas. We show that under suitable conditions on the coefficients of the stochastic differential equation, these integration formulas converge to the true value of the integral. The proofs rely on the use of differential inequalities, Helly's theorems on weak compactness of measures and the spectral theory of linear operators.; Numerical examples are presented that illustrate close agreements between the numerical and theoretical results. |
