Hermite Expansions in Monte-Carlo Computation*

Monte-Carlo computations often yield numerical answers of limited accuracy, and are therefore employed as a last resort. It has been found, however, that some of the limitations of Monte-Carlo methods can be overcome through a judicious use of orthogonal expansions. When a numerical answer is obtained as the expected value of an estimator, expansion of that estimator in a series of orthogonal functions (or functionals) can reduce the variance of the estimate. Expansion of the estimand in orthogonal polynomials can increase accuracy and efficiency and simplify the solution of nonlinear problems. Hermite polynomials will be seen to play a particularly important role, and the purpose of this paper is to explain various aspects of their use through the solution of simple problems in one dimension. Generalization to multidimensional problems is immediate, and justifies the introduction of these methods. The Hermite polynomials