On the numerical expansion of a second order stochastic process

Abstract

Abstract The utility of the classical Karhunen‐Loève expansion of a second order process is limited to its practical derivation, because it depends upon the solution of a Fredholm integral equation associated with it whose kernel is the covariance function of the process. So, in this paper we study two numerical procedures for solving such equations, the Rayleigh‐Ritz and the collocation methods, and also essay two different bases of L 2 ‐orthogonal functions in order to perform the algorithms, Legendre polynomials and trigonometric functions, on two well‐known processes, the Wiener‐Lévy process and the Brownian‐bridge. The accuracy of the numerical results in relation to the real ones, as well as comparative studies among both procedures are also included.