Karhunen-Loéve expansion of Brownian motion for approximate solutions of linear stochastic differential models using Picard iteration

Abstract

This work presents an application of the Picard Iterative Method (PIM) to a class of Stochastic Differential Equations where the randomness in the equation is considered in terms of the Karhunen-Loéve Expansion finite series. Two applicable numerical examples are considered to illustrate the convergence of the approximate solutions to the exact solutions and also to check the efficiency of the method. The results obtained show clearly that accuracy will be more visible by increasing the number of terms in the iteration. Thus, it is recommended for nonlinear financial models of different classes of Stochastic Differential Equations.