Numerical Evaluation of the Stochastic Integral Using Mechanical Quadrature Rules

Many authors have investigated numerical methods for solving stochastic differential equations and their boundary value problems. The situation with numerical stochastic integration, as a subject of its own, has so far been different. It has not been studied with the same vigor as of investigating ordinary numerical integration. Consequently, a timely question appears to arise here on whether usual numerical quadrature methods can be employed to evaluate the stochastic integral. This paper is an attempt to answer this question by presenting a modified mechanical quadrature rule for this purpose. Convergence of this numerical integration method is also investigated and a pertaining result is reported.