Random Generation of Stochastic Area Integrals

Abstract

The authors describe a method of random generation of the integrals \[ A_{1,2} ( {t,t + h} ) = \int _t^{t + h} \int _t^s dw_1 ( r )dw_2 ( s ) - \int _t^{t + h} \int _{t}^s dw_2 ( r )dw_1 ( s ), \] together with the increments$w_1 ( {t + h} ) - w_1 ( t )$ and $w_2 ( {t + h} ) - w_2 ( t )$ of a two-dimensional Brownian path $( {w_1 ( t ),w_2 ( t )} )$. The method chosen is based on Marsaglia's “rectangle-wedge-tail” method, generalised to higher dimensions. The motivation is the need for a numerical scheme for simulation of strong solutions of general multidimensional stochastic differential equations with an order of convergence $O( h )$, where h is the stepsize. Previously, no method has obtained an order of convergence better than $O( {\sqrt h } )$ in the general case.