Random Generation of Stochastic Area Integrals

The authors describe a method of random generation of the integrals \[ A1,2 ( {t,t + h} ) = ∫ _tt + h ∫ _t^s dw_1 ( r )dw_2 ( s ) - ∫ _tt + h ∫ ₜ^s dw_2 ( r )dw_1 ( s ), \] together with the incrementsw₁ ( t + h ) - w₁ ( t ) and w₂ ( t + h ) - w₂ ( t ) of a two-dimensional Brownian path ( w₁ ( t ),w₂ ( 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( √ h ) in the general case.