| The optimal coupling distance is a metric on a space of probability measures. This metric can be used to measure the accuracy of numerical methods for stochastic differential equations. The accuracy of the quadratic Milstein method and the Runge Kutta method is investigated in the optimal coupling distance sense. For a stochastic differential equation dX=aX,tdt+b X,tdW, the quadratic Milstein method is XQM at+Dt= XQMa t+bab DtZb+Dt 4mad,eb egb-1bd &parl0;ZbZg-d bg&parr0;+aaDt where Z is a multivariate standard normal random variable, mu(x, t) = b( x, t)bT (x, t) and the coefficient functions a,b,mu are evaluated at ( X(t), t). The Runge Kutta method is DX1 =bXt ,ttZ 1 DX2 =b&parl0;Xt +lDX1 ,t&parr0;tZ2 XRKt+D t=XRK t+aX t,tDt+c 1DX1 +c2DX2 where c12 + c22 = 1, lambdac 1c22 = 1/2, and Z(1) and Z(2) are independent multivariate standard normal random variables.;If there is a natural coupling between the numerical solution and the exact solution of an SDE, the accuracy of the numerical solution in the pathwise sense can be assessed by the strong convergence measure. The strong convergence measure is the expectation of the distance in the supremum norm, that is Esup0≤ t≤TXt -Xt where X(t) is the exact solution and X( t) is the numerical solution. The optimal coupling distance is a generalization of the strong convergence measure. The optimal coupling distance implies the same pathwise accuracy as does the strong convergence measure. The quadratic Milstein method and the Runge Kutta method have the accuracy of the Milstein method in the optimal coupling distance sense under a nondegeneracy condition, which is that the condition number of the diffusion matrix b(x, t) is bounded. Degenerate diffusions of the form b(x, t) = Db˜( x, t) are allowed where D is a bounded diagonal matrix and b˜(x, t) is a nondegenerate diffusion. |
