Numerical Treatment of Stochastic Differential Equations

We define general Runge–Kutta approximations for the solution of stochastic differential equations (sde). These approximations are proved to converge in quadratic mean to the solution of an sde with a corrected drift. The explicit form of the correction term is given.Concerning the order of convergence, we show that in general it is impossible for the quadratic mean of the one step error to be of an order greater than $O(h^3 )$. This order is attained, e.g., by the stochastic analogue of Heun’s method. In the n-dimensional case, the highest order of convergence is in general only $O(h^2 )$, attained by Euler’s method. The order $O(h^3 )$ can only be reached if $(\nabla _x \sigma ^r )\sigma ^s = (\nabla _x \sigma ^s )\sigma ^r $ for the diffusion matrix.