Variable Step Size Control in the Numerical Solution of Stochastic Differential Equations

Abstract

We introduce a variable step size method for the numerical approximation of pathwise solutions to stochastic differential equations (SDEs). The method, which is dependent on a representation of Brownian paths as binary trees, involves estimation of local errors and of their contribution to the global error. We advocate controlling the variance of the one-step errors, conditional on knowledge of the Brownian path, in such a way that after propagation along the trajectory the error over each step will provide an equal contribution to the variance of the global error. Discretization schemes can be chosen that reduce the mean local error so that it is negligible beside the standard deviation. We show that to obtain convergence of variable step size methods for SDEs, in general it is not sufficient to evaluate the Brownian path only at the points in time where one tries to approximate the solution. We prove that convergence of such methods is guaranteed if the Lévy area is approximated well enough by further subdivision of the Brownian path and the discretization scheme employed uses appropriately both the approximate Lévy areas and increments of the Brownian path.