Error in Methods for Stochastic Differential Equations

We briefly present some standard material on the development of numerical schemes for SDEs. The interested reader is pointed to the standard textbooks [200,270] for a more thorough discussion of foundational concepts. Let us begin with a discussion of the first order stochastic differential equation 

The numerical analysis of stochastic differential equations is traditionally based on the concepts of weak and strong accuracy. Let a numerical method be given for solving an SDE in the form of a discrete stochastic process X +i = 0(X , h) for n = 0, l.v — 1, where vh = x is, fixed. We denote the stochastic solution of the SDE by X(t). In the case of strong accuracy, our measure of the global error is the quantity [200] 

For a = 1 this implies convergence of expectation. The more natural choice from the probabilistic point of view is a = 2 (mean square convergence). Having settled on a choice of a, we say that a method has strong order r if Err C x) for some coefficient C(t) depending on the time interval. 

The weak error is usually defined with reference to suitable test functions (observables) cp (x) by 

As the focus of this chapter is on the computation of averages, the concept of weak order is more appropriate in evaluating methods. Yet, even the weak order of convergence may not be suitable in molecular dynamics applications, or rather, we are interested in the very long term behavior of the weak error which is governed by the coefficient T)(t) as r oo [270,356]. 

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