A brief review on stability investigations of numerical methods for systems of stochastic differential equations

A modest review on stability investigations of numerical methods for systems of Ito-interpreted stochastic differential equations (SDEs) driven by m-dimensional Wiener processes W = (W1, W2, ..., Wm) is presented in Rd. Since the problem of relevance of 1D test equations for multidimensional numerical methods has not completely been solved so far, we suggest to use the Krein-Perron-Frobenius theory of positive operators on positive cones of Rdxd, instead of classic stability functions with values in C1, which is only relevant for the very restricted case of "simultaneously diagonalizable" SDEs. Our main focus is put on the concept of asymptotic mean square and almost sure (a.s.) stability for systems with state-dependent noise (multiplicative case), and the concept of exact preservation of asymptotic probabilistic quantities for systems with stateindependent noise (additive case). The asymptotic exactness of midpoint methods with any equidistant step size h is worked out in order to underline their superiority within the class of all drift-implicit, classic theta methods for multidimensional, bilinear systems of SDEs. Balanced implicit methods with appropriate weights can also provide a.s. exact, asymptotically stable numerical methods for pure diffusions. The review on numerical stability is based on "major breakthroughs" of research for systems of SDEs over the last 35 years, with emphasis on applicability to all dimensions d >= 1.