Henri Schurz ¹

Several convergence and stability issues of the balanced implicit methods (BIMs) for systems of real-valued ordinary stochastic differential equations are thoroughly discussed. These methods are linear-implicit ones, hence easily implementable and computationally more efficient than commonly known nonlinear-implicit methods. In particular, we relax the so far known convergence condition on its weight matrices c(j). The presented convergence proofs extend to the case of nonrandom variable step sizes and show a dependence on certain Lyapunov-functionals V : IRd -> IR+1. The proof of L-2-convergence with global rate 0.5 is based on the stochastic Kantorovich-Lax-Richt meyer principle proved by the author (2002). Eventually, p-th mean stability and almost sure stability results for martingale-type test equations document some advantage of BlMs. The problem of weak convergence with respect to the test class C-b(k)(2) (IRd, IR1) and with global rate 1.0 is tackled too.