Convergence and Stability of Balanced Implicit Methods for Systems of Sdes

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 cj . The presented convergence proofs extend to the case of nonrandom variable step sizes and show a dependence on certain Lyapunov-functionals V : IRd → IR+. The proof of L2-convergence with global rate 0.5 is based on the stochastic Kantorovich-Lax-Richtmeyer 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 BIMs. The problem of weak convergence with respect to the test class C2 b(κ) (IR, IR) and with global rate 1.0 is tackled too.