Convergence and stability of balanced implicit methods for systems of SDES
Henri Schurz 11 Department of Mathematics, Southern Illinois University, 1245 Lincoln Drive, Carbondale, IL 62901-4408, USA
Received by the editors April 11, 2004 and, in revised form, July 30, 2004
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.
AMS subject classifications: 65C30, 65L20, 65D30, 34F05, 37H10, 60H10
Key words: balanced implicit methods; linear-implicit methods; conditional mean consistency; conditional mean square consistency; weak V-stability; stochastic Kantorovich-Lax-Richtmeyer principle; L-2-convergence; weak convergence; almost sure stability; p-th mean stability.