Year: 2005
Author: Henri Schurz
International Journal of Numerical Analysis and Modeling, Vol. 2 (2005), Iss. 2 : pp. 197–220
Abstract
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 $V$ : $\rm{IR}^d \rightarrow \rm{IR}_+^1$. The proof of $L^2$-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 BlMs. The problem of weak convergence with respect to the test class $C_{b(\kappa)}^2 (\rm{IR}^d, \rm{IR}^1)$ and with global rate 1.0 is tackled too.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2005-IJNAM-929
International Journal of Numerical Analysis and Modeling, Vol. 2 (2005), Iss. 2 : pp. 197–220
Published online: 2005-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 24
Keywords: 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.