Volume 2, Issue 2
Convergence and Stability of Balanced Implicit Methods for Systems of SDEs

Int. J. Numer. Anal. Mod., 2 (2005), pp. 197-220.

Published online: 2005-02

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• 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.

• 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.

65C30, 65L20, 65D30, 34F05, 37H10, 60H10

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@Article{IJNAM-2-197, author = {Schurz , Henri}, title = {Convergence and Stability of Balanced Implicit Methods for Systems of SDEs}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2005}, volume = {2}, number = {2}, pages = {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.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/929.html} }
TY - JOUR T1 - Convergence and Stability of Balanced Implicit Methods for Systems of SDEs AU - Schurz , Henri JO - International Journal of Numerical Analysis and Modeling VL - 2 SP - 197 EP - 220 PY - 2005 DA - 2005/02 SN - 2 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/929.html KW - 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. AB -

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.

Henri Schurz. (1970). Convergence and Stability of Balanced Implicit Methods for Systems of SDEs. International Journal of Numerical Analysis and Modeling. 2 (2). 197-220. doi:
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