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 - <p style="text-align: justify;">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.</p>