Mean square A- and L-stability of balanced midpoint Milstein methods for one-dimensional bi-linear stochastic differential equations

• Published in: European journal of mathematics Vol. 11; no. 2
• Main Author: Schurz, Henri
• Format: Journal Article
• Language: English
• Published: Heidelberg Springer Nature B.V 01.06.2025
• Subjects: Asymptotic methods; Differential equations; Mathematics; Mean square values; Numerical methods; Stability
• ISSN: 2199-675X; 2199-6768
• DOI: 10.1007/s40879-025-00829-6

Adequate preservation of asymptotic mean square stability of balanced midpoint Milstein methods (BMMMs) applied to stochastic differential equations (SDEs) driven by standard Wiener processes is shown whenever the underlying SDE has an asymptotically mean square stable equilibrium. These are certain numerical methods built up by the class of balanced implicit Milstein methods combined with midpoint drift-implicitness. The paper verifies that it is indeed possible to construct higher order numerical methods, which are mean square A-stable (i.e. mean square stable for all possible step sizes h>0) for the test class of bi-linear, 1D complex-valued SDEs with multiplicative noise. This fact is remarkable since most of the authors have only addressed a conditional preservation of mean square stability subject to additional step size restrictions for Milstein-type methods. Besides, we prove L-stability of mean square evolutions and mean square exponential stability of BMMMs for bi-linear 1D test equations.