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Global asymptotic stability of solutions of cubic stochastic difference equations
Advances in Difference Equations volume 2004, Article number: 513569 (2004)
Global almost sure asymptotic stability of solutions of some nonlinear stochastic difference equations with cubic-type main part in their drift and diffusive part driven by square-integrable martingale differences is proven under appropriate conditions in ℝ1. As an application of this result, the asymptotic stability of stochastic numerical methods, such as partially drift-implicit θ-methods with variable step sizes for ordinary stochastic differential equations driven by standard Wiener processes, is discussed.
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Rodkina, A., Schurz, H. Global asymptotic stability of solutions of cubic stochastic difference equations. Adv Differ Equ 2004, 513569 (2004). https://doi.org/10.1155/S1687183904309015