Asymptotical mean stability of numerical solutions with multiplicative noise.
- Schurz, Henri
2010 Mathematics Subject Classification
- 65C20 65U05 60H10 65L20
- Stochastic differential equations, numerical methods, weak and strong convergence, asymptotically absolute stability, p-th and weak mean stability, stability function and domain, simulation experiments
As an extent of asymptotically absolute stability of numerical methods in deterministic situation, in this report the asymptotically absolute mean stability of the null solution for stochastic differential equations with respect to different criterions will be examined, both for the exact solution and for its numerical approximations. Among the considered criterions the mean square stability plays the main role in the examinations. For the class of scalar, bilinear, complex-valued stochastic differential equations, comparison studies for different numerical schemes are provided and show their different stability features. However the balanced implicit methods have proved to be rich enough to possess appropriately large stability domains. Finally, experiments for the Kubo oscillator indicate how efficient the asymptotical mean stability examinations could be for the reality.