Stability of numerical methods for solving stochastic differential equations
Use of numerical methods of solving stochastic differential equations (SDE) for statistically modeling the dynamics of various stochastic objects often encounters the problem of instability of a numerical solution; this fact stimulates further development of stability theory for these methods. The primary attention of the present article focuses on the question of mean square stability of numerical methods for solving SDE. Also, a definition is exhibited for a system stiff in mean square, verification of whose conditions in practice causes no difficulties. As numerical methods to be studied, we take the Rosenbrock-type generalized methods. We obtain a sufficient condition for mean square stability of numerical methods in their application to linear mean square asymptotically stable SDE systems with multiplicative noise. We bring an example of a mean square \(A\)-stable numerical method for solving the SDE systems in the sense of Itô and study mean square stability for known numerical methods of solving SDE.
scientific article; zbMATH DE number 917129