Summary: Stability analysis for numerical solutions of stochastic differential equations (SDEs) is discussed. Similar to deterministic ordinary differential equations (ODEs), various numerical schemes, mainly generating a discretized random sequence to approximate the exact solution, are proposed for SDEs. Although convergence issue has been discussed in many literatures, a few results have been known about stability analysis in spite of its significance for numerical SDEs as well. We have proposed the mean-square stability \((MS\)-stability) of numerical single-step schemes for a scalar SDE, that is, the numerical stability with respect to the mean-square norm when it is applied to the linear test equation.
A big barrier has been observed to extend the concept to multi-step schemes or to multi-dimensional test equations, different from the ODE case. We show a way to overcome the difficulties by tackling the two-dimensional case as well as two-step numerical schemes. The underlying idea of considering the self- or mutual-correlation of the solution components is promising for the \(MS\)-stability analysis beyond the single-step methods of one-dimensional case.
For the entire collection see [Zbl 1125.00009].