Exponential stability of θ-method for stochastic differential equations in the G-framework

Abstract

Exponential stability of the stochastic θ -method for stochastic differential equations with G -Brownian motion (called G-SDEs for brevity) is investigated. It is proved that under the global Lipschitz condition, a G-SDE is p th ( p ∈ ( 0 , 1 ) ) moment exponentially stable if and only if the stochastic θ -method with a sufficiently small step size is also p th moment exponentially stable. The stochastic θ -method for a G-SDE is p th ( p ∈ ( 0 , 1 ) ) moment exponentially stable, then it is also quasi surely exponentially stable, and furthermore, the G-SDE is quasi surely exponentially stable. Numerical examples are demonstrated to illustrate the obtained theoretical results.