Choice of θ and mean-square exponential stability in the stochastic theta method of stochastic differential equations

This paper examines the relationship of choice of θ and mean-square exponential stability in the stochastic theta method (STM) of stochastic differential equations (SDEs) and mainly includes the following three results: (i) under the linear growth condition for the drift term, when θ∈[0,1/2), the STM may preserve the mean-square exponential stability of the exact solution, but the counterexample shows that the STM cannot reproduce this stability without this linear growth condition; (ii) when θ∈(1/2,1), without the linear growth condition for the drift term, the STM may reproduce the mean-square exponential stability of the exact solution, but the bound of the Lyapunov exponent cannot be preserved; (iii) when θ=1 (this STM is called as the backward Euler-Maruyama (BEM) method), the STM can reproduce not only the mean-square exponential stability, but also the bound of the Lyapunov exponent. This paper also gives the sufficient and necessary conditions of the mean-square exponential stability of the STM for the linear SDE when θ∈[0,1/2) and θ∈[1/2,1], respectively, and the simulations also illustrate these theoretical results.