Almost sure exponential stability of the θθ-method for stochastic differential equations

Our previous work shows that the backward Euler–Maruyama (BEM) method may reproduce the almost sure stability of stochastic differential equations (SDEs) without the linear growth condition for the drift coefficient (see Wu et al. (2010)) but the Euler–Maruyama (EM) method cannot. It is well known that the θθ-method is more general and may be specialized as the BEM and EM by choosing θ=1θ=1 and θ=0θ=0. Then it is very interesting to examine the interval in which the θθ-method holds the same stability property as the BEM method. This paper shows that when θ∈(1/2,1]θ∈(1/2,1], the θθ-method may reproduce the almost sure stability of the exact solution of SDEs. Finally, a two-dimensional example is presented to illustrate this result.