Exponential mean square stability of the theta approximations for neutral stochastic differential delay equations

In this paper, a split-step theta (SST) method is introduced and analyzed for neutral stochastic differential delay equations (NSDDEs). It is proved that the SST method with θ∈[0,1/2]θ∈[0,1/2] can recover the exponential mean square stability of the exact solution with some restrictive conditions on stepsize and the drift coefficient, but for θ∈(1/2,1]θ∈(1/2,1], the SST can reproduce the exponential mean square stability unconditionally. Then, based on the stability results of SST scheme, we examine the exponential mean square stability of the stochastic linear theta (SLT) approximation for NSDDEs and obtain the similar stability results to that of the SST method. Moreover, for sufficiently small stepsize, we show that the decay rate as measured by the Lyapunov exponent can be reproduced arbitrarily accurately. These results show that different values of theta will drastically affect the exponential mean square stability of the two classes of theta approximations for NSDDEs.