TT-stability of the split-step θθ-methods for linear stochastic delay integro-differential equations

In this paper, TT(Trajectory)-stability of the split-step θθ-methods for linear stochastic delay integro-differential equations is studied. The split-step θθ-methods for stochastic differential equations were introduced in Ding et al. (2010) [18] and the TT-stability of the semi-implicit Euler method for delay differential equations with multiplicative noise has recently been discussed in Cao (2010) [17]. Motivated by the work of Ding et al. (2010) [18] and Cao (2010) [17], we investigate the TT-stability of the split-step θθ-methods for linear stochastic delay integro-differential equations. The Wiener increment is approximated by a discrete random variable with two-point distribution. Numerical experiments are also provided to illustrate the theory.