Convergence and stability of numerical solutions to a class of index 1 stochastic differential algebraic equations with time delay

In this paper, we study the convergence and stability of the stochastic theta method (STM) for a class of index 1 stochastic delay differential algebraic equations. First, in the case of constrained mesh, i.e., the stepsize is a submultiple of the delay, it is proved that the method is strongly consistent and convergent with order 1/2 in the mean-square sense. Then, the result is further extended to the case of non-constrained mesh where we employ linear interpolation to approximate the delay argument. Later, under a sufficient condition for mean-square stability of the analytical solution, it is proved that, when the stepsizes are sufficiently small, the STM approximations reproduce the stability of the analytical solution. Finally, some numerical experiments are presented to illustrate the theoretical findings.