Numerical solution of stochastic state-dependent delay differential equations: convergence and stability

Numerical analysis of stochastic delay differential equations has been widely developed but frequently for the cases where the delay term has a simple feature. In this paper, we aim to study a more general case of delay term which has not been much discussed so far. We mean the case where the delay term takes random values. For this purpose, a new continuous split-step scheme is introduced to approximate the solution and then convergence in the mean-square sense is investigated. Moreover, given a test equation, the mean-square asymptotic stability of the scheme is presented. Numerical examples are provided to further illustrate the obtained theoretical results.