A General Class of One-Step Approximation for Index-1 Stochastic Delay-Differential-Algebraic Equations

Abstract

This paper develops a class of general one-step discretization methods for solving the index-1 stochastic delay differential-algebraic equations. The existence and uniqueness theorem of strong solutions of index-1 equations is given. A strong convergence criterion of the methods is derived, which is applicable to a series of one-step stochastic numerical methods. Some specific numerical methods, such as the Euler-Maruyama method, stochastic  $θ$-methods, split-step $θ$-methods are proposed, and their strong convergence results are given. Numerical experiments further illustrate the theoretical results.