The existence and averaging principle for second order stochastic differential systems with pure delay

Stochastic delay differential systems are widely applied in various fields, featuring stochasticity, time delay, and nonlinearity, which makes their analysis highly challenging. This paper investigates the existence, uniqueness, and averaging principle of solutions for a class of stochastic delay differential systems. First, by using delay matrix functions and rigorous theoretical derivation, we establish a theorem on the existence and uniqueness of solutions, which lays a foundation for further analysis. Second, under classical assumptions combined with inequality techniques and Itǒ's formula, an averaging principle is derived, showing that the solution of the original system can be well approximated by that of the averaged system. Finally, numerical simulations are conducted to verify the correctness and practicality of the theoretical results.