| Abstract: |
| This paper establishes the averaging principle for a class of fully coupled two-time-scale stochastic functional differential equations involving infinite integral-type delays with exponential kernels. The presence of delays leads to the lack of the Markov property, which introduces essential difficulties. Noting the special structure of the infinite delay, by using the variable substitution technique, this paper transforms the original system into a higher-dimensional system without delay. For this higher-dimensional system, we establish the existence and uniqueness of the invariant probability measure for the fixed-$X$ equation and continuous dependence on the parameter $X$. Moreover, the uniform moment boundedness of the fast-varying and slow-varying variables is also proved. Based on these results, by using the martingale problem formulation, we prove the weak convergence of the slow-varying process. As an important application, we investigate a two-time-scale stochastic optimal control problem. By using the established averaging principle and the relaxed control framework, we prove the convergence of the value function and construct near-optimal controls for the original system by using the optimal control of the limit problem. Finally, the example of an optimal advertising problem is presented to demonstrate the applicability of the obtained results. |
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