Averaging principle for McKean-Vlasov SDDEs driven by α-stable time-changed Lévy processes

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Abstract This paper investigates the existence, uniqueness, p-th moment boundedness, and p-th order averaging principle for McKean-Vlasov stochastic delay differential equations driven by α-stable time-changed Lévy processes. These equations involve four levels of complexity: α-stable processes, time-changes, distribution dependence, and time delay, making them challenging for existing methods. To address these challenges, we develop a novel Carathéodory approximation framework that extends key lemmas for classical α-stable Lévy processes to the time-changed setting. Under non-Lipschitz conditions, we employ Ou-Iang inequalities instead of traditional Grönwall lemmas and Bihari-type inequalities to overcome difficulties caused by nonsmooth coefficients and distribution dependence, establishing the existence-uniqueness and p-th moment boundedness of solutions. Furthermore, we study the p-th order averaging principle for the equation. Using time-scale analysis and stochastic inequality techniques, we prove that the solution converges in the p-th moment to that of a simplified averaged equation and provide an explicit convergence rate. Our results unify and extend existing averaging principles for Lévy-driven stochastic delay systems and offer new tools for the asymptotic analysis of multiscale McKean-Vlasov systems under complex non-Gaussian noises.