Well-posedness and stochastic averaging for neutral Hadamard–Itô–Doob fractional differential equations under non-Lipschitz conditions

This paper investigates neutral fractional stochastic Itô-Doob differential equations (NFSIDDEs) under a non-Lipschitz framework, with particular attention to existence, uniqueness, and averaging results. The neutral part is built from several Hadamard fractional integrals of distinct orders, so that the model couples an ultraslow logarithmic memory with the three-time-scale Itô-Doob calculus. The main findings are the following. First, using the Picard iteration method (PIM) together with Bihari’s inequality, we prove that the system admits a unique solution in the space of square-integrable adapted processes as soon as the coefficients obey a Bihari-type concave modulus of continuity, a hypothesis strictly weaker than the Lipschitz condition. Second, we derive an explicit a priori bound for the second moment of the Picard iterates, in which every constant is written in closed form in terms of the horizon, of the fractional orders and of the number of memory channels. Third, we establish an averaging principle for the associated standard form: the solution of the original system and the solution of the averaged system remain mean-square close, the approximation error being of the first order in the small perturbation parameter, uniformly on time windows whose length is inversely proportional to that parameter. These results, obtained by combining classical analytical tools, especially Gronwall’s, Hölder’s and Bihari’s inequalities with Doob-type maximal estimates, provide further insight into the qualitative dynamics and stability features of such equations, at the intersection of neutral fractional calculus and stochastic analysis.