Emergent fractional dynamics from fine-grained stochastic processes: A consistent coarse-graining framework

We derive fractional (nonlocal-in-time) inventory dynamics as a coarse-grained limit of heavy-tailed renewal demand processes. Starting from a renewal process in the domain of attraction of an α -stable law, the dynamics are formulated in an operational time and mapped to physical time via an inverse α -stable subordinator. This stochastic time-change induces a fractional Volterra equation, providing a direct mechanism for the emergence of memory from temporal heterogeneity at the microscopic level. A central result is that the coarse-graining procedure induces a coherent transformation of the entire dynamical system. While the time derivative transforms universally into a Caputo fractional operator, the forcing term is not invariant under subordination and must be modified. In particular, when the forcing is governed by the same operational clock, replacing the classical derivative by a fractional operator while leaving the forcing unchanged is, in general, inconsistent with the underlying subordinated dynamics. The theoretical predictions are validated by Monte Carlo simulations based on inverse-subordinator sampling, confirming agreement at the level of both mean behaviour and, where examined explicitly, variance. The framework provides a consistent route for deriving fractional models from stochastic dynamics, identifying stochastic time-change as the mechanism underlying emergent nonlocal behaviour.