A Musielak–Orlicz approach for modeling uncertainties in long-memory processes

This paper proposes a novel mathematical framework for modeling uncertainties in supOU processes, a common model for long-memory phenomena. We address uncertainties as distortions in reversion and Lévy measures, evaluating them simultaneously via state-dependent divergence functions on Musielak–Orlicz spaces. The core of our approach involves solving optimization problems to determine the upper- and lower-bounds of cumulants under a prescribed uncertainty set. Notably, we demonstrate that while classical measures like Kullback–Leibler divergence fail in this context, Musielak–Orlicz spaces effectively resolve these issues. Along with providing sufficient conditions for the well-posedness of these optimizations, we demonstrate the framework's practical utility through a water environmental application, modeling streamflow discharge. This work offers both a theoretical advancement and a robust tool for long-memory process analysis.