Young differential equations driven by Besov–Orlicz paths

In the article, we consider nonlinear differential equations driven by paths from the exponential Besov–Orlicz space $ B_{\varPhi_\beta,q}^\alpha $ for$ \alpha \in (1/2,1),\quad \varPhi_\beta(x) \sim \mathrm{e}^{x^\beta}-1\quad \mbox{with}\quad \beta\in (0,\infty), \quad\mbox{and}\quad q\in (0,\infty]. $By appealing to the recently obtained sewing lemma for such paths, we construct a Young-type integral and show that such equations admit a unique solution that is again of exponential Besov–Orlicz regularity. The results cover equations driven by paths of a large number of stochastic processes that exhibit long-range dependence, e.g. fractional Brownian motion with Hurst parameter $ H\in (1/2,1) $ or, more generally, any Hermite process.