Volterra equations driven by rough signals 2: Higher-order expansions

We extend the recently developed rough path theory for Volterra equations from [F. Harang and S. Tindel, driven by signals, Stoch. Process. Appl. 142 (2021) 34–78] to case of more noise and/or singular kernels. It was already observed in that introduced there did not satisfy any geometric relation, similar classical theory. Thus, an extension irregular driving signals requires a deeper understanding specific algebraic structure arising path. Inspired elements “non-geometric paths” [M. Gubinelli, Ramification paths, J. Differential Equations 248 (2010) 693–721; M. Hairer D. Kelly, Geometric versus non-geometric path, Ann. Inst. Henri Poincaré-Probab. Stat. 51 (2015) 207–251], we provide simple description controlled process terms rooted trees, with this are able solve signals.