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

Abstract

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