Cascaded analytic nonlinear systems driven by rough paths

It was recently shown that the lack of a suitable probabilistic characterization of the input process for a system of interconnected analytic nonlinear input-output maps is an obstacle to well-posedness. For example, the cascade connection of two such systems is only known to be well-posed when a certain independence property is preserved by the first system in the connection. Hence, it appears that some alternative characterization of an input process is needed in this setting. One possibility is to employ T. Lyons´ construction of a rough path. This concept employs the p-variation of a path and Chen´s identity in order to extend the notion of integration with respect to paths having finite p-variation for p ≥ 1. The primary advantage of such an approach in the context of system interconnections is that independence is no longer needed for producing well-posed cascaded analytic nonlinear systems.