A Levy-area between Brownian motion and rough paths with applications to robust non-linear filtering and RPDEs

We give meaning to differential equations with a rough path term and a Brownian noise term as driving signals. Such differential equations as well as the question of regularity of the solution map arise naturally and we discuss two applications: one revisits Clark's robustness problem in nonlinear filtering, the other is a Feynman--Kac type representation of linear RPDEs. En passant, we give a short and direct argument that implies integrability estimates for rough differential equations with Gaussian driving signals which is of independent interest.