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We give meaning to differential equations with a rough path term and aBrownian noise term as driving signals. Such differential equations as well asthe question of regularity of the solution map arise naturally and we discusstwo applications: one revisits Clark's robustness problem in nonlinearfiltering, the other is a Feynman--Kac type representation of linear RPDEs. Enpassant, we give a short and direct argument that implies integrabilityestimates for rough differential equations with Gaussian driving signals whichis of independent interest.

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