A Lévy area between Brownian motion and rough paths with applications to robust nonlinear filtering and rough partial differential equations

We give meaning to differential equations with a rough path term and a Brownian noise term and study their regularity, that is we are interested in equations of the type S t η = S 0 + ∫ 0 t a ( S r η ) d r + ∫ 0 t b ( S r η ) ∘ d B r + ∫ 0 t c ( S r η ) d η r where η is a deterministic geometric, step- 2 rough path and B is a multi-dimensional Brownian motion. We then give two applications: a Feynman–Kac formula for RPDEs and a robust version of the conditional expectation that appears in the nonlinear filtering problem. En passant, we revisit the recent integrability estimates of Cass et al. (2013) for rough differential equations with Gaussian driving signals which might be of independent interest.