Rough asymptotics of forward-backward stochastic differential equations

This note is concerned with the asymptotic behavior of adapted solutions to forward-backward SDEs when the forward diffusion contains small noises. We prove the sample path Large Deviation Principle (LDP) for the adapted solutions to the FBSDEs under appropriate conditions stated in terms of a certain type of convergence of the solutions to the associated quasilinear PDEs. As an application we apply the LDP results to a problem of rare event simulation. We provide a necessary condition for the existence of an asymptotically efficient estimator amongst the class of importance sampling estimators.