  This work focuses on the Laplace approximation for the rough differential
equation (RDE) driven by mixed rough path with as . Firstly, based on geometric
rough path lifted from mixed fractional Brownian motion (fBm), the
Schilder-type large deviation principle (LDP) for the law of the first level
path of the solution to the RDE is given. Due to the particularity of mixed
rough path, the main difficulty in carrying out the Laplace approximation is to
prove the Hilbert-Schmidt property for the Hessian matrix of the It\^o map
restricted on the Cameron-Martin space of the mixed fBm. To this end, we imbed
the Cameron-Martin space into a larger Hilbert space, then the Hessian is
computable. Subsequently, the probability representation for the Hessian is
shown. Finally, the Laplace approximation is constructed, which asserts the
more precise asymptotics in the exponential scale.
