  This paper provides a large deviation principle for Non-Markovian, Brownian
motion driven stochastic differential equations with random coefficients.
Similar to Gao and Liu \cite{GL}, this extends the corresponding results
collected in Freidlin and Wentzell \cite{FreidlinWentzell}. However, we use a
different line of argument, adapting the PDE method of Fleming \cite{Fleming}
and Evans and Ishii \cite{EvansIshii} to the path-dependent case, by using
backward stochastic differential techniques. Similar to the Markovian case, we
obtain a characterization of the action function as the unique bounded solution
of a path-dependent version of the Eikonal equation. Finally, we provide an
application to the short maturity asymptotics of the implied volatility surface
in financial mathematics.
