Large deviations for non-Markovian diffusions and a path-dependent Eikonal equation

This paper provides a large deviation principle for non-Markovian, Brownian motion driven stochastic differential equations with random coefficients. Similar to Gao and Liu (Stoch. Dyn. 6 (2006) 487-520), this extends the corresponding results collected in Freidlin and Wentzell (Random Perturbations of Dynamical Systems (1984) Springer). However, we use a different line of argument, adapting the PDE method of Fleming (Appl. Math. Optim. 4 (1978) 329-346) and Evans and Ishii (Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985) 1-20) 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 pathdependent version of the Eikonal equation. Finally, we provide an application to the short maturity asymptotics of the implied volatility surface in financial mathematics.