## Abstract

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.

Original language | English (US) |
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Pages (from-to) | 1196-1216 |

Number of pages | 21 |

Journal | Annales de l'institut Henri Poincare (B) Probability and Statistics |

Volume | 52 |

Issue number | 3 |

DOIs | |

State | Published - Aug 2016 |

## Keywords

- Backward stochastic differential equations
- Large deviations
- Viscosity solutions of path-dependent PDEs

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty