Large Deviations for Non-Markovian Diffusions and a Path-Dependent Eikonal Equation
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 \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.
Cite
@article{arxiv.1407.5314,
title = {Large Deviations for Non-Markovian Diffusions and a Path-Dependent Eikonal Equation},
author = {Jin Ma and Zhenjie Ren and Nizar Touzi and Jianfeng Zhang},
journal= {arXiv preprint arXiv:1407.5314},
year = {2014}
}