Large Deviations for Non-Markovian Diffusions and a Path-Dependent Eikonal Equation

TL;DR

This paper establishes a large deviation principle for non-Markovian diffusions driven by Brownian motion with random coefficients, extending classical results using PDE methods and backward SDE techniques, and applies it to financial implied volatility.

Contribution

It introduces a novel approach combining PDE and backward SDE methods to analyze large deviations for non-Markovian diffusions, including a path-dependent Eikonal equation.

Findings

Characterization of the action function as a solution to a path-dependent Eikonal equation

Extension of large deviation principles to non-Markovian stochastic differential equations

Application to short maturity implied volatility asymptotics in finance

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.