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

Jin Ma (1) , Zhenjie Ren (2) , Nizar Touzi (2) , Jianfeng Zhang (1)

1 USC - University of Southern California
2 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

This paper provides a large deviation principle for Non-Markovian, Brownian motion driven stochastic differential equations with random coefficients. Similar to Gao \& Liu \cite{GL}, this extends the corresponding results collected in Freidlin \& Wentzell \cite{FreidlinWentzell}. However, we use a different line of argument, adapting the PDE method of Fleming \cite{Fleming} and Evans \& 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.

Mots clés: viscosity solutions of path dependent PDEs; Large deviations; backward stochastic differential equations; viscosity solutions of path dependent PDEs.

Domaines: Probabilités [math.PR]

Jin Ma, Zhenjie Ren, Nizar Touzi, Jianfeng Zhang. Large Deviations for Non-Markovian Diffusions and a Path-Dependent Eikonal Equation. 2014. ⟨hal-01026115⟩