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arXiv · 1407.5314

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

Jin Ma·Zhenjie Ren·Nizar Touzi·Jianfeng Zhang
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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.

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Jin Ma, Zhenjie Ren, Nizar Touzi, Jianfeng Zhang. 2014-07-20. Large Deviations for Non-Markovian Diffusions and a Path-Dependent Eikonal Equation. https://arxiv.org/abs/1407.5314

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