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

2014/07/20 by Jin Ma, Zhenjie Ren, Ma, Jin +5 · 1 citation
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR

paper · pdf · doi:10.48550/arxiv.1407.5314

arxiv created 2014/07/20 · arxiv updated 2014/07/22

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 \citeGL, this extends the corresponding results collected in Freidlin and Wentzell \citeFreidlinWentzell. However, we use a different line of argument, adapting the PDE method of Fleming \citeFleming and Evans and Ishii \citeEvansIshii 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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