Large deviations for non-Markovian diffusions and a path-dependent Eikonal equation

Jin Ma, Zhenjie Ren, Nizar Touzi, Jianfeng Zhang

Research output: Contribution to journalArticlepeer-review


This paper provides a large deviation principle for non-Markovian, Brownian motion driven stochastic differential equations with random coefficients. Similar to Gao and Liu (Stoch. Dyn. 6 (2006) 487-520), this extends the corresponding results collected in Freidlin and Wentzell (Random Perturbations of Dynamical Systems (1984) Springer). However, we use a different line of argument, adapting the PDE method of Fleming (Appl. Math. Optim. 4 (1978) 329-346) and Evans and Ishii (Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985) 1-20) 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 pathdependent version of the Eikonal equation. Finally, we provide an application to the short maturity asymptotics of the implied volatility surface in financial mathematics.

Original languageEnglish (US)
Pages (from-to)1196-1216
Number of pages21
JournalAnnales de l'institut Henri Poincare (B) Probability and Statistics
Issue number3
StatePublished - Aug 2016


  • Backward stochastic differential equations
  • Large deviations
  • Viscosity solutions of path-dependent PDEs

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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