Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

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

Jin Ma, Zhenjie Ren, Nizar Touzi, and 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 (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 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.

Résumé

Nous montrons un principe de grandes déviations pour les équations différentielles stochastiques non-markoviennes, dirigées par un mouvement brownien, et à coefficients aléatoires dépendant de l’ensemble du passé. Comme dans Gao et Liu (Stoch. Dyn. 6 (2006) 487–520), ceci étend les résultats correspondants dans Freidlin et Wentzell (Random Perturbations of Dynamical Systems (1984) Springer). Cependant, nous utilisons un argument différent, adaptant la méthode d’EDP de Fleming (Appl. Math. Optim. 4 (1978) 329–346) et Evans et Ishii (Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985) 1–20) au cas des équations dépendant des trajectoires, en utilisant des techniques d’équations différentielles stochastiques rétrogrades. Comme dans le cas markovien, nous obtenons une caractérisation de la fonction d’action comme l’unique solution bornée d’une version non markovienne de l’équation eikonale. Enfin, nous proposons une application à l’analyse asymptotique, en maturité courte, de la surface de volatilité implicite en mathématiques financières.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 3 (2016), 1196-1216.

Dates
Received: 19 July 2014
Revised: 27 January 2015
Accepted: 7 April 2015
First available in Project Euclid: 28 July 2016

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1469723517

Digital Object Identifier
doi:10.1214/15-AIHP678

Mathematical Reviews number (MathSciNet)
MR3531706

Zentralblatt MATH identifier
1351.35276

Subjects
Primary: 35D40: Viscosity solutions 35K10: Second-order parabolic equations 60H10: Stochastic ordinary differential equations [See also 34F05] 60H30: Applications of stochastic analysis (to PDE, etc.)

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

Citation

Ma, Jin; Ren, Zhenjie; Touzi, Nizar; Zhang, Jianfeng. Large deviations for non-Markovian diffusions and a path-dependent Eikonal equation. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 3, 1196--1216. doi:10.1214/15-AIHP678. https://projecteuclid.org/euclid.aihp/1469723517


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