Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Large Deviations Principle by viscosity solutions: The case of diffusions with oblique Lipschitz reflections

Magdalena Kobylanski

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Abstract

We establish a Large Deviations Principle for diffusions with Lipschitz continuous oblique reflections on regular domains. The rate functional is given as the value function of a control problem and is proved to be good. The proof is based on a viscosity solution approach. The idea consists in interpreting the probabilities as the solutions to some PDEs, make the logarithmic transform, pass to the limit, and then identify the action functional as the solution of the limiting equation.

Résumé

Nous établissons un principe de Grandes Déviations pour des diffusions réfléchies obliquement sur le bord d’un domaine régulier lorsque la direction de la réflection est Lipschitz. La fonction de taux s’exprime comme la fonction valeur d’un problème d’arrêt optimal et est compacte. Nous utilisons des techniques de solutions de viscosité. Les probabilités recherchées sont interprétées comme des solutions de certaines EDPs, leur transformées logarithmiques donnent lieu à de nouvelles équations dans lesquelles il est aisé de passer à la limites. Enfin les fonctionnelles d’action sont identifiées comme étant les solutions des dites équations limite.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 49, Number 1 (2013), 160-181.

Dates
First available in Project Euclid: 29 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1359470130

Digital Object Identifier
doi:10.1214/11-AIHP444

Mathematical Reviews number (MathSciNet)
MR3060152

Zentralblatt MATH identifier
1270.60032

Subjects
Primary: 60F10: Large deviations 49L25: Viscosity solutions 49J15: Optimal control problems involving ordinary differential equations
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 49L20S

Keywords
Large Deviations Principle Diffusions with oblique reflections Viscosity solutions Optimal control Optimal stopping

Citation

Kobylanski, Magdalena. Large Deviations Principle by viscosity solutions: The case of diffusions with oblique Lipschitz reflections. Ann. Inst. H. Poincaré Probab. Statist. 49 (2013), no. 1, 160--181. doi:10.1214/11-AIHP444. https://projecteuclid.org/euclid.aihp/1359470130


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