Large deviations principle by viscosity solutions: the case of diffusions with oblique Lipschitz reflections

Kobylanski, Magdalena

Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 1, pp. 160-181.

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.

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.

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