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.
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.
Keywords: large deviations principle, diffusions with oblique reflections, viscosity solutions, optimal control, optimal stopping
@article{AIHPB_2013__49_1_160_0, author = {Kobylanski, Magdalena}, title = {Large deviations principle by viscosity solutions: the case of diffusions with oblique {Lipschitz} reflections}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {160--181}, publisher = {Gauthier-Villars}, volume = {49}, number = {1}, year = {2013}, doi = {10.1214/11-AIHP444}, mrnumber = {3060152}, zbl = {1270.60032}, language = {en}, url = {http://archive.numdam.org/articles/10.1214/11-AIHP444/} }
TY - JOUR AU - Kobylanski, Magdalena TI - Large deviations principle by viscosity solutions: the case of diffusions with oblique Lipschitz reflections JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2013 SP - 160 EP - 181 VL - 49 IS - 1 PB - Gauthier-Villars UR - http://archive.numdam.org/articles/10.1214/11-AIHP444/ DO - 10.1214/11-AIHP444 LA - en ID - AIHPB_2013__49_1_160_0 ER -
%0 Journal Article %A Kobylanski, Magdalena %T Large deviations principle by viscosity solutions: the case of diffusions with oblique Lipschitz reflections %J Annales de l'I.H.P. Probabilités et statistiques %D 2013 %P 160-181 %V 49 %N 1 %I Gauthier-Villars %U http://archive.numdam.org/articles/10.1214/11-AIHP444/ %R 10.1214/11-AIHP444 %G en %F AIHPB_2013__49_1_160_0
Kobylanski, Magdalena. Large deviations principle by viscosity solutions: the case of diffusions with oblique Lipschitz reflections. Annales de l'I.H.P. Probabilités et statistiques, Volume 49 (2013) no. 1, pp. 160-181. doi : 10.1214/11-AIHP444. http://archive.numdam.org/articles/10.1214/11-AIHP444/
[1] Large deviations and queing networks: Methods for rate functional identification. Stochastic Process. Appl. 84 (1999) 255-296. | MR | Zbl
and .[2] Grandes déviations et applications. In Ecole d'Eté de Probabilités de Saint-Flour VIII-1978 1-176. Lecture Notes in Math. 774. Springer, Berlin, 1980. | MR | Zbl
.[3] Solutions de viscosité des équations de Hamilton-Jacobi. Mathématiques et Applications 17. Springer, Berlin, 1994. | MR | Zbl
.[4] Fully nonlinear Neumann type boundary conditions for second-order elliptic and parabolic equations J. Differential Equations 106 (1993) 90-106. | MR | Zbl
.[5] Large deviations estimates for the exit probabilities of a diffusion process through some vanishing parts of the boundary. Adv. Differential Equations 2 (1997) 39-84. | MR | Zbl
and .[6] Remarques sur les problèmes de réflexion oblique. C. R. Math. Acad. Sci. Paris 320 (1995) 69-74. | MR | Zbl
and .[7] Discontinuous solutions of deterministic optimal stopping time problems. Math. Modelling Numer. Anal. 21 (1987) 557-579. | EuDML | Numdam | MR | Zbl
and .[8] Exit time problems in optimal control and vanishing viscosity method. SIAM J. Control Optim. 26 (1988) 1133-1148. | MR | Zbl
and .[9] Comparison principle for Dirichlet type Hamilton-Jacobi equations and singular perturbations of degenerated elliptic equations. Appl. Math. Optim. 21 (1990) 21-44. | MR | Zbl
and .[10] Weak dynamic programming principle for viscosity solutions. SIAM J. Control Optim. 49 (2011) 948-962. | MR | Zbl
and .[11] User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27 (1992) 1-67. | MR | Zbl
, and .[12] Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 277 (1983) 1-42. | MR | Zbl
and .[13] Large Deviations Techniques and Applications. Jones and Bartlett Publishers, Boston, MA, 1993. | MR | Zbl
and .[14] A Weak Convergence Approch to the Theory of Large Deviations. Wiley Ser. Probab. Stat. Wiley, New York, 1997. | MR | Zbl
and .[15] On oblique derivative problems for fully nonlinear second-order equations on nonsmooth domains. Nonlinear Anal. 15 (1990) 1123-1138. | MR | Zbl
and .[16] On Lipschitz continuity of the solution mapping to the Skorokhod problem, with applications. Stochastics Stochastics Rep. 35 (1991) 31-62. | MR | Zbl
and .[17] On oblique derivative problems for fully nonlinear second-order elliptic PDE's on domains with corners. Hokkaido Math. J. 20 (1991) 135-164. | MR | Zbl
and .[18] SDEs with oblique reflection on nonsmooth domains. Ann. Probab. 21 (1993) 554-580. | MR | Zbl
and .[19] A probabilistic approach to the reduite in optimal stopping. Probab. Math. Statist. 13 (1992) 97-121. | MR | Zbl
, and .[20] A PDE approach to some aymptotic problems concerning random differential equations with small noise intensities. Ann. Inst. H. Poncaré Anal. Non Linéaire 2 (1985) 1-20. | EuDML | Numdam | MR | Zbl
and .[21] Large Deviations for Stochastic Processes. Mathematical Surv. Monogr. 131. Amer. Math. Soc., Providence, RI, 2006. | MR | Zbl
and .[22] Exit probabilities and optimal stochastic control. Appl. Math. Optim. 4 (1978) 329-346. | MR | Zbl
.[23] Controlled Markov Processes and Viscosity Solutions. Applications of Math. 25. Springer, New York, 1993. | MR | Zbl
and .[24] A PDE-viscosity solution approach to some problems of large deviations. Ann. Sc. Norm. Super. Pisa Cl. Sci. 4 (1986) 171-192. | EuDML | Numdam | MR | Zbl
and .[25] Random Perturbations of Dynamical Systems. Comp. Studies in Math. 260. Springer, New York, 1984. | MR | Zbl
and .[26] Fully nonlinear oblique derivative problems for nonlinear second-order elliptic PDE's. Duke Math. J. 62 (1991) 633-661. | MR | Zbl
.[27] Quelques applications de méthodes d'analyse non-linéaire à la théorie des processus stochastiques. Ph.D. dissertation, l'Université de Tours, 1998.
.[28] Optimal multiple stopping time problem. Ann. Appl. Probab. 21 (2011) 1365-1399. | MR | Zbl
, and .[29] Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations, Part I: The dynamic programming principle and applications; Part II: Viscosity solutions and uniqueness. Comm. Partial Defferential Equations 8 (1983) 1101-1174; 1229-1276. | MR | Zbl
.[30] Stochastic differential equations with reflecting boundary conditions. Comm. Pure Appl. Math. 37 (1984) 511-537. | MR | Zbl
and .[31] A large deviations approach to optimal long term investment. Finance Stoch. 7 (2003) 169-195. | MR | Zbl
.[32] An Introduction to the Theory of Large Deviations. Springer, New York, 1984. | MR | Zbl
.[33] Large Deviations and Applications. CBMS-NSF Regional Conf. Ser. Appl. Math. 46. SIAM, Philadelphia, PA, 1984. | MR | Zbl
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