A. Altarovici, O. Bokanowski, and H. Zidani, A general Hamilton-Jacobi framework for non-linear state-constrained control problems, ESAIM: Control, Optimisation and Calculus of Variations, vol.19, issue.2, pp.337-357, 2013.
DOI : 10.1051/cocv/2012011

M. Assellaou, O. Bokanowski, and H. Zidani, Error estimates for second order hamilton-jacobibellman equations. approximation of probabilistic reachable sets. Discrete and Continuous Dynamical Systems -Serie A, pp.3933-3964, 2015.

J. Aubin and H. Frankowska, The Viability Kernel Algorithm for Computing Value Functions of Infinite Horizon Optimal Control Problems, Journal of Mathematical Analysis and Applications, vol.201, issue.2, pp.555-576, 1996.
DOI : 10.1006/jmaa.1996.0273

M. Bardi and P. Goatin, Invariant Sets for Controlled Degenerate Diffusions: A Viscosity Solutions Approach, Stochastic Analysis, Control, Optimization and Applications: A Volume in Honor of W. H. Fleming, pp.191-208, 1999.
DOI : 10.1007/978-1-4612-1784-8_11

G. Barles and E. Rouy, A strong comparison result for the bellman equation arising in stochastic exit time control problems and its applications, Communications in Partial Differential Equations, vol.43, issue.11-12, pp.11-121945, 1998.
DOI : 10.1137/0324032

O. Bokanowski, B. Brüder, S. Maroso, and H. Zidani, Numerical approximation for a superreplication problem under gamma contraints, SIAM J. of Numer. Analysis, issue.3, pp.472289-2320, 2009.

O. Bokanowski, N. Forcadel, and H. Zidani, Reachability and Minimal Times for State Constrained Nonlinear Problems without Any Controllability Assumption, SIAM Journal on Control and Optimization, vol.48, issue.7, pp.4292-4316, 2010.
DOI : 10.1137/090762075
URL : https://hal.archives-ouvertes.fr/hal-00395589

O. Bokanowski, A. Picarelli, and H. Zidani, Dynamic Programming and Error Estimates for Stochastic Control Problems with Maximum Cost, Applied Mathematics & Optimization, vol.9, issue.3, pp.1-39
DOI : 10.1007/s00245-014-9255-3
URL : https://hal.archives-ouvertes.fr/hal-00931025

B. Bouchard and N. M. Dang, Optimal control versus stochastic target problems: An equivalence result, Systems & Control Letters, vol.61, issue.2, pp.343-346, 2012.
DOI : 10.1016/j.sysconle.2011.11.010
URL : https://hal.archives-ouvertes.fr/hal-00556509

B. Bouchard, R. Elie, and C. Imbert, Optimal Control under Stochastic Target Constraints, SIAM Journal on Control and Optimization, vol.48, issue.5, pp.3501-3531, 2010.
DOI : 10.1137/090757629
URL : https://hal.archives-ouvertes.fr/hal-00373306

B. Bouchard, R. Elie, and N. Touzi, Stochastic Target Problems with Controlled Loss, SIAM Journal on Control and Optimization, vol.48, issue.5, pp.3123-3150, 2009.
DOI : 10.1137/08073593X
URL : https://hal.archives-ouvertes.fr/hal-00323383

B. Bouchard and M. Nutz, Weak Dynamic Programming for Generalized State Constraints, SIAM Journal on Control and Optimization, vol.50, issue.6, pp.3344-3373, 2012.
DOI : 10.1137/110852942
URL : https://hal.archives-ouvertes.fr/hal-00590874

B. Brüder, Super-replication of European options with a derivative asset under constrained finite variation strategies, 2005.

I. , C. Dolcetta, and P. L. Lions, Hamilton-jacobi equations with state constraints, Trans. Amer. Math. Soc, vol.318, pp.643-683, 1990.

P. Cardaliaguet, M. Quincampoix, and P. Saint-pierre, Optimal times for constrained nonlinear control problems without local controllability, Applied Mathematics & Optimization, vol.8, issue.2, pp.21-42, 1997.
DOI : 10.1007/BF02683336

P. Cardaliaguet, M. Quincampoix, and P. Saint-pierre, Numerical schemes for discontinuous value function of optimal control. Set-valued Analysis, pp.111-126, 2000.

M. G. Crandall, H. Ishii, and P. L. Lions, user's guide to viscosity solutions\\ of second order\\ partial differential equations, Bulletin of the American Mathematical Society, vol.27, issue.1, pp.1-67, 1992.
DOI : 10.1090/S0273-0979-1992-00266-5

M. Falcone, T. Giorgi, and P. Loreti, Level Sets of Viscosity Solutions: some Applications to Fronts and Rendez-vous Problems, SIAM Journal on Applied Mathematics, vol.54, issue.5, pp.1335-1354, 1994.
DOI : 10.1137/S0036139992233069

H. Frankowska and R. B. Vinter, Existence of Neighboring Feasible Trajectories: Applications to Dynamic Programming for State-Constrained Optimal Control Problems, Journal of Optimization Theory and Applications, vol.35, issue.1, pp.20-40, 2000.
DOI : 10.1023/A:1004668504089

U. G. Haussmann and J. P. Lepeltier, On the Existence of Optimal Controls, SIAM Journal on Control and Optimization, vol.28, issue.4, pp.851-902, 1990.
DOI : 10.1137/0328049

C. Hermosilla and H. Zidani, Infinite horizon problems on stratifiable state-constraints sets, Journal of Differential Equations, vol.258, issue.4, pp.1420-1460, 2015.
DOI : 10.1016/j.jde.2014.11.001
URL : https://hal.archives-ouvertes.fr/hal-00955921

H. Ishii and S. Koike, A New Formulation of State Constraint Problems for First-Order PDEs, SIAM Journal on Control and Optimization, vol.34, issue.2, pp.554-571, 1996.
DOI : 10.1137/S0363012993250268

H. Ishii and P. Loreti, A class of stochastic optimal control problems with state constraint, Indiana University Mathematics Journal, vol.51, issue.5, pp.1167-1196, 2002.
DOI : 10.1512/iumj.2002.51.2079

A. B. Kurzhanski and P. Varaiya, Ellipsoidal Techniques for Reachability Under State Constraints, SIAM Journal on Control and Optimization, vol.45, issue.4, pp.1369-1394, 2006.
DOI : 10.1137/S0363012903437605

J. M. Lasry and P. L. Lions, Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints, Mathematische Annalen, vol.56, issue.4, pp.583-630, 1989.
DOI : 10.1007/BF01442856

P. D. Loewen, Existence Theory for a Stochastic Bolza Problem, IMA Journal of Mathematical Control and Information, vol.4, issue.4, pp.301-320, 1987.
DOI : 10.1093/imamci/4.4.301

K. Margellos and J. Lygeros, Hamilton???Jacobi Formulation for Reach???Avoid Differential Games, IEEE Transactions on Automatic Control, vol.56, issue.8, pp.1849-1861, 2011.
DOI : 10.1109/TAC.2011.2105730

M. Motta and C. Sartori, Uniqueness results for boundary value problems arising from finite fuel and other singular and unbounded stochastic control problems, Discrete Contin. Dynam. Systems, vol.21, issue.2, pp.513-535, 2008.

S. Osher and A. J. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, Journal of Computational Physics, vol.79, issue.1, pp.12-49, 1988.
DOI : 10.1016/0021-9991(88)90002-2

H. Pham, On some recent aspects of stochastic control and their applications, Probability Surveys, vol.2, issue.0, pp.506-549, 2005.
DOI : 10.1214/154957805100000195
URL : https://hal.archives-ouvertes.fr/hal-00009267

F. Rampazzo and C. Sartori, The Minimum Time Function with Unbounded Controls, J. Math. Systems, vol.8, issue.2, pp.1-34, 1998.

F. Rampazzo and C. Sartori, Hamilton-Jacobi-Bellman equations with fast gradientdependence, Indiana Univ. Math. J, vol.49, issue.3, pp.1043-1077, 2000.

H. M. Soner, Optimal Control with State-Space Constraint I, SIAM Journal on Control and Optimization, vol.24, issue.3, pp.552-561, 1986.
DOI : 10.1137/0324032

H. M. Soner, Optimal Control with State-Space Constraint. II, SIAM Journal on Control and Optimization, vol.24, issue.6, pp.1110-1122, 1986.
DOI : 10.1137/0324067

H. M. Soner and N. Touzi, Dynamic programming for stochastic target problems and geometric flows, Journal of the European Mathematical Society, vol.4, issue.3, pp.201-236, 2002.
DOI : 10.1007/s100970100039

H. M. Soner and N. Touzi, Stochastic Target Problems, Dynamic Programming, and Viscosity Solutions, SIAM Journal on Control and Optimization, vol.41, issue.2, pp.404-424, 2002.
DOI : 10.1137/S0363012900378863

N. Touzi, Optimal Stochastic Control, Stochastic Target Problems, and Backward SDE. Fields Institute Monographs, 2012.

J. Yong and X. Y. Zhou, Stochastic controls, Applications of Mathematics, vol.43, 1999.
DOI : 10.1007/978-1-4612-1466-3