I. Asiminoaei and A. Rascanu, Approximation and simulation of stochastic variational inequalitie-splitting up method, Numer. Funct. Anal. Optim, vol.18, pp.3-4251, 1996.

J. P. Aubin and H. Frankowska, Set-valued analysis, 1990.

V. Bally and V. Rabiet, Asymptotic behavior for multi-scale PDMP's. working paper or preprint, 2015.

V. Barbu, Optimal control of variational inequalities, Research Notes in Mathematics, vol.100, 1984.

G. Barles, R. Buckdahn, and E. Pardoux, Backward stochastic differential equations and integral-partial differential equations, Stochastics An International Journal of Probability and Stochastic Processes, vol.60, issue.1, pp.57-83, 1997.
DOI : 10.1080/17442509708834099

A. G. Bhatt and V. S. Borkar, Occupation measures for controlled Markov processes: Characterization and optimality, Ann. of Probability, vol.24, pp.1531-1562, 1996.

P. Billingsley, Convergence of probability measures Wiley Series in Probability and Statistics: Probability and Statistics, 1999.

J. Bismut, Conjugate convex functions in optimal stochastic control, Journal of Mathematical Analysis and Applications, vol.44, issue.2, pp.384-404, 1973.
DOI : 10.1016/0022-247X(73)90066-8

V. Borkar and V. Gaitsgory, Averaging of Singularly Perturbed Controlled Stochastic Differential Equations, Applied Mathematics and Optimization, vol.56, issue.2, pp.169-209, 2007.
DOI : 10.1007/s00245-007-0893-6

P. Brémaud, Point processes and queues : martingale dynamics. Springer series in statistics, 1981.
DOI : 10.1007/978-1-4684-9477-8

H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, 1973.

R. Buckdahn, D. Goreac, and M. Quincampoix, Stochastic Optimal Control and Linear Programming Approach, Applied Mathematics & Optimization, vol.18, issue.4, pp.257-276, 2011.
DOI : 10.1007/s00245-010-9120-y
URL : https://hal.archives-ouvertes.fr/inria-00636557

R. Carbone, B. Ferrario, and M. Santacroce, Backward stochastic differential equations driven by ca??dla??g martingales, Teoriya Veroyatnostei i ee Primeneniya, vol.52, issue.2, pp.375-385, 2007.
DOI : 10.4213/tvp181

N. Samuel, R. J. Cohen, and . Elliott, Comparisons for backward stochastic differential equations on Markov chains and related no-arbitrage conditions, Ann. Appl. Probab, vol.20, issue.1, pp.267-311, 2010.

F. Confortola and M. Fuhrman, Backward stochastic differential equations associated to jump Markov processes and applications. Stochastic Process, Appl, vol.124, issue.1, pp.289-316, 2014.

F. Confortola, M. Fuhrman, and J. Jacod, Backward stochastic differential equation driven by a marked point process: An elementary approach with an application to optimal control, The Annals of Applied Probability, vol.26, issue.3, 2015.
DOI : 10.1214/15-AAP1132

D. L. Cook, A. N. Gerber, and S. J. Tapscott, Modelling stochastic gene expression: Implications for haploinsufficiency, Proc. Natl. Acad. Sci. USA, pp.15641-15646, 1998.

O. L. Costa and F. Dufour, A linear programming formulation for constrained discounted continuous control for piecewise deterministic Markov processes, Journal of Mathematical Analysis and Applications, vol.424, issue.2, pp.892-914, 2015.
DOI : 10.1016/j.jmaa.2014.11.036
URL : https://hal.archives-ouvertes.fr/hal-01246215

A. Crudu, A. Debussche, A. Muller, and O. Radulescu, Convergence of stochastic gene networks to hybrid piecewise deterministic processes. The Annals of Applied Probability, pp.1822-1859
URL : https://hal.archives-ouvertes.fr/hal-00553482

A. Crudu, A. Debussche, and O. Radulescu, Hybrid stochastic simplifications for multiscale gene networks, BMC Systems Biology, vol.3, issue.1, pp.3-89, 2009.
DOI : 10.1186/1752-0509-3-89
URL : https://hal.archives-ouvertes.fr/hal-00784449

M. H. Davis, Piecewise-deterministic Markov-processes -A general-class of non-diffusion stochastic-models, Journal of the Royal Statistical Society Series B-Methodological, vol.46, issue.3, pp.353-388, 1984.

M. H. Davis, Markov models and optimization, volume 49 of Monographs on Statistics and Applied Probability, 1993.

F. Dufour and R. H. Stockbridge, On the existence of strict optimal controls for constrained, controlled Markov processes in continuous time, Stochastics An International Journal of Probability and Stochastic Processes, vol.6, issue.1, pp.55-78, 2012.
DOI : 10.1137/S0363012995295516
URL : https://hal.archives-ouvertes.fr/hal-00648221

V. Gaitsgory and M. Quincampoix, Linear Programming Approach to Deterministic Infinite Horizon Optimal Control Problems with Discounting, SIAM Journal on Control and Optimization, vol.48, issue.4, pp.2480-2512, 2009.
DOI : 10.1137/070696209

M. Anouar, E. Gassous, and . Rotenstein, Multivalued backward stochastic differential equations with oblique subgradients, Stochastic Process. Appl, vol.125, issue.8, pp.3170-3195, 2015.

T. Daniel and . Gillespie, Exact stochastic simulation of coupled chemical reactions, The Journal of Physical Chemistry, vol.81, issue.25, pp.2340-2361, 1977.

R. Goebel and R. T. Rockafellar, Generalized conjugacy in hamilton-jacobi theory for fully convex lagrangians, Journal of Convex Analysis, vol.9, issue.2, 2002.

D. Goreac and . Viability, Invariance and Reachability for Controlled Piecewise Deterministic Markov Processes Associated to Gene Networks. ESAIM-Control Optimisation and Calculus of Variations, pp.401-426, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00727725

D. Goreac and O. Serea, Linearization Techniques for Controlled Piecewise Deterministic Markov Processes; Application to Zubov???s Method, Applied Mathematics & Optimization, vol.21, issue.3, pp.209-238, 2012.
DOI : 10.1007/s00245-012-9169-x

D. Goreac and O. Serea, Optimality Issues for a Class of Controlled Singularly Perturbed Stochastic Systems, Journal of Optimization Theory and Applications, vol.84, issue.1, pp.22-52, 2015.
DOI : 10.1007/s10957-015-0738-4
URL : https://hal.archives-ouvertes.fr/hal-00987521

C. M. Guldberg and P. Waage, Studies Concerning Affinity. C. M. Forhandlinger: Videnskabs- Selskabet i Christiana, p.1864

J. Hasty, J. Pradines, M. Dolnik, and J. J. Collins, Noise-based switches and amplifiers for gene expression, Proceedings of the National Academy of Sciences, vol.97, issue.5, pp.2075-2080, 2000.
DOI : 10.1073/pnas.040411297

Y. Hu and S. Tang, Multi-dimensional BSDE with oblique reflection and optimal switching. Probab. Theory Related Fields, pp.89-121, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00158569

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 1981.

I. Kharroubi and T. Lim, Progressive Enlargement of Filtrations and Backward Stochastic Differential Equations with Jumps, Journal of Theoretical Probability, vol.120, issue.10, pp.683-724, 2014.
DOI : 10.1007/s10959-012-0428-1

T. G. Kurtz and R. H. Stockbridge, Existence of Markov Controls and Characterization of Optimal Markov Controls, SIAM Journal on Control and Optimization, vol.36, issue.2, pp.609-653, 1998.
DOI : 10.1137/S0363012995295516

J. B. Lasserre, D. Henrion, C. Prieur, and T. E. , Nonlinear Optimal Control via Occupation Measures and LMI-Relaxations, SIAM Journal on Control and Optimization, vol.47, issue.4, pp.1643-1666, 2008.
DOI : 10.1137/070685051
URL : https://hal.archives-ouvertes.fr/hal-00136032

E. Pardoux and S. G. Peng, Adapted solution of a backward stochastic differential equation, Systems & Control Letters, vol.14, issue.1, pp.55-61, 1990.
DOI : 10.1016/0167-6911(90)90082-6

E. Pardoux, Backward stochastic differential equations with subdifferential operator and related variational inequalities. Stochastic Process, Appl, vol.76, issue.2, pp.191-215, 1998.

E. Pardoux, Backward stochastic variational inequalities, Stochastics and Stochastic Reports, vol.293, issue.3-4, pp.159-167, 1999.
DOI : 10.1080/17442509908834208

R. Aurel, E. A¸scanua¸scanu, and . Rotenstein, A non-convex setup for multivalued differential equations driven by oblique subgradients, Nonlinear Anal, vol.111, pp.82-104, 2014.

R. H. Stockbridge, Time-Average Control of Martingale Problems: Existence of a Stationary Solution, The Annals of Probability, vol.18, issue.1, pp.190-205, 1990.
DOI : 10.1214/aop/1176990944

S. Tang and X. Li, Necessary Conditions for Optimal Control of Stochastic Systems with Random Jumps, SIAM Journal on Control and Optimization, vol.32, issue.5, pp.1447-1475, 1994.
DOI : 10.1137/S0363012992233858

J. Xia, Backward stochastic differential equation with random measures, Acta Math. Appl. Sinica (English Ser.), vol.16, issue.3, pp.225-234, 2000.

G. G. Yin and Q. Zhang, Continuous-Time Markov Chains and Applications. A singular Perturbation Approach, 1997.