M. Arisawa, Ergodic problem for the Hamilton-Jacobi-Bellman equation. II. Annales de l'Institut Henri Poincare (C) Non Linear Analysis, pp.1-24, 1998.

M. Bardi and P. Goatin, Invariant sets for controlled degenerate di¤usions: A viscosity solutions approach, Stochastic Analysis, Control, Optimization and Applications, Systems Control: Foundations and Applications, pp.191-208, 1999.

. Imranh, . Biswas, . Espenr, K. Jakobsen, and . Karlsen, Viscosity solutions for a system of integro-pdes and connections to optimal switching and control of jump-di¤usion processes, Applied Mathematics and Optimization, vol.62, issue.1, pp.47-80, 2010.

B. Bouchard and N. Touzi, Weak Dynamic Programming Principle for Viscosity Solutions, SIAM Journal on Control and Optimization, vol.49, issue.3, pp.948-962, 2011.
DOI : 10.1137/090752328
URL : https://hal.archives-ouvertes.fr/hal-00367355

R. Buckdahn, D. Goreac, and M. Quincampoix, Existence of Asymptotic Values for Nonexpansive Stochastic Control Systems, Applied Mathematics & Optimization, vol.18, issue.9, pp.1-28
DOI : 10.1007/s00245-013-9230-4
URL : https://hal.archives-ouvertes.fr/hal-00879274

R. Buckdahn, S. Peng, M. Quincampoix, and C. Rainer, Existence of stochastic control under state constraints, Comptes Rendus de l'Acad??mie des Sciences - Series I - Mathematics, vol.327, issue.1, pp.17-22, 1998.
DOI : 10.1016/S0764-4442(98)80096-7

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

W. Feller, An Introduction to Probability Theory and its Applications, 1971.

D. Goreac, Asymptotic Control for a Class of Piecewise Deterministic Markov Processes Associated to Temperate Viruses, SIAM Journal on Control and Optimization, vol.53, issue.4, p.1089528, 2014.
DOI : 10.1137/140998913
URL : https://hal.archives-ouvertes.fr/hal-01089528

D. Goreac, . Serea, and . Oana-silvia, Uniform Assymptotics in the Average Continuous Control of Piecewise Deterministic Markov Processes : Vanishing Approach, ESAIM: ProcS, pp.168-177, 2014.
DOI : 10.1051/proc/201445017
URL : https://hal.archives-ouvertes.fr/hal-01069146

G. H. Hardy and J. E. Littlewood, Tauberian theorems concerning power series and dirichlet's series whose coe¢cients are positive, Proceedings of the, pp.2-13174, 1914.
DOI : 10.1112/plms/s2-13.1.174
URL : http://plms.oxfordjournals.org/cgi/content/short/s2-13/1/174

N. V. Krylov, On the rate of convergence of ?nite-di¤erence approximations for Bellman's equations with variable coe¢cients. Probab. Theory Related Fields, pp.1-16, 2000.

X. Li, M. Quincampoix, and J. Renault, Generalized limit value in optimal control, 2015.

A. Øksendal and B. Sulem, Applied Stochastic Control of Jump Di¤usions. Universitext, 2007.

M. Oliu-barton and G. Vigeral, A uniform Tauberian theorem in optimal control Advances in Dynamic Games, Annals of the International Society of Dynamic Games, 2013.

P. Shi-ge and X. H. Zhu, The viability property of controlled jump di¤usion processes, Acta Math. Sin. (Engl. Ser, issue.8, pp.241351-1368, 2008.

H. Pham, Optimal stopping of controlled jump di¤usion processes: a viscosity solution approach, J. Math. Systems Estim. Control, vol.8, issue.1, p.27, 1998.

J. Renault, General limit value in dynamic programming, Journal of Dynamics and Games, vol.1, issue.3, pp.471-484, 2014.
DOI : 10.3934/jdg.2014.1.471
URL : https://hal.archives-ouvertes.fr/hal-00769763