Mixed generalized Dynkin game and stochastic control in a Markovian framework

References

• M. Alario-Nazaret, J.P. Lepeltier, and B. Marchal, Dynkin Games (Bad Honnef Workshop on Stochastic Processes), Lecture Notes in Control and Information Sciences Vol. 43. Springer-Verlag, Berlin, 1982.
   Google Scholar
• E. Bayraktar and Y.J. Huang, On the multi-dimensional controller-and-stopper game, SIAM J. Control Optim. 51 (2013), pp. 1263–1297.
   Web of Science ®Google Scholar
• A. Bensoussan and A. Friedman, Non-linear variational inequalities and differential games with stopping times, J. Funct. Anal. 16 (1974), pp. 305–352.
   Google Scholar
• A. Bensoussan and J.-L. Lions, Applications des in\’{e}quations variationnelles en contr{\^o}le stochastique [Applications of Variational Inequalities in Stochastic Control], M\’{e}thodes math\’{e}matiques de l’informatique, Vol. 6. Dunod, Paris, 1978.
   Google Scholar
• D. Bertsekas and S. Shreve, Stochastic Optimal Control: The Discrete Time Case, Academic Press, Orlando, 1978.
   Google Scholar
• J.M. Bismut, Sur un problème de Dynkin[On a Dynkin problem], Z. Wahrsch. Verw. Gebiete 39 (1977), pp. 31–53.
   Web of Science ®Google Scholar
• B. Bouchard and N. Touzi, Weak dynamic programming principle for viscosity solutions, SIAM J. Control Optim. 49(3) (2011), pp. 948–962.
   Web of Science ®Google Scholar
• R. Buckdahn and J. Li, Probabilistic interpretation for systems of Isaacs equations with two reflecting barriers, Nonlinear Differ. Equ. Appl. 16 (2009), pp. 381–420.
   Web of Science ®Google Scholar
• R. Buckdahn and J. Li, Stochastic differential games with reflection and related obstacle problems for Isaacs equations, Acta Math. Appl. Sinica Eng. Ser. 27 (2011), p. 647–678.
   Google Scholar
• R. Buckdahn and T. Nie, Generalized Hamilton--Jacobi--Bellman equations with Dirichlet boundary and stochastic exit time optimal control problem. SIAM J. Control. Optim. 54(2) (2016), pp. 602–631.
   Google Scholar
• S. Choukroun, A. Cosso, and H. Pham, Relected BSDEs with nonpositive jumps, and controller-and-stopper games, Stoch. Process. Appl. 125 (2015), pp. 597–633.
   Web of Science ®Google Scholar
• S. Crépey and A. Matoussi, Reflected and doubly reflected BSDEs with jumps, Ann. Appl. Prob. 18 (2008), pp. 2041–2069.
   Web of Science ®Google Scholar
• J. Cvitanić and I. Karatzas, Backward stochastic differential equations with reflection and Dynkin games, Ann. Prob. 24(4) (1996), pp. 2024–2056.
   Web of Science ®Google Scholar
• C. Dellacherie and P.-A. Meyer, Probabilités et Potentiel [Probabilities and Potential], Chap. I--IV. Nouvelle ed., Hermann, Paris, 1975. MR0488194.
   Google Scholar
• R. Dumitrescu, M.C. Quenez, and A. Sulem, Generalized Dynkin games and Doubly reflected BSDEs with jumps, 2013. Available at http://arxiv.org/pdf/1310.2764v2.pdf.
   Google Scholar
• R. Dumitrescu, M.C. Quenez, and A. Sulem, Optimal stopping for dynamic risk measures with jumps and obstacle problems, J. Optim. Theory Appl. 167(1) (2015), pp. 219–242. doi:10.1007/s10957-014-0635-2.
   Web of Science ®Google Scholar
• R. Dumitrescu, M.C. Quenez, and A. Sulem, Game options in an imperfect market with default (2015). Available at arXiv:1511.09041.
   Google Scholar
• R. Dumitrescu, M.C. Quenez, and A. Sulem, A weak dynamic programming principle for combined optimal stopping/stochastic control with Ef-expectations. SIAM J. Control. Optim. 54(4) (2016), pp. 2090–2115.
   Google Scholar
• S. Hamadène, Mixed zero-sum stochastic differential game and American game options, SIAM J. Control Optim. 45(2) (2006), pp. 496–518.
   Web of Science ®Google Scholar
• S. Hamadène and M. Hassani, BSDEs with two reacting barriers driven by a Brownian motion and an independent Poisson noise and related Dynkin game, Electron. J. Probab. 11(5) (2006), pp. 121–145.
   Google Scholar
• S. Hamadène and J.-P. Lepeltier, Reflected BSDEs and mixed game problem, Stochastic Process. Appl. 85 (2000), pp. 177–188.
   Web of Science ®Google Scholar
• S. Hamadène and Y. Ouknine, Reflecting backward SDEs with general jumps, Teor. Veroyatnost. i Primenen. 60(2) (2015), pp. 357–376.
   Web of Science ®Google Scholar
• S. Hamadène and H. Wang, BSDEs with two RCLL reflecting obstacles driven by a Brownian motion and Poisson random measure and related mixed zero-sum games, Stoch. Process. Appl. 119 (2009), pp. 2881–2912.
   Web of Science ®Google Scholar
• J. Jacod, Calcul Stochastique et Problèmes de martingales [Stochastic Calculus and Martingale Problems]. Lecture Notes in Mathematics, 714, Berlin: Springer, 1979.
   Google Scholar
• I. Karatzas and I. Zamfirescu, Martingale approach to stochastic differential games of control and stopping, Ann. Probab. 36(4) (2008), pp. 1495–1527.
   Web of Science ®Google Scholar
• Y. Kifer, Dynkin games and Israeli options, ISRN Probab. Stat. (2013). ID 856458, 17 pages.
   Google Scholar
• Y. Kifer and I. Yu, Game options, Finance Stoch. 4(4) (2012), pp. 443–463.
   Google Scholar
• M. Kobylanski, M.-C. Quenez, and M. Roger de Campagnolle, Dynkin games in a general framework, Stochastics. 86(2) (2014), pp. 304–329.
   Google Scholar
• B. Øksendal and A. Sulem, Applied Stochastic Control of Jump Diffusions, 2nd ed., Universitext, Springer, Berlin, 2007.
   Google Scholar
• S. Peng, A generalized dynamic programming principle and Hamilton--Jacobi--Bellman-Equation, Stoch. Stoch. Rep. 38 (1992), pp. 119–134.
   Google Scholar
• M.-C. Quenez and A. Sulem, BSDEs with jumps, optimization and applications to dynamic risk measures, Stoch. Process. Appl. 123 (2013), pp. 3328–3357.
   Web of Science ®Google Scholar
• M.-C. Quenez and A. Sulem, Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps, Stoch. Process. Appl. 124 (2014), pp. 3031–3054.
   Web of Science ®Google Scholar