## Electronic Journal of Probability

### Generalized Dynkin games and doubly reflected BSDEs with jumps

#### Abstract

We introduce a game problem which can be seen as a generalization of the classical Dynkin game problem to the case of a nonlinear expectation ${\cal E}^g$, induced by a Backward Stochastic Differential Equation (BSDE) with jumps with nonlinear driver $g$. Let $\xi , \zeta$ be two RCLL adapted processes with $\xi \leq \zeta$. The criterium is given by ${\cal J}_{\tau , \sigma }= {\cal E}^g_{0, \tau \wedge \sigma } \left (\xi _{\tau }\textbf{1} _{\{ \tau \leq \sigma \}}+\zeta _{\sigma }\textbf{1} _{\{\sigma <\tau \}}\right ),$ where $\tau$ and $\sigma$ are stopping times valued in $[0,T]$. Under Mokobodzki’s condition, we establish the existence of a value function for this game, i.e. $\inf _{\sigma }\sup _{\tau } {\cal J}_{\tau , \sigma } = \sup _{\tau } \inf _{\sigma } {\cal J}_{\tau , \sigma }$. This value can be characterized via a doubly reflected BSDE. Using this characterization, we provide some new results on these equations, such as comparison theorems and a priori estimates. When $\xi$ and $\zeta$ are left upper semicontinuous along stopping times, we prove the existence of a saddle point. We also study a generalized mixed game problem when the players have two actions: continuous control and stopping. We then study the generalized Dynkin game in a Markovian framework and its links with parabolic partial integro-differential variational inequalities with two obstacles.

#### Article information

Source
Electron. J. Probab., Volume 21 (2016), paper no. 64, 32 pp.

Dates
Accepted: 5 October 2016
First available in Project Euclid: 25 October 2016

https://projecteuclid.org/euclid.ejp/1477395747

Digital Object Identifier
doi:10.1214/16-EJP4568

Mathematical Reviews number (MathSciNet)
MR3580030

Zentralblatt MATH identifier
1351.93170

#### Citation

Dumitrescu, Roxana; Quenez, Marie-Claire; Sulem, Agnès. Generalized Dynkin games and doubly reflected BSDEs with jumps. Electron. J. Probab. 21 (2016), paper no. 64, 32 pp. doi:10.1214/16-EJP4568. https://projecteuclid.org/euclid.ejp/1477395747

#### References

• [1] Alario-Nazaret, M. Lepeltier, J.P. and Marchal, B. Dynkin games. (Bad Honnef Workshop on stochastic processes), Lecture Notes in control and Information Sciences 43, (1982), 23–32. Springer-Verlag, Berlin. MR0814103
• [2] Bahlali K., Hamadène, S. and Mezerdi, B., Backward stochastic differential equations with two reflecting barriers and continuous with quadratic growth coefficient, Stoch Proc Appl 115, (2005) 1107–1129.
• [3] Barles, G., Imbert, C.: Second-order elliptic integro-differential equations: viscosity solutions theory revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25, (2008), 567–585
• [4] Bensoussan, A., Lions, J.L., Applications des inéquations variationnelles en contrôle stochastique. Dunod, Paris (1978)
• [5] Bismut J.M., Sur un problème de Dynkin, Z.Wahrsch. Verw. Gebiete 39 (1977) 31–53.
• [6] Buckdahn, R. and Li R. Stochastic Differential Games and Viscosity Solutions of Hamilton-Jacobi-Bellman-Isaacs Equations, SIAM J Control Optim 47(1), (2008), 444–475.
• [7] Crépey, S., Matoussi, A., Reflected and Doubly Reflected BSDEs with jumps, Annals of App. Prob. 18(5), (2008), 2041–2069.
• [8] Cohn, D. (2013). Measure Theory, second edition, Birkhauser.
• [9] Cvitanić J. and Karatzas, I., Backward stochastic differential equations with reflection and Dynkin games, Annals of Prob. 24, n.4, (1996), 2024–2056.
• [10] Dellacherie, C. and Meyer, P.-A. (1975). Probabilités et Potentiel, Chap. I-IV. Nouvelle édition. Hermann.
• [11] Dellacherie, C. and Meyer, P.-A. (1980): Probabilités et Potentiel, Théorie des Martingales, Chap. V-VIII. Nouvelle édition. Hermann.
• [12] Dumitrescu R., Quenez M.-C. and Sulem A., Optimal Stopping for Dynamic Risk Measures with Jumps and Obstacle Problems, J. Optim. Theory Applic., Vol. 167, Issue 1 (2015), pp 219–242.
• [13] Dumitrescu, R., Quenez M.C., Sulem A., A Weak Dynamic Programming Principle for Combined Optimal Stopping/ Stochastic Control with ${\cal E}^f$-Expectations, SIAM J. Control and Optimization, (2016) Vol. 54, No. 4, pp. 2090–2115.
• [14] Dumitrescu, R., Quenez M.C., Sulem A., Game options in an imperfect market with default, arXiv:1511.09041, 2015.
• [15] Dumitrescu R., Quenez M.-C. and Sulem A., Mixed generalized Dynkin game and stochastic control in a Markovian framework, Stochastics (2016), pp. 1–30, DOI 10.1080/17442508.2016.1230614
• [16] Ekeland I., Temam R., Convex Analysis and Variational Problems, Front Cover. North-Holland Publishing Company, 1976.
• [17] El Karoui, N. (1981): Les aspects probabilistes du contrôle stochastique. École d’été de Probabilités de Saint-Flour IX-1979 Lect. Notes in Math. 876 73–238.
• [18] El Karoui N. and M.C. Quenez. Non-linear Pricing Theory and Backward Stochastic Differential Equations, Financial Mathematics, Lectures Notes in Mathematics 1656, Bressanone, 1996, Editor: W.J.Runggaldier, collection Springer,1997.
• [19] El Karoui N., Kapoudjian C., Pardoux E., Peng S. and M.C. Quenez. Reflected solutions of Backward SDE’s and related obstacle problems for PDE’s, The Annals of Probability, 25(2), (1997), 702–737.
• [20] Essaky, E.H., Harraj, N., Ouknine, Backward stochastic differential equation with two reflecting barriers and jumps, Stoch Anal Appl, 23, no. 5, (2005), 921–938.
• [21] Hamadène, S., Mixed zero-sum stochastic differential game and American game options, SIAM J. Control Optim., 45(2), (2006), 496–518.
• [22] Hamadène, S. and Hassani, M., 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), 121–145.
• [23] Hamadène, S., Hassani, M. and Ouknine, Y., Backward SDEs with two rcll reflecting barriers without Mokobodzki’s hypothesis, Bull. Sci. math. 134 (2010) 874–899.
• [24] Hamadène, S. and Hdhiri, I., BSDEs with two reflecting barriers and quadratic growth coefficient without Mokobodzki’s condition, J. Appl. Math. Stoch. Anal. (2006) Art. ID 95818, 28 pp.
• [25] Hamadène, S. and Lepeltier, J.-P.,Reflected BSDEs and mixed game problem, Stochastic Process. Appl. 85(2) (2000) 177–188.
• [26] Hamadène, S. J.-P. Lepeltier, A. Matoussi, Double barrier backward SDEs with continuous coefficient. Backward stochastic differential equations (Paris, 1995?1996), 161–175, Pitman Res. Notes Math. Ser., 364, Longman, Harlow, 1997.
• [27] Hamadène, S. and Ouknine, Y., Reflected Backward SDEs with general jumps, Teor. Veroyatnost. i Primenen. (2015), 60(2), 357–376.
• [28] Karatzas I. and S. E. Shreve, Methods of mathematical finance, Applications of Mathematics (New York), 39, Springer, New York, 1998.
• [29] Kifer. Y., Game options, Finance and Stochastics, 4(4), 2000. 443–463,
• [30] Kobylanski M. and Quenez M.-C., Optimal stopping in a general framework, Electron. J. Probab., 17(72), (2012), 1–28,
• [31] Kobylanski M., Quenez M.-C. and Roger de Campagnolle M., Dynkin games in a general framework, Stochastics 86(2), (2014), pp. 304–329.
• [32] Lepeltier, J.-P. and Xu, M., Reflected backward stochastic differential equations with two rcll barriers, ESAIM: Probability and Statistics, February 2007, 11, 3–22. MR2299643
• [33] Peng, S. Nonlinear expectations, nonlinear evaluations and risk measures, 165–253, Lecture Notes in Math., 1858, Springer, Berlin, (2004),
• [34] Peng, S. and Xu M., The smallest $g$-supermartingale and reflected BSDE with single and double $L^2$ obstacles, Ann. I. H. Poincaré, Probab. Statist. 41 (2005), no. 3, 605–630.
• [35] Quenez M.-C. and Sulem A., BSDEs with jumps, optimization and applications to dynamic risk measures, Stoch Proc Appl 123 (2013), 3328–3357.
• [36] Quenez M.-C. and Sulem A., Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps, Stoch Proc Appl 124 (9), (2014), 3031–3054.
• [37] Royer M., Backward stochastic differential equations with jumps and related non-linear expectations, Stoch Proc Appl, 116 (2006), no. 10, 1358–1376.