Stochastic differential games involving impulse controls
ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 3, p. 749-760

A zero-sum stochastic differential game problem on infinite horizon with continuous and impulse controls is studied. We obtain the existence of the value of the game and characterize it as the unique viscosity solution of the associated system of quasi-variational inequalities. We also obtain a verification theorem which provides an optimal strategy of the game.

DOI : https://doi.org/10.1051/cocv/2010023
Classification:  91A15,  49N25,  49L20
Keywords: stochastic differential game, impulse control, quasi-variational inequalities, viscosity solution
@article{COCV_2011__17_3_749_0,
     author = {Zhang, Feng},
     title = {Stochastic differential games involving impulse controls},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {17},
     number = {3},
     year = {2011},
     pages = {749-760},
     doi = {10.1051/cocv/2010023},
     zbl = {1223.93121},
     mrnumber = {2826978},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2011__17_3_749_0}
}
Zhang, Feng. Stochastic differential games involving impulse controls. ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 3, pp. 749-760. doi : 10.1051/cocv/2010023. http://www.numdam.org/item/COCV_2011__17_3_749_0/

[1] K.E. Breke and B. Øksendal, A verification theorem for combined stochastic control and impulse control, in Stochastic analysis and related topics VI, J. Decreusefond, J. Gjerde, B. Øksendal and A. Üstünel Eds., Birkhauser, Boston (1997) 211-220. | MR 1652344 | Zbl 0894.93039

[2] R. Buckdahn and J. Li, Stochastic differential games and viscosity solutions of Hamiltonian-Jacobi-Bellman-Isaacs equations. SIAM J. Control Optim. 47 (2008) 444-475. | MR 2373477 | Zbl 1157.93040

[3] A. Cadenillas and F. Zapatero, Classical and impulse stochastic control of the exchange rate using interest rates and reserves. Math. Finance 10 (2000) 141-156. | MR 1802595 | Zbl 1034.91036

[4] M.G. Crandall, H. Ishii and P-L. Lions, User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27 (1992) 1-67. | MR 1118699 | Zbl 0755.35015

[5] L.C. Evans and P.E. Souganidis, Differential games and representation formulas for Hamilton-Jacobi equations. Indiana Univ. Math. J. 33 (1984) 773-797. | MR 756158 | Zbl 1169.91317

[6] W.H. Fleming and H.M. Soner, Controlled Markov processes and viscosity solutions. Springer-Verlag, New York (2005). | MR 2179357 | Zbl 0773.60070

[7] W.H. Fleming and P.E. Souganidis, On the existence of value functions of two-player, zero-sum stochastic differential games. Indiana Univ. Math. J. 38 (1989) 293-314. | MR 997385 | Zbl 0686.90049

[8] R. Korn, Some applications of impulse control in mathematical finance. Math. Meth. Oper. Res. 50 (1999) 493-518. | MR 1731297 | Zbl 0942.91048

[9] B. Øksendal and A. Sulem, Optimal stochastic impulse control with delayed reaction. Appl. Math. Optim. 58 (2008) 243-255. | MR 2439661 | Zbl 1161.93029

[10] L.C.G. Rogers and D. Williams, Diffusions, Markov processes, and martingales. John Wiley & Sons, New York (1987). | MR 921238 | Zbl 0627.60001 | Zbl 0826.60002

[11] A.J. Shaiju and S. Dharmatti, Differential games with continuous, switching and impulse controls. Nonlinear Anal. 63 (2005) 23-41. | MR 2167312 | Zbl 1132.91356

[12] J. Yong, Systems governed by ordinary differential equations with continuous, switching and impulse controls. Appl. Math. Optim. 20 (1989) 223-235. | MR 1004708 | Zbl 0691.49031

[13] J. Yong, Zero-sum differential games involving impulse controls. Appl. Math. Optim. 29 (1994) 243-261. | MR 1264011 | Zbl 0808.90142