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.
Mots clés : 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},
pages = {749--760},
publisher = {EDP-Sciences},
volume = {17},
number = {3},
year = {2011},
doi = {10.1051/cocv/2010023},
mrnumber = {2826978},
zbl = {1223.93121},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv/2010023/}
}
TY - JOUR AU - Zhang, Feng TI - Stochastic differential games involving impulse controls JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2011 SP - 749 EP - 760 VL - 17 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2010023/ DO - 10.1051/cocv/2010023 LA - en ID - COCV_2011__17_3_749_0 ER -
%0 Journal Article %A Zhang, Feng %T Stochastic differential games involving impulse controls %J ESAIM: Control, Optimisation and Calculus of Variations %D 2011 %P 749-760 %V 17 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2010023/ %R 10.1051/cocv/2010023 %G en %F COCV_2011__17_3_749_0
Zhang, Feng. Stochastic differential games involving impulse controls. ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 3, pp. 749-760. doi : 10.1051/cocv/2010023. http://archive.numdam.org/articles/10.1051/cocv/2010023/
[1] and , 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 | Zbl
[2] and , Stochastic differential games and viscosity solutions of Hamiltonian-Jacobi-Bellman-Isaacs equations. SIAM J. Control Optim. 47 (2008) 444-475. | MR | Zbl
[3] and , Classical and impulse stochastic control of the exchange rate using interest rates and reserves. Math. Finance 10 (2000) 141-156. | MR | Zbl
[4] , and , User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27 (1992) 1-67. | MR | Zbl
[5] and , Differential games and representation formulas for Hamilton-Jacobi equations. Indiana Univ. Math. J. 33 (1984) 773-797. | MR | Zbl
[6] and , Controlled Markov processes and viscosity solutions. Springer-Verlag, New York (2005). | MR | Zbl
[7] and , On the existence of value functions of two-player, zero-sum stochastic differential games. Indiana Univ. Math. J. 38 (1989) 293-314. | MR | Zbl
[8] , Some applications of impulse control in mathematical finance. Math. Meth. Oper. Res. 50 (1999) 493-518. | MR | Zbl
[9] and , Optimal stochastic impulse control with delayed reaction. Appl. Math. Optim. 58 (2008) 243-255. | MR | Zbl
[10] and , Diffusions, Markov processes, and martingales. John Wiley & Sons, New York (1987). | MR | Zbl
[11] and , Differential games with continuous, switching and impulse controls. Nonlinear Anal. 63 (2005) 23-41. | MR | Zbl
[12] , Systems governed by ordinary differential equations with continuous, switching and impulse controls. Appl. Math. Optim. 20 (1989) 223-235. | MR | Zbl
[13] , Zero-sum differential games involving impulse controls. Appl. Math. Optim. 29 (1994) 243-261. | MR | Zbl
Cité par Sources :
