# Stochastic differential games involving impulse controls

ESAIM: Control, Optimisation and Calculus of Variations (2011)

- Volume: 17, Issue: 3, page 749-760
- ISSN: 1292-8119

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topZhang, Feng. "Stochastic differential games involving impulse controls." ESAIM: Control, Optimisation and Calculus of Variations 17.3 (2011): 749-760. <http://eudml.org/doc/272918>.

@article{Zhang2011,

abstract = {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.},

author = {Zhang, Feng},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {stochastic differential game; impulse control; quasi-variational inequalities; viscosity solution; stochastic differential games; quasi-variational inequality; Elliott-Kalton value; verification theorem},

language = {eng},

number = {3},

pages = {749-760},

publisher = {EDP-Sciences},

title = {Stochastic differential games involving impulse controls},

url = {http://eudml.org/doc/272918},

volume = {17},

year = {2011},

}

TY - JOUR

AU - Zhang, Feng

TI - Stochastic differential games involving impulse controls

JO - ESAIM: Control, Optimisation and Calculus of Variations

PY - 2011

PB - EDP-Sciences

VL - 17

IS - 3

SP - 749

EP - 760

AB - 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.

LA - eng

KW - stochastic differential game; impulse control; quasi-variational inequalities; viscosity solution; stochastic differential games; quasi-variational inequality; Elliott-Kalton value; verification theorem

UR - http://eudml.org/doc/272918

ER -

## References

top- [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. Zbl0894.93039MR1652344
- [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. Zbl1157.93040MR2373477
- [3] A. Cadenillas and F. Zapatero, Classical and impulse stochastic control of the exchange rate using interest rates and reserves. Math. Finance10 (2000) 141–156. Zbl1034.91036MR1802595
- [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. Zbl0755.35015MR1118699
- [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. Zbl1169.91317MR756158
- [6] W.H. Fleming and H.M. Soner, Controlled Markov processes and viscosity solutions. Springer-Verlag, New York (2005). Zbl0773.60070MR2179357
- [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. Zbl0686.90049MR997385
- [8] R. Korn, Some applications of impulse control in mathematical finance. Math. Meth. Oper. Res.50 (1999) 493–518. Zbl0942.91048MR1731297
- [9] B. Øksendal and A. Sulem, Optimal stochastic impulse control with delayed reaction. Appl. Math. Optim.58 (2008) 243–255. Zbl1161.93029MR2439661
- [10] L.C.G. Rogers and D. Williams, Diffusions, Markov processes, and martingales. John Wiley & Sons, New York (1987). Zbl0826.60002MR921238
- [11] A.J. Shaiju and S. Dharmatti, Differential games with continuous, switching and impulse controls. Nonlinear Anal.63 (2005) 23–41. Zbl1132.91356MR2167312
- [12] J. Yong, Systems governed by ordinary differential equations with continuous, switching and impulse controls. Appl. Math. Optim.20 (1989) 223–235. Zbl0691.49031MR1004708
- [13] J. Yong, Zero-sum differential games involving impulse controls. Appl. Math. Optim.29 (1994) 243–261. Zbl0808.90142MR1264011

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