Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-22T17:16:23.003Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic differential games involving impulse controls*

Published online by Cambridge University Press:  23 April 2010

Feng Zhang
Feng Zhang*
Affiliation:
School of Mathematics, Shandong University, Jinan 250100, P.R. China. [email protected]
Get access

Abstract

A zero-sum stochastic differential gameproblem on infinite horizon with continuous and impulse controls isstudied. We obtain the existence of the value of the game andcharacterize it as the unique viscosity solution of the associatedsystem of quasi-variational inequalities. We also obtain averification theorem which provides an optimal strategy of the game.

Keywords

Stochastic differentialgameimpulse controlquasi-variational inequalitiesviscositysolution
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 3 , July 2011 , pp. 749 - 760
Copyright
© EDP Sciences, SMAI, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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.
Buckdahn, R. and Stochastic di, J. Lifferential games and viscosity solutions of Hamiltonian-Jacobi-Bellman-Isaacs equations. SIAM J. Control Optim. 47 (2008) 444–475. CrossRef
Cadenillas, A. and Zapatero, F., Classical and impulse stochastic control of the exchange rate using interest rates and reserves. Math. Finance 10 (2000) 141–156. CrossRef
Crandall, M.G., Ishii, H. and Lions, P-L., User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. 27 (1992) 1–67. CrossRef
Evans, L.C. and Souganidis, P.E., Differential games and representation formulas for Hamilton-Jacobi equations. Indiana Univ. Math. J. 33 (1984) 773–797. CrossRef
W.H. Fleming and H.M. Soner, Controlled Markov processes and viscosity solutions. Springer-Verlag, New York (2005).
Fleming, W.H. and Souganidis, P.E., On the existence of value functions of two-player, zero-sum stochastic differential games. Indiana Univ. Math. J. 38 (1989) 293–314. CrossRef
Korn, R., Some applications of impulse control in mathematical finance. Math. Meth. Oper. Res. 50 (1999) 493–518. CrossRef
Øksendal, B. and Sulem, A., Optimal stochastic impulse control with delayed reaction. Appl. Math. Optim. 58 (2008) 243–255. CrossRef
L.C.G. Rogers and D. Williams, Diffusions, Markov processes, and martingales. John Wiley & Sons, New York (1987).
Shaiju, A.J. and Dharmatti, S., Differential games with continuous, switching and impulse controls. Nonlinear Anal. 63 (2005) 23–41. CrossRef
Yong, J., Systems governed by ordinary differential equations with continuous, switching and impulse controls. Appl. Math. Optim. 20 (1989) 223–235. CrossRef
Yong, J., Zero-sum differential games involving impulse controls. Appl. Math. Optim. 29 (1994) 243–261. CrossRef