Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-27T19:22:09.022Z Has data issue: false hasContentIssue false
  • English
  • Français

Deterministic minimax impulse control in finite horizon: theviscosity solution approach∗∗∗

Published online by Cambridge University Press:  22 March 2012

Brahim El Asri
Brahim El Asri*
Affiliation:
Institut für Stochastik, Friedrich-Schiller-Universität Jena, Ernst-Abbe-Platz 2, 07743 Jena, Germany. [email protected]
Get access

Abstract

We study here the impulse control minimax problem. We allow the cost functionals anddynamics to be unbounded and hence the value functions can possibly be unbounded. We provethat the value function of the problem is continuous. Moreover, the value function ischaracterized as the unique viscosity solution of an Isaacs quasi-variational inequality.This problem is in relation with an application in mathematical finance.

Keywords

Impulse controlrobust controldifferential gamesquasi-variational inequalityviscosity solution
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 1 , January 2013 , pp. 63 - 77
Copyright
© EDP Sciences, SMAI, 2012

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

Barles, G., Deterministic impulse control problems. SIAM J. Control Optim. 23 (1985) 419–432. Google Scholar
Barron, E.N., Evans, L.C. and Jensen, R., Viscosity solutions of Isaaes’ equations and differential games with Lipschitz controls. J. Differential Equations 53 (1984) 213–233. Google Scholar
A. Bensoussan and J.L. Lions, Impulse Control and Quasi-Variational Inequalities. Bordes, Paris (1984)
Bernhard, P., A robust control approach to option pricing including transaction costs. Annals of International Society of Dynamic Games, Birkäuser, Boston 7 (2005) 391–416. CrossRefGoogle Scholar
Bernhard, P., El Farouq, N. and Thiery, S., An impulsive differential game arising in finance with interesting singularities. Annals of International Society of Dynamic Games, Birkäuser, Boston 8 (2006) 335–363. Google Scholar
Crandall, M., Ishii, H. and Lions, P.L., Users guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. 27 (1992) 1–67. Google Scholar
Dharmatti, S. and Ramaswamy, M., Zero-sum differential games involving hybrid controls. J. Optim. Theory Appl. 128 (2006) 75–102. Google Scholar
Dharmatti, S. and Shaiju, A.J., Infinite dimensional differential games with hybrid controls. Proc. Indian Acad. Sci. Math. 117 (2007) 233–257. Google Scholar
El Asri, B., Optimal multi-modes switching problem in infinite horizon. Stoc. Dyn. 10 (2010) 231–261. Google Scholar
N. El Farouq, G. Barles and P. Bernhard, Deterministic minimax impulse control. Appl. Math. Optim. (2010) DOI: 10.1007/s00245-009-9090-0. CrossRef
Evans, L.C. and Souganidis, P.E., Differential games and representation formulas for the solution of Hamilton-Jacobi-Isaacs equations. Indiana Univ. J. Math. 33 (1984) 773–797. Google Scholar
Fleming, W.H., The convergence problem for differential games. Ann. Math. Study 52 (1964) 195–210. Google Scholar
P.L. Lions, Generalized Solutions of Hamilton-Jacobi Equations. Pitman, London (1982)
Lions, P.L. and Souganidis, P.E., Differential games, optimal control and directional derivatives of viscosity solutions of Bellmans and Isaacs equations. SIAM J. Control Optim. 23 (1985) 566–583. Google Scholar
Shaiju, A.J. and Dharmatti, S., Differential games with continuous, switching and impulse controls. Nonlinear Anal. 63 (2005) 23–41. Google Scholar
Souganidis, P.E., Max-min representations and product formulas for the viscosity solutions of Hamilton-Jacobi equations with applications to differential games. Nonlinear Anal. 9 (1985) 217–57. Google Scholar
Yong, J.M., Systems governed by ordinary differential equations with continuous, switching and impulse controls. Appl. Math. Optim. 20 (1989) 223–235. Google Scholar
Yong, J.M., Optimal switching and impulse controls for distributed parameter systems. Systems Sci. Math. Sci. 2 (1989) 137–160. Google Scholar
Yong, J.M., Differential games with switching strategies. J. Math. Anal. Appl. 145 (1990) 455–469. Google Scholar
Yong, J.M., A zero-sum differential game in a finite duration with switching strategies. SIAM J. Control Optim. 28 (1990) 1234–1250. Google Scholar
Yong, J.M., Zero-sum differential games involving impulse controls. Appl. Math. Optim. 29 (1994) 243–261. Google Scholar