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Applied Mathematics and Optimization

, Volume 61, Issue 3, pp 353–378 | Cite as

Deterministic Minimax Impulse Control

  • Naïma El FarouqEmail author
  • Guy Barles
  • Pierre Bernhard
Article

Abstract

We prove the uniqueness of the viscosity solution of an Isaacs quasi-variational inequality arising in an impulse control minimax problem, motivated by an application in mathematical finance.

Keywords

Impulse control Robust control Differential games Quasi-variational inequality Viscosity solution 

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References

  1. 1.
    Bardi, M., Capuzzo-Dolcetta, I.: Optimal and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhaüser, Basel (1997) zbMATHCrossRefGoogle Scholar
  2. 2.
    Barles, G.: Solutions de Viscosité des Équations de Hamilton-Jacobi. Mathématiques & Applications. Springer, Berlin, Heidelberg, New York (1994) zbMATHGoogle Scholar
  3. 3.
    Barles, G.: Deterministic impulse control problems. SIAM J. Control Optim. 23, 419–432 (1985) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bernhard, P.: A robust control approach to option pricing including transaction costs. In: Annals of the ISDG, vol. 7, pp. 391–416. Birkhaüser, Basel (2005) Google Scholar
  5. 5.
    Bernhard, P., El Farouq, N., Thiery, S.: An impulsive differential game arising in finance with interesting singularities. In: Annals of the ISDG, vol. 8, pp. 335–363. Birkhaüser, Basel (2006) Google Scholar
  6. 6.
    Crandall, M.G., Lions, P.L.: Viscosity solutions of Hamilton-Jacobi equations. Trans. Am. Math. Soc. 177, 1–42 (1983) CrossRefMathSciNetGoogle Scholar
  7. 7.
    Dharmatti, S., Shaiju, A.J.: Infinite dimensional differential games with hybrid controls. Proc. Indian Acad. Sci. Math. 117, 233–257 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Dharmatti, S., Ramaswamy, M.: Zero-sum differential games involving hybrid controls. J. Optim. Theory Appl. 128, 75–102 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Evans, L.C., Souganidis, P.E.: Differential games and representation formulas for the solution of Hamilton-Jacobi-Isaacs equations. Indiana Univ. J. Math. 33, 773–797 (1984) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Fleming, W.H.: The convergence problem for differential games, 2. Ann. Math. Study 52, 195–210 (1964) zbMATHMathSciNetGoogle Scholar
  11. 11.
    Lions, P.L.: Generalized Solutions of Hamilton-Jacobi Equations. Pitman, London (1982) zbMATHGoogle Scholar
  12. 12.
    Lions, P.L., Souganidis, P.E.: Differential games, optimal control and directional derivatives of viscosity solutions of Bellman’s and Isaacs’ equations. SIAM J. Control Optim. 23, 566–583 (1985) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Shaiju, A.J., Dharmatti, S.: Differential games with continuous, switching and impulse controls. Nonlinear Anal. 63, 23–41 (2005) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Souganidis, P.E.: Max-min representations and product formulas for the viscosity solutions of Hamilton-Jacobi equations with applications to differential games. Nonlinear Anal. Theory Methods Appl. 9, 217–257 (1985) CrossRefMathSciNetGoogle Scholar
  15. 15.
    Yong, J.M.: Zero-sum differential games involving impulse controls. Appl. Math. Optim. 29, 243–261 (1994) zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Naïma El Farouq
    • 1
    Email author
  • Guy Barles
    • 2
  • Pierre Bernhard
    • 3
  1. 1.Laboratoire d’Informatique, de Modélisation et d’Optimisation des Systèmes, UMR CNRS/UBP 6158Université Blaise Pascal (Clermont-Ferrand II)Aubière cedexFrance
  2. 2.Laboratoire de Mathématiques et Physique Théorique, Fédération Denis PoissonUniversité de ToursToursFrance
  3. 3.INRIA-Sophia Antipolis-MéditerranéeSophia Antipolis cedexFrance

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