Advertisement

Applied Mathematics & Optimization

, Volume 78, Issue 1, pp 25–60 | Cite as

Solution to HJB Equations with an Elliptic Integro-Differential Operator and Gradient Constraint

  • Harold A. Moreno-Franco
Article
  • 242 Downloads

Abstract

The main goal of this paper is to establish existence, regularity and uniqueness results for the solution of a Hamilton–Jacobi–Bellman (HJB) equation, whose operator is an elliptic integro-differential operator. The HJB equation studied in this work arises in singular stochastic control problems where the state process is a controlled d-dimensional Lévy process.

Keywords

HJB equation NIDD problem Integro-differential operator Stochastic control problem Lévy process 

Mathematics Subject Classification

49L99 45K05 93E20 

Notes

Acknowledgements

The results in this paper are part of the Ph.D. thesis of the author H. A. Moreno-Franco [31], under the supervision of Dr. Daniel Hernández-Hernández and Dr. Víctor Rivero. The author would like to thank: CONACyT and CIMAT for the Ph.D. fellowship and facilities provided; National Research University Higher School of Economics for the financial support in finishing this project; his doctoral advisors of thesis Dr. Daniel Hernández-Hernández and Dr. Víctor Rivero, for their guidance on this work; and finally, his readers of thesis Dr. Jose Luis Menaldi, Dr. Renato Iturriaga, Dr. Hector Sanchez and Dr. Juan Carlos Pardo, for their advice and suggestions.

References

  1. 1.
    Adams, R.A., Fournier, J.J.F.: Sobolev Spaces. Pure and Applied Mathematics, vol. 140, 2nd edn. Elsevier/Academic Press, Amsterdam (2003)zbMATHGoogle Scholar
  2. 2.
    Asmussen, S., Højgaard, B., Taksar, M.: Optimal risk control and dividend distribution policies. Example of excess-of loss reinsurance for an insurance corporation. Finance Stoch. 4(3), 299–324 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Asmussen, S., Taksar, M.: Controlled diffusion models for optimal dividend pay-out. Insur. Math. Econ. 20(1), 1–15 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Avram, F., Palmowski, Z., Pistorius, M.R.: On the optimal dividend problem for a spectrally negative Lévy process. Ann. Appl. Probab. 17(1), 156–180 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Azcue, P., Muler, N.: Optimal reinsurance and dividend distribution policies in the Cramér–Lundberg model. Math. Financ. 15(2), 261–308 (2005)CrossRefzbMATHGoogle Scholar
  6. 6.
    Bayraktar, E., Emmerling, T., Menaldi, J.L.: On the impulse control of jump diffusions. SIAM J. Control Optim. 51(3), 2612–2637 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bensoussan, A., Lions, J.L.: Applications des inéquations variationnelles en contrôle stochastique, Méthodes Mathématiques de l’Informatique, No. 6, Dunod, Paris (1978)Google Scholar
  8. 8.
    Bony, J.M.: Problème de Dirichlet et semi-groupe fortement fellérien associés à un opérateur intégro-differentiel. C. R. Acad. Sci. Paris Sér A-B 265, 362–364 (1967)zbMATHGoogle Scholar
  9. 9.
    Bühlmann, H.: Mathematical Methods in Risk Theory. Fundamental Principles of Mathematical Sciences, vol. 172. Springer, Berlin (1996). Reprint of the 1970 originalGoogle Scholar
  10. 10.
    Csató, G., Dacorogna, B., Kneuss, O.: The Pullback Equation for Differential Forms. Progress in Nonlinear Differential Equations and their Applications, vol. 83. Birkhäuser/Springer, New York (2012)CrossRefzbMATHGoogle Scholar
  11. 11.
    Davis, M.H.A., Guo, X., Wu, G.: Impulse control of multidimensional jump diffusions. SIAM J. Control Optim. 48(8), 5276–5293 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Evans, L.C.: A second-order elliptic equation with gradient constraint. Commun. Partial Differ. Equ. 4(5), 555–572 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions, vol. 25, 2nd edn. Springer, New York (2006)zbMATHGoogle Scholar
  14. 14.
    Garroni, M.G., Menaldi, J.L.: Second Order Elliptic Integro-Differential Problems. Chapman & Hall/CRC Research Notes in Mathematics, vol. 430. Chapman & Hall/CRC, Boca Raton (2002)CrossRefzbMATHGoogle Scholar
  15. 15.
    Gerber, H.U.: Entscheidungskriterien für den zusammengesetzten Poisson-Prozess. Mitt. Ver. Schweiz. Vers. Math. 69, 185–227 (1969)zbMATHGoogle Scholar
  16. 16.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin (2001). Reprint of the 1998 editionzbMATHGoogle Scholar
  17. 17.
    Gimbert, F., Lions, P.L.: Existence and regularity results for solutions of second-order, elliptic integro-differential operators. Ricerche Mat. 33(2), 315–358 (1984)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Hynd, R.: The eigenvalue problem of singular ergodic control. Commun. Pure Appl. Math. 65(5), 649–682 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hynd, R.: Analysis of Hamilton–Jacobi–Bellman equations arising in stochastic singular control. ESAIM Control Optim. Calc. Var. 19(1), 112–128 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ishii, H., Koike, S.: Boundary regularity and uniqueness for an elliptic equation with gradient constraint. Commun. Partial Differ. Equ. 8(4), 317–346 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kruk, L.: Optimal policies for \(n\)-dimensional singular stochastic control problems. I. The Sko-rokhod problem. SIAM J. Control Optim. 38(5), 1603–1622 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kyprianou, A.E.: Fluctuations of Lévy Processes with Applications. Universitext. Introductory Lectures, 2nd edn. Springer, Heidelberg (2014)CrossRefzbMATHGoogle Scholar
  23. 23.
    Kyprianou, A.E., Palmowski, Z.: Distributional study of de Finettis dividend problem for a general Lévy insurance risk process. J. Appl. Probab. 44(2), 428–443 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ladyzhenskaya, O.A., Ural’tseva, N.N.: Linear and Quasilinear Elliptic Equations, vol. 46. Academic Press, New York (1986)zbMATHGoogle Scholar
  25. 25.
    Lenhart, S.: /83). Integro-differential operators associated with diffusion processes with jumps. Appl. Math. Optim. 9(2), 177–191 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Lenhart, S.: Correction: “Integro-differential operators associated with diffusion processes with jumps” [Appl. Math. Optim. 9 (1982/83), no. 2, 177–191]. Appl. Math. Optim. 13(3), 283 (1985)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Lions, P.L.: A remark on Bony maximum principle. Proc. Am. Math. Soc. 88(3), 503–508 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Loeffen, R.L.: On optimality of the barrier strategy in de Finettis dividend problem for spectrally negative Lévy processes. Ann. Appl. Probab. 18(5), 1669–1680 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Menaldi, J.L., Robin, M.: Singular ergodic control for multidimensional Gaussian–Poisson processes. Stochastics 85(4), 682–691 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Menaldi, J.L., Taksar, M.I.: Optimal correction problem of a multidimensional stochastic system. Automatica J. IFAC 25(2), 223–232 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Moreno-Franco, H.A. (2015). Solution to HJB equations with an elliptic integro-differential operator and gradient constraint. PhD thesis, CIMAT. https://researchgate.net/profile/Harold_Moreno
  32. 32.
    Protter, P.E.: Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin (2005). Version 2.1, Corrected third printingCrossRefGoogle Scholar
  33. 33.
    Renaud, J.-F., Zhou, X.: Distribution of the present value of dividend payments in a Lévy risk model. J. Appl. Probab. 44(2), 420–427 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Rudin, W.: Podstawy Analizy Matematycznej, 4th edn. Wydawnictwo Naukowe PWN, Warsaw (1996). Translated from the English by Wojciech WojtynśkiGoogle Scholar
  35. 35.
    Sato, K.: Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics, vol. 68. Cambridge University Press, Cambridge (1999). Translated from the 1990 Japanese original, Revised by the authorzbMATHGoogle Scholar
  36. 36.
    Schmidli, H.: Optimisation in non-life insurance. Stoch. Models 22(4), 689–722 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Soner, H.M., Shreve, S.E.: Regularity of the value function for a two-dimensional singular stochastic control problem. SIAM J. Control Optim. 27(4), 876–907 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, vol. 30. Princeton University Press, Princeton, NJ (1970)zbMATHGoogle Scholar
  39. 39.
    Taksar, M.I.: Optimal risk and dividend distribution control models for an insurance company. Math. Methods Oper. Res. 51(1), 1–42 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Zhanblan-Pike, M., Shiryaev, A.N.: Optimization of the flow of dividends. Uspekhi Mat. Nauk 50(2(302)), 25–46 (1995)MathSciNetGoogle Scholar
  41. 41.
    Zhou, X.: On a classical risk model with a constant dividend barrier. N. Am. Actuar. J. 9(4), 95–108 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Zhou, X.: “On optimal dividend strategies in the compound Poisson model” by Hans U. Gerber and Elias S. W. Shiu, April 2006. N. Am. Actuar. J. 10(3), 79–84 (2006)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.National Research University Higher School of EconomicsMoscowRussia

Personalised recommendations