Skip to main content
SpringerLink
Log in

Relationship Between MP and DPP for the Stochastic Optimal Control Problem of Jump Diffusions

  • Published:
Applied Mathematics & Optimization Submit manuscript

Abstract

This paper is concerned with the stochastic optimal control problem of jump diffusions. The relationship between stochastic maximum principle and dynamic programming principle is discussed. Without involving any derivatives of the value function, relations among the adjoint processes, the generalized Hamiltonian and the value function are investigated by employing the notions of semijets evoked in defining the viscosity solutions. Stochastic verification theorem is also given to verify whether a given admissible control is optimal.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alvarez, D., Tourin, A.: Viscosity solutions of nonlinear integral-differential equations. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 13(3), 293–317 (1996)

    MATH  MathSciNet  Google Scholar 

  2. Arisawa, M.: A new definition of viscosity solutions for a class of second-order degenerate elliptic integro-differential equations. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 23(5), 695–711 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Arisawa, M.: A new definition of viscosity solutions for a class of second-order degenerate elliptic integro-differential equations. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 24(1), 167–169 (2007) (Errata)

    Article  MATH  MathSciNet  Google Scholar 

  4. Barles, G., Imbert, C.: Second-order elliptic integral-differential equations: Viscosity solutions’ theory revisited. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 25(3), 567–585 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  5. Barles, G., Buckdahn, R., Pardoux, E.: Backward stochastic differential equations and integral-partial differential equations. Stoch. Stoch. Rep. 60, 57–83 (1997)

    MATH  MathSciNet  Google Scholar 

  6. Bensoussan, A.: Lectures on Stochastic Control. Lecture Notes in Mathematics, vol. 972. Springer, Berlin (1981)

    Google Scholar 

  7. Bismut, J.M.: An introductory approach to duality in optimal stochastic control. SIAM J. Control Optim. 20(1), 62–78 (1978)

    MATH  MathSciNet  Google Scholar 

  8. Biswas, I.H., Jakobsen, E.R., Karlsen, K.H.: Viscosity solutions for a system of integro-PDEs and connections to optimal switching and control of jump-diffusion processes. Appl. Math. Optim. 62(1), 47–80 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  9. Framstad, N.C., Øksendal, B., Sulem, A.: A sufficient stochastic maximum principle for optimal control of jump diffusions and applications to finance. J. Optim. Theory Appl. 121(1), 77–98 (2004)

    Article  MathSciNet  Google Scholar 

  10. Framstad, N.C., Øksendal, B., Sulem, A.: A sufficient stochastic maximum principle for optimal control of jump diffusions and applications to finance. J. Optim. Theory Appl. 124(2), 511–512 (2005) (Errata)

    Article  MathSciNet  Google Scholar 

  11. Gozzi, F., Swiech, A., Zhou, X.Y.: A corrected proof of the stochastic verification theorem within the framework of viscosity solutions. SIAM J. Control Optim. 43(6), 2009–2019 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  12. Haussmann, U.G.: On the adjoint process for optimal control of diffusion processes. SIAM J. Control Optim. 19(2), 221–243 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  13. Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes. North-Holland, Kodansha (1989)

    MATH  Google Scholar 

  14. Jakobsen, E.R., Karlsen, K.H.: A “maximum principle for semicontinuous functions” applicable to integro-differential equations. Nonlinear Differ. Equ. Appl. 13(2), 137–165 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Li, J., Peng, S.G.: Stochastic optimization theory of backward stochastic differential equations with jumps and viscosity solutions of Hamilton-Jacobi-Bellman equations. Nonlinear Anal. 70, 1776–1796 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  16. Øksendal, B., Sulem, A.: Applied Stochastic Control of Jump Diffusions. Springer, Berlin (2005)

    Google Scholar 

  17. Pham, H.: Optimal stopping of controlled jump diffusion processes: A viscosity solution approach. J. Math. Syst. Estim. Control 8(1), 1–27 (1998)

    MathSciNet  Google Scholar 

  18. Sayah, A.: Équations d’Hamilton-Jacobi du premier ordre avec termes intégre-différentiels, I. Unicité des solutions de viscosité, II. Existence de solutions de viscosité. Commun. Partial Differ. Equ. 16(6–7), 1057–1093 (1991)

    MATH  Google Scholar 

  19. Situ, R.: Theory of Stochastic Differential Equations with Jumps and Applications. Springer, New York (2005)

    MATH  Google Scholar 

  20. Soner, H.M.: Optimal control with state-space constraint II. SIAM J. Control Optim. 24(6), 1155–1174 (1986)

    Article  MathSciNet  Google Scholar 

  21. Soner, H.M.: Optimal control of jump-Markov processes and viscosity solutions. In: Fleming, W.H., Lions, P.L. (eds.) Stochastic Differential Systems, Stochastic Control Theory and Applications. IMA Math. Appl., vol. 10, pp. 501–511. Springer, New York (1988)

    Google Scholar 

  22. Tang, S.J., Li, X.J.: Necessary conditions for optimal control of stochastic systems with random jumps. SIAM J. Control Optim. 32(5), 1447–1475 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  23. Yong, J.M.: Dynamic Programming Method and the Hamilton-Jacobi-Bellman Equations. Shanghai Sci. Technol., Shanghai (1992) (in Chinese)

    Google Scholar 

  24. Yong, J.M., Zhou, X.Y.: Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer, New York (1999)

    MATH  Google Scholar 

  25. Zhou, X.Y.: The connection between the maximum principle and dynamic programming in stochastic control. Stoch. Stoch. Rep. 31, 1–13 (1990)

    MATH  Google Scholar 

  26. Zhou, X.Y., Yong, J.M., Li, X.J.: Stochastic verification theorem within the framework of viscosity solutions. SIAM J. Control Optim. 35(1), 243–253 (1997)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Shandong University, Jinan, 250100, People’s Republic of China

    Jing-Tao Shi & Zhen Wu

Authors
  1. Jing-Tao Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhen Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhen Wu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shi, JT., Wu, Z. Relationship Between MP and DPP for the Stochastic Optimal Control Problem of Jump Diffusions. Appl Math Optim 63, 151–189 (2011). https://doi.org/10.1007/s00245-010-9115-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00245-010-9115-8

Keywords

Search

Navigation