Advertisement

Generalized semiconcavity of the value function of a jump diffusion optimal control problem

  • Ermal FeleqiEmail author
Article
  • 120 Downloads

Abstract

Generalized semiconcavity results for the value function of a jump diffusion optimal control problem are established, in the state variable, uniformly in time. Moreover, the semiconcavity modulus of the value function is expressed rather explicitly in terms of the semiconcavity or regularity moduli of the data (Lagrangian, terminal cost, and terms comprising the controlled SDE), at least under appropriate restrictions either on the class of the moduli, or on the SDEs. In particular, if the moduli of the data are of power type, then the semiconcavity modulus of the value function is also of power type. An immediate corollary are analogous regularity properties for (viscosity) solutions of certain integro-differential Hamilton-Jacobi-Bellman equations, which may be represented as value functions of appropriate optimal control problems for jump diffusion processes.

Mathematics Subject Classification

35D10 35E10 60H30 93E20 

Keywords

Generalized semiconcavity Value function Optimal control Partial integro-differential equations Hammilton-Jacobi-Bellman equations Jump diffusions 

References

  1. 1.
    Adams R.A., Fournier J.J.F.: Sobolev Spaces Vol 140 of Pure and Applied Mathematics (Amsterdam). 2nd edn. Elsevier/Academic Press, Amsterdam (2003)Google Scholar
  2. 2.
    Applebaum D.: Lévy Processes and Stochastic Calculus, Volume 116 of Cambridge Studies in Advanced Mathematics. 2nd edn. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar
  3. 3.
    Barles G., Chasseigne E., Ciomaga A., Imbert C.: Lipschitz regularity of solutions for mixed integro-differential equations. J. Differ. Equs. 252(11), 6012–6060 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Barles G., Chasseigne E., Imbert C.: Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations. J. Euro. Math. Soc. (JEMS) 13(1), 1–26 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Barles G., Imbert C.: Second-order elliptic integro-differential equations: viscosity solutions’ theory revisited. Ann. Inst. H. Poincaré Anal. Non Linéaire 25(3), 565–585 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bensoussan, A., Lions, J.-L.: Impulse control and quasivariational inequalities. μ. Gauthier-Villars, Montrouge, 1984. Translated from the French by J. M. Cole.Google Scholar
  7. 7.
    Bian B., Guan P.: Convexity preserving for fully nonlinear parabolic integro-differential equations. Methods Appl. Anal. 15(1), 39–51 (2008)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Buckdahn R., Cannarsa P., Quincampoix M.: Lipschitz continuity and semiconcavity properties of the value function of a stochastic control problem. NoDEA Nonlinear Differ. Equs. Appl. 17(6), 715–728 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Buckdahn R., Huang J., Li J.: Regularity properties for general HJB equations: a backward stochastic differential equation method. SIAM J. Control Optim. 50(3), 1466–1501 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Caffarelli L., Chan C.H., Vasseur A.: Regularity theory for parabolic nonlinear integral operators. J. Am. Math. Soc. 24(3), 849–869 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Caffarelli L., Silvestre L.: Regularity theory for fully nonlinear integro-differential equations. Commun. Pure Appl. Math. 62(5), 597–638 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Caffarelli L., Silvestre L.: The Evans-Krylov theorem for nonlocal fully nonlinear equations. Ann. Math. (2) 174(2), 1163–1187 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Caffarelli L., Silvestre L.: Regularity results for nonlocal equations by approximation. Arch. Ration. Mech. Anal. 200(1), 59–88 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Caffarelli L.A., Vasseur A.: Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. Ann. Math. (2) 171(3), 1903–1930 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Caffarelli L.A., Vasseur A.F.: The De Giorgi method for regularity of solutions of elliptic equations and its applications to fluid dynamics. Discret. Contin. Dyn. Syst. Ser. S 3(3), 409–427 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Cannarsa P., Sinestrari C.: Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control. Progress in Nonlinear Differential Equations and their Applications. Birkhäuser Boston Inc., Boston (2004)Google Scholar
  17. 17.
    Fleming W.H., Soner H.M.: Controlled Markov Processes and Viscosity Solutions, Volume 25 of Stochastic Modelling and Applied Probability. 2nd edn. Springer, New York (2006)Google Scholar
  18. 18.
    Fujiwara T., Kunita H.: Stochastic differential equations of jump type and Lévy processes in diffeomorphisms group. J. Math. Kyoto Univ. 25(1), 71–106 (1985)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Garroni M.G., Menaldi J.L.: Second Order Elliptic Integro-Differential Problems, Volume 430 of Chapman & Hall/CRC Research Notes in Mathematics. 2nd edn. Chapman & Hall/CRC, Boca Raton, FL (2002)CrossRefGoogle Scholar
  20. 20.
    Giga Y., Goto S, Ishii H., Sato M.-H.: Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains. Indiana Univ. Math. J. 40(2), 443–470 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    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
  22. 22.
    Ishii H., Lions P.-L.: Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J. Differ. Equs. 83(1), 26–78 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Jing S.: Regularity properties of viscosity solutions of integro-partial differential equations of Hamilton-Jacobi-Bellman type. Stoch. Process. Appl. 123(2), 300–328 (2013)CrossRefzbMATHGoogle Scholar
  24. 24.
    Kunita, H.: Stochastic differential equations based on Lévy processes and stochastic flows of diffeomorphisms. In: Real and stochastic analysis, Trends Math., pp. 305–373. Birkhäuser Boston, Boston, MA (2004)Google Scholar
  25. 25.
    Øksendal B., Sulem A.: pplied stochastic control of jump diffusions. 2nd edn. Universitext. Springer, Berlin (2007)CrossRefGoogle Scholar
  26. 26.
    Protter P.E.: Stochastic integration and differential equations, volume 21 of Applications of Mathematics (New York). Stochastic Modelling and Applied Probability. 2nd edn. Springer, Berlin (2004)Google Scholar
  27. 27.
    Yong J., Zhou X.Y.: Stochastic controls, volume 43 of Applications of Mathematics (New York). Hamiltonian systems and HJB equations. Springer, Berlin (1999)Google Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità di PadovaPadovaItaly

Personalised recommendations