Skip to main content
SpringerLink
Log in

Optimal stochastic control of diffusion type processes and Hamilton-Jacobi-Bellman equations

  • Conference paper
  • First Online:
Advances in Filtering and Optimal Stochastic Control

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 42))

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. A.D. ALEXANDROV: Almost everywhere existence of the second differential of a convex function and some properties of convex surfaces connected with it. Ucen. Zap. Leningrad Gos. Univ. No. 37 (Ser. Mat., vyp 6) (1939), p. 3–35 (in Russian).

    Google Scholar 

  2. R. BELLMAN: Dynamic programming. Princeton Univ. Press, Princeton, N.J. (1957).

    Google Scholar 

  3. A. BENSOUSSAN and J.L. LIONS: Applications des inéquations variationnelles en contrôle stochastique. Dunod, Paris (1978).

    Google Scholar 

  4. H. BREZIS and L.C. EVANS: A variational approach to the Bellman-Dirichlet equation for two elliptic operators, Arch. Rat. Mech. Anal., 71 (1979), p. 1–14.

    Google Scholar 

  5. H. BUSEMANN: Convex surfaces, Interscience, New-York (1958).

    Google Scholar 

  6. E. CALABI: Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens. Michigan Math. J., 5 (1958), p. 105–126.

    Google Scholar 

  7. M.G. CRANDALL and P.L. LIONS: Condition d'unicité pour les solutions généralisées des équations de Hamilton-Jacobi du ler ordre. Comptes-Rendus Paris, 252 (1981), p. 183–186.

    Google Scholar 

  8. M.G. CRANDALL and P.L. LIONS: Viscosity solutions of Hamilton-Jacobi equations. To appear in Trans. Amer. Math. Soc.

    Google Scholar 

  9. M.G. CRANDALL, L.C. EVANS and P.L. LIONS: Some properties of viscosity solutions of Hamilton-Jacobi equations. Preprint.

    Google Scholar 

  10. L.C. EVANS: A convergence theorem for solutions of nonlinear second-order elliptic equations. Ind. Univ. Math. J., 27 (1978), p. 875–887.

    Google Scholar 

  11. L.C. EVANS: On solving certain nonlinear partial differential equations by accretive operator methods. Israel J. Math., 36 (1981), p. 225–247.

    Google Scholar 

  12. L.C. EVANS: Classical solutions of fully nonlinear, convex, second-order elliptic equations. To appear in Comm. Pure Appl. Math.

    Google Scholar 

  13. L.C. EVANS: Classical solutions of the Hamilton-Jacobi-Bellman equation for uniformly elliptic operators. Preprint.

    Google Scholar 

  14. L.C. EVANS and A. FRIEDMAN: Optimal stochastic switching and the Dirichlet problem for the Bellman equation. Trans. Amer. Math. Soc., 253 (1979), p. 365–389.

    Google Scholar 

  15. L.C. EVANS and P.L. LIONS: Résolution des équations de Hamilton-Jacobi-Bellman pour des opérateurs uniformément elliptiques. Comptes-Rendus Paris, 290 (1980), p. 1049–1052.

    Google Scholar 

  16. W.H. FLEMING and R. RISHEL: Deterministic and stochastic optimal control. Springer, Berlin (1975).

    Google Scholar 

  17. B. GAVEAU: Méthodes de contrôle optimal en analyse complexe. I. Résolution d'équations de Monge-Ampère. J. Funct. Anal., 25 (1977), p. 391–411.

    Google Scholar 

  18. I.L. GENIS and N.V. KRYLOV: An example of a one dimensional controlled process. Th. Proba. Appl., 21 (1976), p. 148–152.

    Google Scholar 

  19. K. ITO: personnal communication.

    Google Scholar 

  20. R. JENSEN and P.L. LIONS: Some asymptotic problems in optimal stochastic control and fully nonlinear elliptic equations. Preprint.

    Google Scholar 

  21. N.V. KRYLOV: Control of diffusion type processes, Moscou (1979) — in Russian.

    Google Scholar 

  22. N.V. KRYLOV: Control of the diffusion type processes. Proceedings of the International Congress of Mathematicians, Helsinki, 1978.

    Google Scholar 

  23. N.V. KRYLOV: On control of the solution of a stochastic integral equation. Th. Proba. Appl., 17 (1972), p. 114–131.

    Google Scholar 

  24. N.V. KRYLOV: On control of the solution of a stochastic integral equation with degeneration. Math. USSR Izv., 6 (1972), p. 249–262.

    Google Scholar 

  25. N.V. KRYLOV: On equations of minimax type in the theory of elliptic and parabolic equations in the plane. Math. USSR Sbornik, 10 (1970), p. 1–19.

    Google Scholar 

  26. P.L. LIONS: Generalized solutions of Hamilton-Jacobi equations. Pitman, London (1982).

    Google Scholar 

  27. P.L. LIONS: Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. To appear.

    Google Scholar 

  28. P.L. LIONS: Résolution analytique des problèmes de Bellman-Dirichlet, 146 (1981), p. 151–166.

    Google Scholar 

  29. P.L. LIONS: Bifurcation and optimal stochastic control. Preprint; see also MRC report, University of Wisconsin-Madison (1982).

    Google Scholar 

  30. P.L. LIONS: Contrôle de diffusions dans RN. Comptes-Rendus Paris. 288, (1979), p. 339–342.

    Google Scholar 

  31. P.L. LIONS: Control of diffusion processes in RN. Comm. Pure Appl. Math., 34 (1981), p. 121–147.

    Google Scholar 

  32. P.L. LIONS: Equations de Hamilton-Jacobi-Bellman dégénérées. Comptes-Rendus, 289 (1979), p. 329–332.

    Google Scholar 

  33. P.L. LIONS: Equations de Hamilton-Jacobi-Bellman. In Séminaire Goulaouic-Schwartz 1979–1980, Ecole Polytechnique, Palaiseau.

    Google Scholar 

  34. P.L. LIONS: Optimal stochastic control and Hamilton-Jacobi-Bellman equations. In Mathematical Control Theory, Banach Center Publications, Warsaw.

    Google Scholar 

  35. P.L. LIONS: Une méthode nouvelle pour l'existence de solutions régulières de l'équation de Monge-Ampère. Comptes-Rendus Paris, 293 (1981), p. 589–592.

    Google Scholar 

  36. P.L. LIONS: Sur les équations de Monge-Ampère. I, II and III. Preprint.

    Google Scholar 

  37. P.L. LIONS and J.L. MENALDI: Optimal control of stochastic integrals and Hamilton-Jacobi-Bellman equations. I, II. SIAM J. Control Optim., 20 (1982), p. 58–95.

    Google Scholar 

  38. P.L. LIONS and B. MERCIER: Approximation numérique des équations de Hamilton-Jacobi-Bellman. R.A.I.R.O., 14 (1980), p. 369–393.

    Google Scholar 

  39. P.L. LIONS and M. NISIO: A uniqueness result for the semi-group associated with the Hamilton-Jacobi-Bellman operator. Preprint.

    Google Scholar 

  40. P.L. LIONS, G. PAPANICOLAOU and S.R.S. VARADHAN: work in preparation.

    Google Scholar 

  41. M. NISIO: On stochastic optimal controls and envelope of Markovian semi-groups. Proc. Intern. Symp. SDE, Kyoto (1976).

    Google Scholar 

  42. PERTHAME: Thèse de 3e cycle, Université P. et M. Curie, 1982–83.

    Google Scholar 

  43. J.P. QUADRAT: Analyse numérique de l'équation de Bellman stochastique. Rapport Laboria no 140 (1975), INRIA, Rocquencourt.

    Google Scholar 

  44. M.V. SAFONOV: On the Dirichlet problem for Bellman's equation in a plane domain. Math. USSR Sbornik, 31 (1977), p. 231–248.

    Google Scholar 

  45. M.V. SAFONOV: On the Dirichlet problem for Bellman's equation in a plane domain. II. Math. USSR Sbornik, 34 (1978), p. 521–526.

    Google Scholar 

  46. E. STEIN: Singular integrals and differentiability properties of functions. Princeton Univ. Press, Princeton (1970)

    Google Scholar 

  47. D.W. STROOCK and S.R.S. VARADHAN: On degenerate elliptic-parabolic operators of second-order and their associated diffusions. Comm. Pure Appl. Math., 25 (1972), p. 651–713.

    Google Scholar 

  48. S.R.S. VARADHAN: personnal communication.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Place de Lattre de Tassigny, Ceremade, Paris IX University, 75775, Paris Cedex 16, France

    P. L. Lions

Authors
  1. P. L. Lions
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Wendell H. Fleming Luis G. Gorostiza

Rights and permissions

Reprints and permissions

Copyright information

© 1982 Springer-Verlag

About this paper

Cite this paper

Lions, P.L. (1982). Optimal stochastic control of diffusion type processes and Hamilton-Jacobi-Bellman equations. In: Fleming, W.H., Gorostiza, L.G. (eds) Advances in Filtering and Optimal Stochastic Control. Lecture Notes in Control and Information Sciences, vol 42. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0004539

Download citation

  • DOI: https://doi.org/10.1007/BFb0004539

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-11936-4

  • Online ISBN: 978-3-540-39517-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation