Acta Mathematica

, Volume 132, Issue 1, pp 111–151 | Cite as

An optimal stopping problem with linear reward

  • Pierre van Moerbeke
Article

Keywords

Maximum Principle Heat Equation Free Boundary Problem Reward Function Exit Time 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1].
    Bather, J. A., Optimal stopping problems for brownian motion.Advances in Appl. Prob., 2 (1970), 259–286.MATHMathSciNetCrossRefGoogle Scholar
  2. [2].
    Beurling, A., An extension of the Riemann mapping theorem.Acta Math., 90 (1953), 117–130.MATHMathSciNetCrossRefGoogle Scholar
  3. [3].
    Beurling, A. Comformal mapping and Schlicht functions. Seminar, The Institute for Advanced Study, Princeton, N.J.Google Scholar
  4. [4].
    Blumenthal, R. M. &Getoor, R. K.,Markov Processes and Potential Theory. Academic Press, New York, 1968.MATHGoogle Scholar
  5. [5].
    Breiman, L., First exit times from a square root boundary.Fifth Berkeley Symposium, Calif., (1965), 9–16.Google Scholar
  6. [6].
    Brown, A. B., Functional dependence.Trans. Amer. Math. Soc., October (1934), 379–394.Google Scholar
  7. [7].
    Cannon, J. R. &Hill, C. D., Existence, uniqueness, stability and monotone dependence in a Stefan problem.J. Math. Mech., 17 (1967), 1–14.MATHMathSciNetGoogle Scholar
  8. [8].
    Doob, J. L., A probability approach to the heat equation.Trans. Amer. Math. Soc., 80 (1955), 216–280.MATHMathSciNetCrossRefGoogle Scholar
  9. [9].
    Dynkin, E. B., Optimal selection of stopping time for a Markov process.Dokl. Akad. Nauk USSR, 150, 2 (1963), 238–240 (Russian),Soviet Math. (English translation), 4 (1963), 627–629.MATHMathSciNetGoogle Scholar
  10. [10].
    Friedman, A., Free boundary problems for parabolic equations I: Melting of Solids.J. Math. Mech., 8 (1959), 499–518.MATHMathSciNetGoogle Scholar
  11. [11].
    Partial Differential Equations of Parabolic Type. Prentice Hall, Englewood Cliffs, N.J., 1964.MATHGoogle Scholar
  12. [12].
    Fulks, W. &Guenther, R. B., A free boundary problem and an extension of Muskat's model.Acta Math., 122 (1969), 273–300.MATHMathSciNetCrossRefGoogle Scholar
  13. [13].
    Gevrey, M., Sur les équations aux dérivées partielles du type parabolique.J. Math. Pure Appl., Ser. 6, Vol. 9 (1913), 305–475.Google Scholar
  14. [14].
    Grigelionis, B. I. &Shiryaev, A. N., On Stefan's problem and optimal stopping rules for Markov processes.Theor. Probability Appl., 9 (1966), 541–558.CrossRefGoogle Scholar
  15. [15].
    Ito, M. &McKean, H. P.,Diffusion Processes, and Their Sample Paths, Springer Verlag, New York, 1965.Google Scholar
  16. [16].
    Kac, M., Can one hear the shape of a drum?Amer. Math. Monthly, 73, 4 II (1966) 1–23.MATHMathSciNetCrossRefGoogle Scholar
  17. [17].
    Kotlow, D. B., A free boundary problem connected with the optimal stopping problem for diffusion processes. To appear inTrans. Amer. Math. Soc. Google Scholar
  18. [18].
    Miranker, W. L., A free boundary value problem for the heat equation.Quart. Appl. Math., 16 2 (1958), 121–130.MATHMathSciNetGoogle Scholar
  19. [19].
    McKean, H. P.,Stochastic Integrals. Academic Press, 1971.Google Scholar
  20. [20].
    —, Appendix: A free boundary problem for the heat equation arising from a problem in mathematical economics.Industrial Management Review, 6 2 (1965), 32–39.MathSciNetGoogle Scholar
  21. [21].
    Polya, G., Qualitatives über Wärme ausgleich.Z. Angew. Math. Mech., 13 (1938), 125–128.Google Scholar
  22. [22].
    Robbins, H. &Siegmund, D. O.,Great expectations: the theory of optimal stopping. Houghton Mifflin, Boston, 1970.Google Scholar
  23. [23].
    Schatz, A., Free boundary problems of Stefan type with prescribed flux.J. Math. Anal. Appl., 28 3 (1969), 569–580.MATHMathSciNetCrossRefGoogle Scholar
  24. [24].
    Shepp, L. A., Explicit solutions to some problems of optimal stopping.Ann. Math. Statist., 40, 3 (1969), 993–1010.MATHMathSciNetGoogle Scholar
  25. [25].
    Snell, J. L., Applications of martingale system theorems,Trans. Amer. Math. Soc. 73 (1952), 293–312.MATHMathSciNetCrossRefGoogle Scholar
  26. [26].
    Sonine, N., Sur la généralisation d'une formule d'Abel.Acta Math., 4 (1884), 171–176.MATHCrossRefGoogle Scholar
  27. [27].
    Sturm, Ch., Mémoire sur une classe d'equations à différences partielles.J. Math. Pures Appl., Ser. 1, Vol. 1 (1836), 373–344.Google Scholar
  28. [28].
    Taylor, H. M., Optimal stopping in a Markov process.Ann. Math. Statist., 39 (1968), 1333–1344.MATHMathSciNetGoogle Scholar
  29. [29].
    Van Moerbeke, P. L. J., Optimal stopping and free boundary problems. To appear in the Proceedings of the Conference on Stochastic Differential Equations, Edmonton, (Alberta),Rocky Mountain J. Math., 1974.Google Scholar
  30. [30].
    Van Moerbeke, P. L. J. About optimal stopping problems.Arch. Rational Mech. Anal., to appear.Google Scholar

Copyright information

© Almqvist & Wiksell 1974

Authors and Affiliations

  • Pierre van Moerbeke
    • 1
  1. 1.The University of LouvainLouvainBelgium

Personalised recommendations