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On optimal stopping and free boundary problems

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References

  1. Appell, P., Sur l'équation ∂ 2 z/∂x 2−∂z/∂y=0 et la théorie de la chaleur. J. Math. Pures Appl. 8, 187–216 (1892).

    Google Scholar 

  2. Bather, J.A., Optimal stopping problems for brownian motion. Adv. Appl. Prob. 2, 259–286 (1970).

    Google Scholar 

  3. Bers, L., F. John, & M. Schechter, Partial Differential Equations, Lectures in Applied Mathematics, Vol 3. New York: Interscience Publishers 1964.

    Google Scholar 

  4. Blumenthal, R. M., & R. K. Getoor, Markov Processes and Potential Theory. New York: Academic Press 1968.

    Google Scholar 

  5. Cannon, J. R., & C. D. Hill, Existence, uniqueness, stability and monotone dependence in a Stefan problem. J. Math. Mech. 17, 1–14 (1967).

    Google Scholar 

  6. Cherkasov, I. D., On the transformation of the diffusion process to a Wiener process. Theory of Prob. and its Appl. 2, 373–377 (1957).

    Google Scholar 

  7. Chernoff, H., Sequential tests for the mean of a normal distribution. Proc. Fourth Berkeley Symposium 1, 79–91 (1961).

    Google Scholar 

  8. Chernoff, H., Sequential tests for the mean of a normal distribution III (small t). Ann. Math. Statist. 36, 28–54 (1965).

    Google Scholar 

  9. Chernoff, H., Sequential tests for the mean of a normal distribution IV (discrete case). Ann. Math. Statist. 36, 55–68 (1965).

    Google Scholar 

  10. Chernoff, H., Optimal stochastic control. Sankhya 30, 221–252 (1968).

    Google Scholar 

  11. Chow, Y.S., & H. Robbins, On optimal stopping rules for sn/n. Illinois J. Math. 9, 444–454 (1965).

    Google Scholar 

  12. Doob, J. L., A probability approach to the heat equation. Trans. Amer. Math. Soc. 80, 216–280 (1955).

    Google Scholar 

  13. Doob, J. L., Conditional Brownian motion and the boundary limits of harmonic functions. Bull. Soc. Math. France 85, 431–458 (1957).

    Google Scholar 

  14. Dvoretzky, A., Existence and properties of certain optimal stopping rules. Proc. Fifth Berkeley Symp. on Math. Stat. and Prob, 1, (1965).

  15. 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, 627–629 (1963).

    Google Scholar 

  16. Dynkin, E. B., & A. A. Yushkevich, Markov processes, theorems and problems. New York: Plenum Press 1969.

    Google Scholar 

  17. Erdélyi, A, W. Magnus, Oberhettinger, & F.G. Tricomi, Bateman Manuscript Project, Higher Transcendental Functions, Vol. 3, Cal. Tech. New York: McGraw-Hill 1953.

  18. Friedman, A., Free boundary problems for parabolic equations I: Melting of solids. J. Math, and Mech. 8, 499–518 (1959).

    Google Scholar 

  19. Friedman, A., Partial Differential Equations of Parabolic Type. Englewood Cliffs, N.J.: Prentice Hall 1964.

    Google Scholar 

  20. Fulks, W., & R.B. Guenther, A free boundary problem and an extension of Muskat's model. Acta Mathematica 122, 273–300 (1969).

    Google Scholar 

  21. Gevrey, M., Sur les équations aux dérivées partielles du type parabolique. J. Math. Pures Appl. 9, 305–475 (1913).

    Google Scholar 

  22. Grigelionis, B.I., & A.N. Shiryaev, On Stefan's problem and optimal stopping rules for Markov processes. Theory of Probability and its Applications 9, 541–558 (1966).

    Google Scholar 

  23. Ito, K., & H.P. McKean, Diffusion Processes and Their Sample Paths. New York: Springer 1965.

    Google Scholar 

  24. Kolodner, I.I., Free boundary problems for the heat equation with applications to problems of change of phase. Comm. Pure Appl. Math. 9, 1–31 (1956).

    Google Scholar 

  25. Kotlow, D.B., A free boundary problem connected with the optimal stopping problem for diffusion processes. To appear in Trans. of Amer. Math. Soc.

  26. Lindley, D.V., Dynamic programming and decision theory. Applied Statistics 10, 81–90 (1961).

    Google Scholar 

  27. Miranker, W.L., A free boundary value problem for the heat equation. Quarterly of Applied Mathematics 16, 2, 121–130 (1958).

    Google Scholar 

  28. McKean, H.P., Stochastic Integrals. New York: Academic Press 1971.

    Google Scholar 

  29. McKean, H.P., Appendix: A free boundary problem for the heat equation arising from a problem in mathematical economics. Industrial Management Review 6, 2, 32–39 (1965).

    Google Scholar 

  30. Robbins, H., & D.O. Siegmund, Boundary crossing probabilities for the Wiener Process. Tech. report, No. 4, June 23, 1969, (Dept. Stat., Stanford University).

  31. Samuelson, P.A., Rational theory of warrant pricing. Industrial Management Review 6, 2, 13–31 (1965).

    Google Scholar 

  32. Schatz, A., Free boundary problems of Stefan type with prescribed flux. J. of Math. Anal. and Appl. 28, 3, 569–580 (1969).

    Google Scholar 

  33. Shepp, L.A., Explicit solutions to some problems of optimal stopping. The Annals of Math. Statist. 40, 3, 993–1010 (1969).

    Google Scholar 

  34. Snell, J.L., Applications of martingale system theorems. Trans. Amer. Math. Soc. 73, 293–312 (1952).

    Google Scholar 

  35. Taylor, H.M., Optimal stopping in a Markov process. Ann. Math. Statist. 39 1333–1344 (1968).

    Google Scholar 

  36. Tychonov, A.N., Théorèmes d'unicité pour l'équation de la chaleur. Rec. Math. (Math. Sbornik) 42, 199–216 (1935).

    Google Scholar 

  37. Van Moerbeke, P.L.J., Optimal stopping and free boundary problems, the Proceedings of the Conference on Stochastic Differential Equations (July 1972), Edmonton (Alberta). Rocky Mountain Math. J. 4, 3, 539–578 (1974).

    Google Scholar 

  38. Van Moerbeke, P.L.J., An optimal stopping problem with linear reward. Acta Mathematica, 1–2, 132, 111–151 (1974).

    Google Scholar 

  39. Walker, L.H., Regarding stopping rules for brownian motion and random walks. Bull. Amer. Math. Soc. 75, 46–50 (1969).

    Google Scholar 

  40. Walter, W., Differential and Integral Inequalities. Berlin-Heidelberg-New York: Springer 1970.

    Google Scholar 

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Authors and Affiliations

  1. The Institute for Advanced Study, Princeton

    Pierre van Moerbeke

  2. The University of Louvain, Belgium

    Pierre van Moerbeke

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  1. Pierre van Moerbeke
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Communicated by M. Kac

This work was supported by ONR Contracts N0014-67-A-0112-0031 and N00014-67-A-0112-0058 and by NSF grants GP-36418X1, GP-31392X2, and GK-18339.

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van Moerbeke, P. On optimal stopping and free boundary problems. Arch. Rational Mech. Anal. 60, 101–148 (1976). https://doi.org/10.1007/BF00250676

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