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The monotone follower problem in stochastic decision theory

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Abstract

We consider the problem of optimally tracking the “random demand”x+w t, w. Brownian motion, by a nondecreasing processξ. adapted to the Brownian past, so as to minimize the expected lossE∫ T0 φ(x+wt−ξt)dt. The decision problem is reduced to a free boundary one, and the latter is studied and solved for a large class of cost functionsφ(⋅).

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References

  1. V. E. Beneš, L. A. Shepp, H. S. Witsenhausen, Some solvable stochastic control problems,Stochastics 4, 39–83 (1980).

    Google Scholar 

  2. A. Bensoussan, J. L. Lions,Applications des inéquations variationelles en contrôle stochastique, Dunod: Paris, 1978.

    Google Scholar 

  3. J. R. Cannon, C. D. Hill, Existence, uniqueness, stability and monotone dependence in a Stefan problem for the heat equation,J. Math. & Mech. 17:1, 1–19 (1967).

    Google Scholar 

  4. J. R. Cannon, C. D. Hill, Remarks on a Stefan problem,J. Math. & Mech. 17:5, 433–440 (1967).

    Google Scholar 

  5. A. Fasano, Alcune osservazioni su una classe di problemi a contorno libero per l'equazione del calore,Le Matematiche 29, 1–15 (1974).

    Google Scholar 

  6. A. Fasano, M. Primicerio, General free boundary problems for the heat equation, Pts I & II,J. Math. Anal. Appl., 57:3, 694–723 (1977), and 58:1, 202–231 (1977).

    Google Scholar 

  7. A. Friedman, Free boundary problems for parabolic equations, Pt I: Melting of solids,J. Math. & Mech. 8:4, 483–498 (1959).

    Google Scholar 

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

    Google Scholar 

  9. A. Friedman, One phase moving boundary problems, in: D. G. Wilson, A. D. Solomon, P. T. Boggs (eds.),Moving Boundary Problems, 25–40, Academic Press, New York, 1978.

    Google Scholar 

  10. P. A. Meyer, Un cours sur les intégrales stochastiques, Séminaire de Probabilités X,Lecture Notes in Mathematics 511, Springer-Verlag, Berlin, 1976.

    Google Scholar 

  11. P. van Moerbeke, Optimal stopping and free boundary problems,Archive Rational Mech. & Anal. 60:2, 101–148 (1976).

    Google Scholar 

  12. A. Schatz, Free boundary problems of Stefan type with prescribed flux,J. Math. Anal. Appl. 28:3, 569–579 (1969).

    Google Scholar 

  13. B. Sherman, General one-phase Stefan problems and free boundary problems for the heat equation with Cauchy data prescribed on the free boundary,SIAM J. Appl. Math. 20:4, 555–570 (1971).

    Google Scholar 

  14. B. Sherman, Free boundary problems for the heat equation in which the moving interface coincides initially with the fixed face,J. Math. Anal. Appl. 33:2, 449–466 (1971).

    Google Scholar 

  15. A. K. Zvonkin, A transformation on the phase space of a diffusion process that removes the drift,Math. USSR (Sbornik) 29:1, 29–49 (1974).

    Google Scholar 

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Authors and Affiliations

  1. Graduate School of Arts and Sciences, Department of Mathematical Statistics, Columbia University, 10027, New York, NY, USA

    Ioannis Karatzas

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  1. Ioannis Karatzas
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Communicated by W. H. Fleming

This research was supported in part by the Air Force Office of Scientific Research, under AF-AFOSR 77-3063.

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Karatzas, I. The monotone follower problem in stochastic decision theory. Appl Math Optim 7, 175–189 (1981). https://doi.org/10.1007/BF01442115

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  • DOI: https://doi.org/10.1007/BF01442115

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