Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-19T02:04:10.682Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal stopping problems with generalized objective functions

Published online by Cambridge University Press:  14 July 2016

T. P. Hill  and
D. P. Kennedy
T. P. Hill*
Affiliation:
Georgia Institute of Technology
D. P. Kennedy*
Affiliation:
University of Cambridge
*
∗ Postal address: School of Mathematics, Georgia Institute of Technology, Atlanta, GA30332, USA.
∗∗ Postal address: Statistical Laboratory, 16 Mill Lane, Cambridge, CB2 1SB, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

Optimal stopping of a sequence of random variables is studied, with emphasis on generalized objectives which may be non-monotone functions of EXt, where t is a stopping time, or may even depend on the entire vector (E[X1I{t=l}], · ··, E[XnI{t=n}]), such as the minimax objective to maximize minj{E[XjI{t=j}]}. Convexity is used to establish a prophet inequality and universal bounds for the optimal return, and a method for constructing optimal stopping times for such objectives is given.

Keywords

PROPHET INEQUALITYCONVEXITY THEOREMVECTOR OBJECTIVE FUNCTIONS
Type
Research Papers
Information
Journal of Applied Probability , Volume 27 , Issue 4 , December 1990 , pp. 828 - 838
Copyright
Copyright © Applied Probability Trust 1990 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

This work was partially supported by NSF grant DMS-8601608, a Fulbright Research Grant, and a grant from the US–UK Educational Commission.

References

Chow, T. S., Robbins, H. and Siegmund, D. (1971) Great Expectations: The Theory of Optimal Stopping. Houghton Mifflin, Boston.Google Scholar
Dubins, L. and Spanier, E. (1961) How to cut a cake fairly. Amer. Math. Monthly 68, 1–17.Google Scholar
Dvoretzky, A., Wald, A. and Wolfowitz, J. (1951) Relations among certain ranges of vector measures. Pacific J. Math. 1, 59–74.Google Scholar
Hill, T. P. (1988) A proportionality principle for partitioning problems. Proc. Amer. Math. Soc. 103, 288–293.Google Scholar
Krengel, U. and Sucheston, L. (1978) On semiamarts, amarts, and processes with finite value. In Probability on Banach Spaces, ed. Kuelbs, J. Marcel Dekker, New York.Google Scholar
Lyapounov, A. (1940) Sur les fonctions-vecteurs complètement additives. Bull. Acad. Sci. URSS 6, 465–478.Google Scholar
Pitman, J. W. and Speed, T. P. (1973) A note on random times. Stoch. Proc. Appl. 1, 369–374.Google Scholar
Steinhaus, H. (1948) The problem of fair division. Econometrica 16, 101–104.Google Scholar