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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.
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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 

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Footnotes

†

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

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