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Prophet regions for independent [0, 1]-valued random variables with random discounting
 

Summary: Prophet regions for independent [0, 1]-valued random
variables with random discounting
Pieter C. Allaart
University of North Texas
November 16, 2004
Abstract
Let X1, X2 . . . and B1, B2 . . . be mutually independent [0, 1]-valued random
variables, with EBj = > 0 for all j. Let Yj = B1 Bj-1Xj for j 1. A com-
plete comparison is made between the optimal stopping value V (Y1, . . . , Yn) :=
sup{E Y : is a stopping rule for Y1, . . . , Yn} and E(max1jn Yj). It is shown
that the set of ordered pairs {(x, y) : x = V (Y1, . . . , Yn), y = E(max1jn Yj)
for some sequence Y1, . . . , Yn obtained as above} is precisely the set {(x, y) :
0 x 1, x y n,(x)}, where n,(x) = [(1 - )n + 2]x - -(n-2)
x2
if
x n-1
, and n,(x) = minj1{(1 - )jx + j
} otherwise. Sharp difference
and ratio prophet inequalities are derived from this result, and an analogous
comparison for infinite sequences is obtained.

  

Source: Allaart, Pieter - Department of Mathematics, University of North Texas

 

Collections: Mathematics