 
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 · · · Bj1Xj 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  (n2)
x2
if
x n1
, 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.
