Prophet Regions for Independent [0, 1]-Valued Random Variables with Random Discounting

Abstract

Let X ₁, X ₂… and B ₁, B ₂… be mutually independent [0, 1]-valued random variables, with EB _j  = β > 0 for all j. Let Y _j  = B ₁ … sB _j−1 X _j for j ≥ 1. A complete comparison is made between the optimal stopping value V(Y ₁,…,Y _n ):=sup{EY _τ:τ is a stopping rule for Y ₁,…,Y _n } and E(max _1≤j≤n Y _j ). It is shown that the set of ordered pairs {(x, y):x = V(Y ₁,…,Y _n ), y = E(max _1≤j≤n Y _j ) for some sequence Y ₁,…,Y _n obtained as described} is precisely the set {(x, y):0 ≤ x ≤ 1, x ≤ y ≤ Ψ _{n, β}(x)}, where Ψ _{n, β}(x) = [(1 − β)n + 2β]x − β⁻⁽ⁿ⁻²⁾ x ² if x ≤ β ⁿ⁻¹, and Ψ _{n, β}(x) = min _j≥1{(1 − β)jx + β ^j } otherwise. Sharp difference and ratio prophet inequalities are derived from this result, and an analogous comparison for infinite sequences is obtained.

Key Words:

• Conjugate duality
• Inequalities for stochastic processes
• Optimal stopping value
• Prophet inequality

Mathematics Subject Classification:

• 60G40
• 60E15
• 62L15

ACKNOWLEDGMENT

The author wishes to thank the referee for raising a question concerning discount factors outside the unit interval. This question led to Proposition 5.3.