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October, 1974 The Optimal Reward Operator in Special Classes of Dynamic Programming Problems
David A. Freedman
Ann. Probab. 2(5): 942-949 (October, 1974). DOI: 10.1214/aop/1176996559

Abstract

Consider a dynamic programming problem with separable metric state space $S$, constraint set $A$, and reward function $r(x, P, y)$ for $(x, P)\in A$ and $y\in S$. Let $Tf$ be the optimal reward in one move, for the reward function $r(x, P, y) + f(y)$. Three results are proved. First, suppose $S$ is compact, $A$ closed, and $r$ upper semi-continuous; then $T^n0$ is upper semi-continuous, and there is an optimal Borel strategy for the $n$-move game. Second, suppose $S$ is compact, $A$ is an $F_\sigma$, and $\{r > a\}$ is an $F_\sigma$ for all $a$; then $\{T^n0 > a\}$ is an $F_\sigma$ for all $a$, and there is an $\varepsilon$-optimal Borel strategy for the $n$-move game. Third, suppose $A$ is open and $r$ is lower semi-continuous; then $T^n0$ is lower semi-continuous, and there is an $\varepsilon$-optimal Borel measurable strategy for the $n$-move game.

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David A. Freedman. "The Optimal Reward Operator in Special Classes of Dynamic Programming Problems." Ann. Probab. 2 (5) 942 - 949, October, 1974. https://doi.org/10.1214/aop/1176996559

Information

Published: October, 1974
First available in Project Euclid: 19 April 2007

zbMATH: 0318.49022
MathSciNet: MR359819
Digital Object Identifier: 10.1214/aop/1176996559

Subjects:
Primary: 49C99
Secondary: 28A05 , 60K99 , 90C99

Keywords: dynamic programming , gambling , optimal reward , optimal strategy

Rights: Copyright © 1974 Institute of Mathematical Statistics

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Vol.2 • No. 5 • October, 1974
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