The Optimal Reward Operator in Special Classes of Dynamic Programming Problems
Freedman, David A.
Ann. Probab., Tome 2 (1974) no. 6, p. 942-949 / Harvested from Project Euclid
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.
Publié le : 1974-10-14
Classification:  Dynamic programming,  optimal reward,  optimal strategy,  gambling,  49C99,  60K99,  90C99,  28A05
@article{1176996559,
     author = {Freedman, David A.},
     title = {The Optimal Reward Operator in Special Classes of Dynamic Programming Problems},
     journal = {Ann. Probab.},
     volume = {2},
     number = {6},
     year = {1974},
     pages = { 942-949},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1176996559}
}
Freedman, David A. The Optimal Reward Operator in Special Classes of Dynamic Programming Problems. Ann. Probab., Tome 2 (1974) no. 6, pp.  942-949. http://gdmltest.u-ga.fr/item/1176996559/