## Annals of Probability

### The Optimal Reward Operator in Dynamic Programming

#### Abstract

Consider a dynamic programming problem with analytic state space $S$, analytic constraint set $A$, and semi-analytic reward function $r(x, P, y)$ for $(x, P)\in A$ and $y\in S$: namely, $\{r > a\}$ is an analytic set for all $a$. Let $Tf$ be the optimal reward in one move, with the modified reward function $r(x, P, y) + f(y)$. The optimal reward in $n$ moves is shown to be $T^n0$, a semi-analytic function on $S$. It is also shown that for any $n$ and positive $\varepsilon$, there is an $\varepsilon$-optimal strategy for the $n$-move game, measurable on the $\sigma$-field generated by the analytic sets.

#### Article information

Source
Ann. Probab., Volume 2, Number 5 (1974), 926-941.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176996558

Digital Object Identifier
doi:10.1214/aop/1176996558

Mathematical Reviews number (MathSciNet)
MR359818

Zentralblatt MATH identifier
0318.49021

JSTOR