## The Annals of Probability

- Ann. Probab.
- Volume 9, Number 2 (1981), 293-301.

### Decision Processes with Total-Cost Criteria

Stephen Demko and Theodore P. Hill

#### Abstract

By a decision process is meant a pair $(X, \Gamma)$, where $X$ is an arbitrary set (the state space), and $\Gamma$ associates to each point $x$ in $X$ an arbitrary nonempty collection of discrete probability measures (actions) on $X$. In a decision process with nonnegative costs depending on the current state, the action taken, and the following state, there is always available a Markov strategy which uniformly (nearly) minimizes the expected total cost. If the costs are strictly positive and depend only on the current state, there is even a stationary strategy with the same property. In a decision process with a fixed goal $g$ in $X$, there is always a stationary strategy which uniformly (nearly) minimizes the expected time to the goal, and, if $X$ is countable, such a stationary strategy exists which also (nearly) maximizes the probability of reaching the goal.

#### Article information

**Source**

Ann. Probab., Volume 9, Number 2 (1981), 293-301.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176994470

**Digital Object Identifier**

doi:10.1214/aop/1176994470

**Mathematical Reviews number (MathSciNet)**

MR606991

**Zentralblatt MATH identifier**

0457.60027

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G99: None of the above, but in this section

Secondary: 62C05: General considerations

**Keywords**

Gambling theory dynamic programming decision theory stationary strategy Markov strategy total-cost criteria

#### Citation

Demko, Stephen; Hill, Theodore P. Decision Processes with Total-Cost Criteria. Ann. Probab. 9 (1981), no. 2, 293--301. doi:10.1214/aop/1176994470. https://projecteuclid.org/euclid.aop/1176994470