## Advances in Applied Probability

- Adv. in Appl. Probab.
- Volume 48, Number 1 (2016), 52-68.

### Optimal stopping rule for the full-information duration problem with random horizon

#### Abstract

The full-information duration problem with a random number *N* of objects is considered. These objects appear sequentially and their values *X*_{k} are observed, where *X*_{k}, independent of *N*, are independent and identically distributed random variables from a known continuous distribution. The objective of the problem is to find a stopping rule that maximizes the duration of holding a relative maximum (e.g. the *k*th object is a relative maximum if *X*_{k} = max{*X*_{1}, *X*_{2}, . . ., *X*_{k}}). We assume that *N* is a random variable with a known upper bound *n*, so two models, Model 1 and Model 2, can be considered according to whether the planning horizon is *N* or *n*. The structure of the optimal rule, which depends on the prior distribution assumed on *N*, is examined. The monotone rule is defined and a necessary and sufficient condition for the optimal rule to be monotone is given for both models. Special attention is paid to the class of priors such that *N* / *n* converges, as *n* → ∞, to a random variable *V*_{m} having density *f*_{Vm}(*v*) = *m*(1 - *v*)^{m-1}, 0 ≤ *v* ≤ 1 for a positive integer *m*. An interesting feature is that the optimal duration (relative to *n*) for Model 2 is just (*m* + 1) times as large as that for Model 1 asymptotically.

#### Article information

**Source**

Adv. in Appl. Probab., Volume 48, Number 1 (2016), 52-68.

**Dates**

First available in Project Euclid: 8 March 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.aap/1457466155

**Mathematical Reviews number (MathSciNet)**

MR3161295

**Zentralblatt MATH identifier**

1337.60076

**Subjects**

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

Secondary: 62L15: Optimal stopping [See also 60G40, 91A60]

**Keywords**

Secretary problem best-choice problem planar Poisson process monotone rule

#### Citation

Tamaki, Mitsushi. Optimal stopping rule for the full-information duration problem with random horizon. Adv. in Appl. Probab. 48 (2016), no. 1, 52--68. https://projecteuclid.org/euclid.aap/1457466155