## Advances in Applied Probability

- Adv. in Appl. Probab.
- Volume 48, Number 3 (2016), 726-743.

### Urn sampling distributions giving alternate correspondences between two optimal stopping problems

#### Abstract

The best-choice problem and the duration problem, known as versions of the secretary problem, are concerned with choosing an object from those that appear sequentially. Let (*B*,p) denote the best-choice problem and (*D*,p) the duration problem when the total number *N* of objects is a bounded random variable with prior p=(*p*_{1}, *p*_{2},...,*p*_{n}) for a known upper bound *n*. Gnedin (2005) discovered the correspondence relation between these two quite different optimal stopping problems. That is, for any given prior p, there exists another prior q such that (*D*,p) is equivalent to (*B*,q). In this paper, motivated by his discovery, we attempt to find the alternate correspondence {p^{(m)},*m*≥0}, i.e. an infinite sequence of priors such that (*D*,p^{(m-1)}) is equivalent to (*B*,p^{(m)}) for all *m*≥1, starting with p^{(0)}=(0,...,0,1). To be more precise, the duration problem is distinguished into (*D*_{1},p) or (*D*_{2},p), referred to as model 1 or model 2, depending on whether the planning horizon is *N* or *n*. The aforementioned problem is model 1. For model 2 as well, we can find the similar alternate correspondence {p^{[m]},*m*≥ 0}. We treat both the no-information model and the full-information model and examine the limiting behaviors of their optimal rules and optimal values related to the alternate correspondences as *n*→∞. A generalization of the no-information model is given. It is worth mentioning that the alternate correspondences for model 1 and model 2 are respectively related to the urn sampling models without replacement and with replacement.

#### Article information

**Source**

Adv. in Appl. Probab., Volume 48, Number 3 (2016), 726-743.

**Dates**

First available in Project Euclid: 19 September 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.aap/1474296312

**Mathematical Reviews number (MathSciNet)**

MR3568889

**Zentralblatt MATH identifier**

1351.60051

**Subjects**

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

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

**Keywords**

Dividend general diffusion optimization optimal financing regime-switching

#### Citation

Tamaki, Mitsushi. Urn sampling distributions giving alternate correspondences between two optimal stopping problems. Adv. in Appl. Probab. 48 (2016), no. 3, 726--743. https://projecteuclid.org/euclid.aap/1474296312