Duration problem on trajectories

Abstract

Imagine that an urn contains m minus balls and p plus balls. We draw balls from the urn one by one without replacement until we wish to stop. Determine , where is the value of the ball chosen at kth draw, . The problem which we consider in this paper is to stop with maximum duration of possessing the maximum value of the trajectory formed by . We derive the optimal stopping rule and then make some comparison of the asymptotics of the optimal rules between our problem and the related problems.

Keywords:

• Optimal stopping
• Urn sampling
• Ballot problem
• Duration

AMS 2000 Subject Classification::

• Primary 60G40

Acknowledgements

The research was carried out while the first author was visiting the Department of Business Administration, Aichi University. He would like to thank the Department for the warm hospitality received here and Japan Society for the Promotion of Science (grant L 01530) and Russian Foundation for Basic Research (project N 06-01-00128) for the support.

Notes

^¶Postal address: Institute of Applied Mathematical Research, Karelian Research Center of Russian Academy of Sciences, Pushkinskaya st. 11, Petrozavodsk 185610, Russia.

^§Postal address: Department of Business Administration, Aichi University, Nagoya campus, 370 Kurozasa Miyoshi, Nishikamo, Aichi 470-0296, Japan. [email protected].

Additional information

Notes on contributors

Vladimir V. Mazalov

¶ ¶Postal address: Institute of Applied Mathematical Research, Karelian Research Center of Russian Academy of Sciences, Pushkinskaya st. 11, Petrozavodsk 185610, Russia.

Mitsushi Tamaki

§ §Postal address: Department of Business Administration, Aichi University, Nagoya campus, 370 Kurozasa Miyoshi, Nishikamo, Aichi 470-0296, Japan. [email protected].