Asymptotic Duration for Optimal Multiple Stopping Problems
We study the asymptotic duration of optimal stopping problems involving a sequence of independent random variables that are drawn from a known continuous distribution. These variables are observed as a sequence, where no recall of previous observations is permitted, and the objective is to form an o...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-02-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/12/5/652 |
| _version_ | 1797264130563899392 |
|---|---|
| author | Hugh N. Entwistle Christopher J. Lustri Georgy Yu. Sofronov |
| author_facet | Hugh N. Entwistle Christopher J. Lustri Georgy Yu. Sofronov |
| author_sort | Hugh N. Entwistle |
| collection | DOAJ |
| description | We study the asymptotic duration of optimal stopping problems involving a sequence of independent random variables that are drawn from a known continuous distribution. These variables are observed as a sequence, where no recall of previous observations is permitted, and the objective is to form an optimal strategy to maximise the expected reward. In our previous work, we presented a methodology, borrowing techniques from applied mathematics, for obtaining asymptotic expressions for the expectation duration of the optimal stopping time where one stop is permitted. In this study, we generalise further to the case where more than one stop is permitted, with an updated objective function of maximising the expected sum of the variables chosen. We formulate a complete generalisation for an exponential family as well as the uniform distribution by utilising an inductive approach in the formulation of the stopping rule. Explicit examples are shown for common probability functions as well as simulations to verify the asymptotic calculations. |
| first_indexed | 2024-04-25T00:24:01Z |
| format | Article |
| id | doaj.art-10af30dcd37942c6bfd689dfe864c1ed |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-04-25T00:24:01Z |
| publishDate | 2024-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-10af30dcd37942c6bfd689dfe864c1ed2024-03-12T16:49:50ZengMDPI AGMathematics2227-73902024-02-0112565210.3390/math12050652Asymptotic Duration for Optimal Multiple Stopping ProblemsHugh N. Entwistle0Christopher J. Lustri1Georgy Yu. Sofronov2School of Mathematical and Physical Sciences, Macquarie University, Sydney, NSW 2109, AustraliaSchool of Mathematics and Statistics, The University of Sydney, Camperdown, NSW 2006, AustraliaSchool of Mathematical and Physical Sciences, Macquarie University, Sydney, NSW 2109, AustraliaWe study the asymptotic duration of optimal stopping problems involving a sequence of independent random variables that are drawn from a known continuous distribution. These variables are observed as a sequence, where no recall of previous observations is permitted, and the objective is to form an optimal strategy to maximise the expected reward. In our previous work, we presented a methodology, borrowing techniques from applied mathematics, for obtaining asymptotic expressions for the expectation duration of the optimal stopping time where one stop is permitted. In this study, we generalise further to the case where more than one stop is permitted, with an updated objective function of maximising the expected sum of the variables chosen. We formulate a complete generalisation for an exponential family as well as the uniform distribution by utilising an inductive approach in the formulation of the stopping rule. Explicit examples are shown for common probability functions as well as simulations to verify the asymptotic calculations.https://www.mdpi.com/2227-7390/12/5/652sequential decision analysisoptimal stoppingmultiple optimal stoppingsecretary problemsasymptotic approximations |
| spellingShingle | Hugh N. Entwistle Christopher J. Lustri Georgy Yu. Sofronov Asymptotic Duration for Optimal Multiple Stopping Problems Mathematics sequential decision analysis optimal stopping multiple optimal stopping secretary problems asymptotic approximations |
| title | Asymptotic Duration for Optimal Multiple Stopping Problems |
| title_full | Asymptotic Duration for Optimal Multiple Stopping Problems |
| title_fullStr | Asymptotic Duration for Optimal Multiple Stopping Problems |
| title_full_unstemmed | Asymptotic Duration for Optimal Multiple Stopping Problems |
| title_short | Asymptotic Duration for Optimal Multiple Stopping Problems |
| title_sort | asymptotic duration for optimal multiple stopping problems |
| topic | sequential decision analysis optimal stopping multiple optimal stopping secretary problems asymptotic approximations |
| url | https://www.mdpi.com/2227-7390/12/5/652 |
| work_keys_str_mv | AT hughnentwistle asymptoticdurationforoptimalmultiplestoppingproblems AT christopherjlustri asymptoticdurationforoptimalmultiplestoppingproblems AT georgyyusofronov asymptoticdurationforoptimalmultiplestoppingproblems |