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The Asymptotics of Multiple Optimal Stopping Times

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posted on 2022-10-19, 03:32 authored by Hugh EntwistleHugh Entwistle

We consider optimal stopping problems, in which a sequence of independent random variables is drawn from a known continuous density and are sequentially observed, with no recall of previous observations. The objective of such problems is to find a procedure which maximizes the expected reward. We first present a methodology for obtaining asymptotic expressions for the expectation and variance of the single optimal stopping time as the number of drawn variables becomes large. We then extend these results to the multiple stopping problem where the objective is now to maximize the expected reward, which is the sum of all variables stopped on. For a family of distributions with exponential tails and for the uniform distribution, we provide the complete generalisation of the multiple stopping problem by computing the inductive behaviour of the stopping time. Explicit calculations are provided for several common probability density functions as well as numerical simulations to support our asymptotic predictions.


Table of Contents

1 Introduction -- 2 Background -- 3 Formulation -- 4 Computing νⁿ,¹ Behaviour -- 5 Calculating Optimal Stopping Statistics (Single Stopping) -- 6 Numerical Comparisons for Single Stopping -- 7 Computing νⁿ,ᵏ Behaviour -- 8 Calculating Multiple Optimal Stopping Expectation -- 9 Discussion and Further Research -- A Appendix -- List of Symbols -- References

Awarding Institution

Macquarie University

Degree Type

Thesis MRes


Thesis (MRes), Macquarie University, Faculty of Science and Engineering, 2021

Department, Centre or School

Department of Mathematics and Statistics

Year of Award


Principal Supervisor

Georgy Sofronov

Additional Supervisor 1

Christopher Lustri


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