Search models

1992 ◽  
Vol 29 (3) ◽  
pp. 605-615 ◽  
Author(s):  
J. A. Bather
Keyword(s):  
Mathematical Models ◽  
Optimal Stopping ◽  
Poisson Processes ◽  
Stopping Rules ◽  
Stopping Times ◽  
Oil Exploration ◽  
Search Models ◽  
Number Of Components ◽  
Optimal Stopping Rules ◽  
Optimal Stopping Times

Mathematical models have been proposed for oil exploration and other kinds of search. They can be used to estimate the amount of undiscovered resources or to investigate optimal stopping times for the search. Here we consider a continuous search for hidden objects using a model which represents the number and values of the objects by mixtures of Poisson processes. The flexibility of the model and its complexity depend on the number of components in the mixture. In simple cases, optimal stopping rules can be found explicitly and more general qualitative results can sometimes be obtained.

Search models

1992 ◽  
Vol 29 (03) ◽  
pp. 605-615 ◽  
Author(s):  
J. A. Bather
Keyword(s):  
Mathematical Models ◽  
Optimal Stopping ◽  
Poisson Processes ◽  
Stopping Rules ◽  
Stopping Times ◽  
Oil Exploration ◽  
Search Models ◽  
Number Of Components ◽  
Optimal Stopping Rules ◽  
Optimal Stopping Times

Mathematical models have been proposed for oil exploration and other kinds of search. They can be used to estimate the amount of undiscovered resources or to investigate optimal stopping times for the search. Here we consider a continuous search for hidden objects using a model which represents the number and values of the objects by mixtures of Poisson processes. The flexibility of the model and its complexity depend on the number of components in the mixture. In simple cases, optimal stopping rules can be found explicitly and more general qualitative results can sometimes be obtained.

CHARACTERIZATIONS OF OPTIMAL POLICIES IN A GENERAL STOPPING PROBLEM AND STABILITY ESTIMATING

2014 ◽  
Vol 28 (3) ◽  
pp. 335-352 ◽  
Author(s):  
Evgueni Gordienko ◽  
Andrey Novikov
Keyword(s):  
Optimal Stopping ◽  
Transition Probabilities ◽  
Measurable Space ◽  
Stopping Rules ◽  
Stopping Times ◽  
Optimal Stopping Problem ◽  
The Stability ◽  
Image Position ◽  
Optimal Stopping Times ◽  
Natural Classes

We consider an optimal stopping problem for a general discrete-time process X1, X2, …, Xn, … on a common measurable space. Stopping at time n (n = 1, 2, …) yields a reward Rn(X1, …, Xn) ≥ 0, while if we do not stop, we pay cn(X1, …, Xn) ≥ 0 and keep observing the process. The problem is to characterize all the optimal stopping times τ, i.e., such that maximize the mean net gain: $$E(R_\tau(X_1,\dots,X_\tau)-\sum_{n=1}^{\tau-1}c_n(X_1,\dots,X_n)).$$ We propose a new simple approach to stopping problems which allows to obtain not only sufficient, but also necessary conditions of optimality in some natural classes of (randomized) stopping rules.In the particular case of Markov sequence X1, X2, … we estimate the stability of the optimal stopping problem under perturbations of transition probabilities.

Optimal stopping rules for correlated random walks with a discount

2004 ◽  
Vol 41 (2) ◽  
pp. 483-496 ◽  
Author(s):  
Pieter Allaart
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Optimal Stopping ◽  
Constant Factor ◽  
Stopping Rules ◽  
Correlated Random Walk ◽  
Numerical Examples ◽  
Optimal Rule ◽  
Correlated Random Walks ◽  
Optimal Stopping Rules

Optimal stopping rules are developed for the correlated random walk when future returns are discounted by a constant factor per unit time. The optimal rule is shown to be of dual threshold form: one threshold for stopping after an up-step, and another for stopping after a down-step. Precise expressions for the thresholds are given for both the positively and the negatively correlated cases. The optimal rule is illustrated by several numerical examples.

Optimal stopping with discount and observation costs

2000 ◽  
Vol 37 (01) ◽  
pp. 64-72 ◽  
Author(s):  
Robert Kühne ◽  
Ludger Rüschendorf
Keyword(s):  
Differential Equations ◽  
Optimal Stopping ◽  
Point Processes ◽  
Numerical Solutions ◽  
Domain Of Attraction ◽  
Approximate Solutions ◽  
Poisson Approximation ◽  
Stopping Times ◽  
Optimal Stopping Problem ◽  
Optimal Stopping Times

For i.i.d. random variables in the domain of attraction of a max-stable distribution with discount and observation costs we determine asymptotic approximations of the optimal stopping values and asymptotically optimal stopping times. The results are based on Poisson approximation of related embedded planar point processes. The optimal stopping problem for the limiting Poisson point processes can be reduced to differential equations for the boundaries. In several cases we obtain numerical solutions of the differential equations. In some cases the analysis allows us to obtain explicit optimal stopping values. This approach thus leads to approximate solutions of the optimal stopping problem of discrete time sequences.

Optimal stopping rules in proofreading

1982 ◽  
Vol 19 (3) ◽  
pp. 723-729 ◽  
Author(s):  
Mark C. K. Yang ◽  
Dennis D. Wackerly ◽  
Andrew Rosalsky
Keyword(s):  
Optimal Stopping ◽  
Stopping Rules ◽  
Optimal Stopping Rules

Optimal stopping rules under various conditions are obtained for a proofreader who has a probability p (known or unknown) of detecting a misprint in proofsheets which contain an unknown but Poisson-distributed number of misprints.

On the best-choice problem when the number of observations is random

1983 ◽  
Vol 20 (1) ◽  
pp. 165-171 ◽  
Author(s):  
Joseph D. Petruccelli
Keyword(s):  
Optimal Stopping ◽  
Random Variable ◽  
Stopping Rules ◽  
Necessary And Sufficient Condition ◽  
Choice Problem ◽  
Best Choice Problem ◽  
Optimal Stopping Rule ◽  
Optimal Stopping Rules ◽  
Bounded Random Variable ◽  
Necessary And Sufficient

We consider the problem of maximizing the probability of choosing the largest from a sequence of N observations when N is a bounded random variable. The present paper gives a necessary and sufficient condition, based on the distribution of N, for the optimal stopping rule to have a particularly simple form: what Rasmussen and Robbins (1975) call an s(r) rule. A second result indicates that optimal stopping rules for this problem can, with one restriction, take virtually any form.

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