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Double Optimal Stopping in the Fishing Problem

Part of: Stochastic processes Special processes

Published online by Cambridge University Press:  14 July 2016

Anna Karpowicz
Anna Karpowicz*
Affiliation:
Wrocław University of Technology
*
∗Postal address: Wrocław University of Technology, Institute of Mathematics and Computer Science, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland. Email address: [email protected]
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Abstract

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In this paper we consider the following problem. An angler buys a fishing ticket that allows him/her to fish for a fixed time. There are two locations to fish at the lake. The fish are caught according to a renewal process, which is different for each fishing location. The angler's success is defined as the difference between the utility function, which is dependent on the size of the fish caught, and the time-dependent cost function. These functions are different for each fishing location. The goal of the angler is to find two optimal stopping times that maximize his/her success: when to change fishing location and when to stop fishing. Dynamic programming methods are used to find these two optimal stopping times and to specify the expected success of the angler at these times.

Keywords

Fishing problemoptimal stoppingdynamic programmingsemi-Markov processinfinitesimal generator

MSC classification

Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 60K15: Markov renewal processes, semi-Markov processes
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 2 , June 2009 , pp. 415 - 428
Copyright
Copyright © Applied Probability Trust 2009 

References

[1] Boshuizen, F. A. and Gouweleeuw, J. M. (1993). General optimal stopping theorems for semi-Markov processes. Adv. Appl. Prob. 25, 825–846.Google Scholar
[2] Brémaud, P. (1981). Point Processes and Queues. Springer, New York.CrossRefGoogle Scholar
[3] Davis, M. H. A. (1993). Markov Models and Optimization (Monogr. Statist. Appl. Prob. 49). Chapman and Hall, New York.CrossRefGoogle Scholar
[4] Ferenstein, E. and Sierociński, A. (1997). Optimal stopping of a risk process. Applicationes Math. 24, 335–342.Google Scholar
[5] Ferguson, T. S. (1997). A Poisson fishing model. In Festschrift for Lucien Le Cam, eds Pollard, D. et al., Springer, New York, pp. 235–244.Google Scholar
[6] Jensen, U. and Hsu, G. H. (1993). Optimal stopping by means of point process observations with applications in reliability. Math. Operat. Res. 18, 645–657.CrossRefGoogle Scholar
[7] Karpowicz, A. and Szajowski, K. (2007). Double optimal stopping of a risk process. Stochastics 79, 155–167.Google Scholar
[8] Kramer, M. and Starr, N. (1990). Optimal stopping in a size dependent search. Sequent. Anal. 9, 59–80.CrossRefGoogle Scholar
[9] Rolski, T., Schmidli, H., Schimdt, V. and Teugels, J. (1998). Stochastic Processes for Insurance and Finance. John Wiley, Chichester.Google Scholar
[10] Starr, N. (1974). Optimal and adaptive stopping based on capture times. J. Appl. Prob. 11, 294–301.CrossRefGoogle Scholar
[11] Starr, N. and Woodroofe, M. (1974). Gone fishin': optimal stopping based on catch times. Tech. Rep. 33, Department of Statistics, University of Michigan.Google Scholar
[12] Starr, N., Wardrop, R. and Woodroofe, M. (1976). Estimating a mean from delayed observations. Z. Wahrscheinlichkeitsth 35, 103–113.Google Scholar