Abstract
The model considered here will be formulated in relation to the “fishing problem,” even if other applications of it are much more obvious. The angler goes fishing, using various techniques, and has at most two fishing rods. He buys a fishing pass for a fixed time. The fish are caught using different methods according to renewal processes. The fish’s value and the interarrival times are given by the sequences of independent, identically distributed random variables with known distribution functions. This forms the marked renewal–reward process. The angler’s measure of satisfaction is given by the difference between the utility function, depending on the value of the fish caught, and the cost function connected with the time of fishing. In this way, the angler’s relative opinion about the methods of fishing is modeled. The angler’s aim is to derive as much satisfaction as possible, and additionally he must leave the lake by a fixed time. Therefore, his goal is to find two optimal stopping times to maximize his satisfaction. At the first moment, he changes his technique, e.g., by discarding one rod and using the other one exclusively. Next, he decides when he should end his outing. These stopping times must be shorter than the fixed time of fishing. Dynamic programming methods are used to find these two optimal stopping times and to specify the expected satisfaction of the angler at these times.
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- 1.
The following convention is used throughout the paper: \(\overrightarrow{x} = ({x}_{1},{x}_{2},\ldots,{x}_{s})\) for the ordered collection of the elements {x i } i = 1 s.
- 2.
For the optimization problem there are two epochs: before the first stop, where there are some payoffs, the model of stream of events, and after the first stop, when there are other payoffs and different streams of events. In Sect. 17.3, this will be emphasized by adopting adequate denotations.
References
Boshuizen, F., Gouweleeuw, J.: General optimal stopping theorems for semi-Markov processes. Adv. Appl. Probab. 4, 825–846 (1993)
Boshuizen, F.A.: A general framework for optimal stopping problems associated with multivariate point processes, and applications. Sequential Anal. 13(4), 351–365 (1994)
Brémaud, P.: Point Processes and Queues. Martingale Dynamics. Springer, Berlin (1981)
Davis, M.H.A.: Markov Models and Optimization. Chapman & Hall, New York (1993)
Ferenstein, E., Pasternak-Winiarski, A.: Optimal stopping of a risk process with disruption and interest rates. In: Brèton, M., Szajowski, K. (eds.) Advances in Dynamic Games: Differential and Stochastic Games: Theory, Application and Numerical Methods, Annals of the International Society of Dynamic Games, vol. 11, 18 pp. Birkhäuser, Boston (2010)
Ferenstein, E., Sierociński, A.: Optimal stopping of a risk process. Appl. Math. 24(3), 335–342 (1997)
Ferguson, T.: A Poisson fishing model. In: Pollard, D., Torgersen, E., Yang, G. (eds.) Festschrift for Lucien Le Cam: Research Papers in Probability and Statistics. Springer, Berlin (1997)
Haggstrom, G.: Optimal sequential procedures when more then one stop is required. Ann. Math. Stat. 38, 1618–1626 (1967)
Jacobsen, M.: Point process theory and applications. Marked point and piecewise deterministic processes. In: Prob. and its Applications, vol. 7. Birkhäuser, Boston (2006)
Jensen, U.: An optimal stopping problem in risk theory. Scand. Actuarial J. 2, 149–159 (1997)
Jensen, U., Hsu, G.: Optimal stopping by means of point process observations with applications in reliability. Math. Oper. Res. 18(3), 645–657 (1993)
Karpowicz, A.: Double optimal stopping in the fishing problem. J. Appl. Prob. 46(2), 415–428 (2009). DOI 10.1239/jap/1245676097
Karpowicz, A., Szajowski, K.: Double optimal stopping of a risk process. GSSR Stochast. Int. J. Prob. Stoch. Process. 79, 155–167 (2007)
Kramer, M., Starr, N.: Optimal stopping in a size dependent search. Sequential Anal. 9, 59–80 (1990)
Muciek, B.K., Szajowski, K.: Optimal stopping of a risk process when claims are covered immediately. In: Mathematical Economics, Toru Maruyama (ed.) vol. 1557, pp. 132–139. Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502 Japan Kôkyûroku (2007)
Nikolaev, M.: Obobshchennye posledovatelnye procedury. Litovskiui Mat. Sb. 19, 35–44 (1979)
Rolski, T., Schmidli, H., Schimdt, V., Teugels, J.: Stochastic Processes for Insurance and Finance. Wiley, Chichester (1998)
Shiryaev, A.: Optimal Stopping Rules. Springer, Berlin (1978)
Starr, N.: Optimal and adaptive stopping based on capture times. J. Appl. Prob. 11, 294–301 (1974)
Starr, N., Wardrop, R., Woodroofe, M.: Estimating a mean from delayed observations. Z. f ür Wahr. 35, 103–113 (1976)
Starr, N., Woodroofe, M.: Gone fishin’: optimal stopping based on catch times. Report No. 33, Department of Statistics, University of Michigan, Ann Arbor, MI (1974)
Szajowski, K.: Optimal stopping of a 2-vector risk process. In: Stability in Probability, Jolanta K. Misiewicz (ed.), Banach Center Publ. 90, 179–191. Institute of Mathematics, Polish Academy of Science, Warsaw (2010), doi:10.4064/bc90-0-12
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Karpowicz, A., Szajowski, K. (2013). Anglers’ Fishing Problem. In: Cardaliaguet, P., Cressman, R. (eds) Advances in Dynamic Games. Annals of the International Society of Dynamic Games, vol 12. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8355-9_17
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