Karpowicz, Anna and Szajowski, Krzysztof (2010): Anglers’ Fishing Problem. Published in: Annals of the International Society of Dynamic Games , Vol. 12, No. Advances in Dynamic Games (August 2012): pp. 327349.

PDF
MPRA_paper_41800.pdf Download (310kB)  Preview 
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.
Item Type:  MPRA Paper 

Original Title:  Anglers’ Fishing Problem 
Language:  English 
Keywords:  Stopping time Optimal stopping Dynamic programming SemiMarkov process Marked renewal process Renewal–reward process Infinitesimal generator Fishing problem Bilateral approach Stopping game 
Subjects:  C  Mathematical and Quantitative Methods > C6  Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C61  Optimization Techniques ; Programming Models ; Dynamic Analysis C  Mathematical and Quantitative Methods > C7  Game Theory and Bargaining Theory > C72  Noncooperative Games C  Mathematical and Quantitative Methods > C7  Game Theory and Bargaining Theory > C73  Stochastic and Dynamic Games ; Evolutionary Games ; Repeated Games 
Item ID:  41800 
Depositing User:  Krzysztof Szajowski 
Date Deposited:  08 Oct 2012 13:28 
Last Modified:  27 Sep 2019 22:59 
References:  Boshuizen, F., Gouweleeuw, J.: General optimal stopping theorems for semiMarkov 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., PasternakWiniarski, 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) CrossRef Jensen, U., Hsu, G.: Optimal stopping by means of point process observations with applications in reliability. Math. Oper. Res. 18(3), 645–657 (1993) CrossRef 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) CrossRef 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 6068502 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 2vector 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/bc90012 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/41800 