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A Stopping Problem

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A stopping time N = f(So,…,Sn) is used to monitor the fluctuation of a random walk (Sn). The random variable N is the stage at which Sn exceeds a predetermined bound for the first time. An extrapolation method is suggested giving monotone upper and lower bounds to the distribution function of N at each stage of iteration. The extrapolation method is based on the Perron-Frobenius theory of positive matrices and its generalization. The stopping problem is applied to some well-known models arising in quality control, risk theory, and queueing theory.

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  • BORCH, K. (1967), “The theory of risk,” Journal of the Royal Statistical Society, Series B, 29, 432–467.

    Google Scholar 

  • BORCH, K. (1974), The Mathematical Theory of Insurance, Heath, Lexington.

    Google Scholar 

  • DE FINETTI, B. (1957), “Su un’impostazione alternativa della teoria colletiva del rischio,” Transactions of the XV International Congress of Actuaries 2, 433–443.

    Google Scholar 

  • KARLIN, S. and TAYLOR, H.M. (1975), A first course in stochastic processes, Academic Press, San Diego.

    Google Scholar 

  • PAGE, E.S. (1954), “Continuous Inspection Schemes,” Biometrika, 41, 100–114.

    Google Scholar 

  • WALDMANN, K.-H. (1986a), “Bounds for the distribution of the run length in general quality control schemes,” Statistische Hefte 27, 37–56.

    CrossRef  Google Scholar 

  • WALDMANN, K.-H. (1986b), “Bounds for the distribution of the run length of one-sided and two-sided CUSUM quality control schemes,” Technometics 28, 61–67.

    CrossRef  Google Scholar 

  • WALDMANN, K.-H. (1988) “On optimal dividend payments and related problems,” Mathematics and Economics 7, 237–249.

    CrossRef  Google Scholar 

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© 1993 Springer-Verlag Berlin · Heidelberg

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Waldmann, KH. (1993). A Stopping Problem. In: Diewert, W.E., Spremann, K., Stehling, F. (eds) Mathematical Modelling in Economics. Springer, Berlin, Heidelberg.

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-78510-8

  • Online ISBN: 978-3-642-78508-5

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