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Generalized Gambler’s Ruin Problem: Explicit Formulas via Siegmund Duality

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  • Paweł Lorek

    (University of Wrocław)

Abstract

We give explicit formulas for ruin probabilities in a multidimensional Generalized Gambler’s ruin problem. The generalization is best interpreted as a game of one player against d other players, allowing arbitrary winning and losing probabilities (including ties) depending on the current fortune with particular player. It includes many previous other generalizations as special cases. Instead of usually utilized first-step-like analysis we involve dualities between Markov chains. We give general procedure for solving ruin-like problems utilizing Siegmund duality in Markov chains for partially ordered state spaces studied recently in context of Möbius monotonicity.

Suggested Citation

  • Paweł Lorek, 2017. "Generalized Gambler’s Ruin Problem: Explicit Formulas via Siegmund Duality," Methodology and Computing in Applied Probability, Springer, vol. 19(2), pages 603-613, June.
    • Handle: RePEc:spr:metcap:v:19:y:2017:i:2:d:10.1007_s11009-016-9507-6
    DOI: 10.1007/s11009-016-9507-6
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    References listed on IDEAS

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    1. Lefebvre, Mario, 2008. "The gambler's ruin problem for a Markov chain related to the Bessel process," Statistics & Probability Letters, Elsevier, vol. 78(15), pages 2314-2320, October.
    2. Blaszczyszyn, Bartlomiej & Sigman, Karl, 1999. "Risk and duality in multidimensions," Stochastic Processes and their Applications, Elsevier, vol. 83(2), pages 331-356, October.
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    Cited by:

    1. Lesław Gajek & Marcin Rudź, 2020. "Finite-horizon general insolvency risk measures in a regime-switching Sparre Andersen model," Methodology and Computing in Applied Probability, Springer, vol. 22(4), pages 1507-1528, December.

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