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Improved Analytical Bounds for Gambler’s Ruin Probabilities

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  • Werner Hürlimann

    (Aon Re and IRMG)

Abstract

Given integer-valued wagers Feller (1968) has established upper and lower bounds on the probability of ruin, which often turn out to be very close to each other. However, the exact calculation of these bounds depends on the unique non-trivial positive root of the equation Φ(ρ) = 1, where Φ is the probability generating function for the wager. In the situation of incomplete information about the distribution of the wager, one is interested in bounds depending only on the first few moments of the wager. Ethier and Khoshnevisan (2002) derive bounds depending explicitly on the first four moments. However, these bounds do not make the best possible use of the available information. Based on the theory of s-convex extremal random variables among arithmetic and real random variables, a substantial improvement can be given. By fixed first four moments of the wager, the obtained new bounds are nearly perfect analytical approximations to the exact bounds of Feller.

Suggested Citation

  • Werner Hürlimann, 2005. "Improved Analytical Bounds for Gambler’s Ruin Probabilities," Methodology and Computing in Applied Probability, Springer, vol. 7(1), pages 79-95, March.
    • Handle: RePEc:spr:metcap:v:7:y:2005:i:1:d:10.1007_s11009-005-6656-4
    DOI: 10.1007/s11009-005-6656-4
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    References listed on IDEAS

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    1. Kozek, Andrzej S., 1995. "A rule of thumb (not only) for gamblers," Stochastic Processes and their Applications, Elsevier, vol. 55(1), pages 169-181, January.
    2. Denuit, Michel & Lefevre, Claude, 1997. "Some new classes of stochastic order relations among arithmetic random variables, with applications in actuarial sciences," Insurance: Mathematics and Economics, Elsevier, vol. 20(3), pages 197-213, October.
    3. S. N. Ethier & Davar Khoshnevisan, 2002. "Bounds on Gambler's Ruin Probabilities in Terms of Moments," Methodology and Computing in Applied Probability, Springer, vol. 4(1), pages 55-68, March.
    4. Denuit, Michel & Vylder, Etienne De & Lefevre, Claude, 1999. "Extremal generators and extremal distributions for the continuous s-convex stochastic orderings," Insurance: Mathematics and Economics, Elsevier, vol. 24(3), pages 201-217, May.
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