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.
Similar content being viewed by others
References
M. Denuit and Cl. Lefevre, “Some new classes of stochastic order relations among arithmetic random variables, with applications in actuarial science,” Insurance: Mathematics & Economics vol. 20 pp. 197–213, 1997.
M. Denuit, Cl. Lefevre, and M. Shaked, “The s-convex orders among real random variables, with applications,” Mathematical Inequalities & Applications vol. 1 pp. 585–613, 1998.
M. Denuit, Cl. Lefevre, and M. Mesfioui, “On s-convex stochastic extrema for arithmetic risks,” Insurance: Mathematics & Economics vol. 25 pp. 143–155, 1999a.
M. Denuit, E. De Vylder, and Cl. Lefevre, “Extremal generators and extremal distributions for the continuous s-convex stochastic orderings,” Insurance: Mathematics & Economics vol. 24 pp. 201–217, 1999b.
S. N. Ethier and D. Khoshnevisan, “Bounds on gambler’s ruin probabilities in terms of moments,” Methodology and Computing in Applied Probability vol. 4 pp. 55–68, 2002.
W. Feller, An Introduction to Probability Theory and Its Applications, J. Wiley: New York, vol. 1, (3rd edition), 1968.
S. Karlin and W. J. Studden, Tchebycheff Systems: With Applications in Analysis and Statistics, J. Wiley: New York, 1966.
A. S. Kozek, “A rule of thumb (not only) for gamblers,” Stochastic Processes and Their Applications vol. 55, pp. 169–181, 1995.
A. A. Markov, Wahrscheinlichkeitstheorie, B.G. Teubner: Leipzig, 1912.
J. V. Uspensky, Introduction to Mathematical Probability, McGraw-Hill: New York, 1937.
Author information
Authors and Affiliations
Corresponding author
Additional information
AMS 2000 Subject Classification: 60E15, 60G40, 91A60
Rights and permissions
About this article
Cite this article
Hürlimann, W. Improved Analytical Bounds for Gambler’s Ruin Probabilities. Methodol Comput Appl Probab 7, 79–95 (2005). https://doi.org/10.1007/s11009-005-6656-4
Received:
Revised:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s11009-005-6656-4