Journal of Statistical Theory and Practice

, Volume 2, Issue 2, pp 199–219 | Cite as

Gambler’s Ruin with Catastrophes and Windfalls

  • B. HunterEmail author
  • A. C. Krinik
  • C. Nguyen
  • J. M. Switkes
  • H. F. von Bremen


We compute ruin probabilities, in both infinite-time and finite-time, for a Gambler’s Ruin problem with both catastrophes and windfalls in addition to the customary win/loss probabilities. For constant transition probabilities, the infinite-time ruin probabilities are derived using difference equations. Finite-time ruin probabilities of a system having constant win/loss probabilities and variable catastrophe/windfall probabilities are determined using lattice path combinatorics. Formulae for expected time till ruin and the expected duration of gambling are also developed. The ruin probabilities (in infinite-time) for a system having variable win/loss/catastrophe probabilities but no windfall probability are found. Finally, the infinite-time ruin probabilities of a system with variable win/loss/catastrophe/windfall probabilities are determined.


Gambler’s Ruin Catastrophe Birth-Death Process 

AMS Subject Classification



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  1. Ash, R. B., 1970. Basic Probability Theory, John Wiley and Sons, Inc., New York.zbMATHGoogle Scholar
  2. Edwards, A. W. F., 1983. Pascal’s problem: the “Gambler’s Ruin”. International Statistical Review, 73–79.Google Scholar
  3. Feller, W., 1968. An Introduction to Probability Theory and its Applications, Volume 1, 3rd edition, John Wiley and Sons, Inc., New York.Google Scholar
  4. Goldberg, S., 1986. Introduction to Difference Equations, with Illustrative Examples from Economics, Psychology, and Sociology, Dover Publications, Inc., New York.zbMATHGoogle Scholar
  5. Harper, J. D., Ross, K. A., 2005. Stopping strategies and Gambler’s ruin. Mathematics Magazine, 255–268.Google Scholar
  6. Hoel, P., Port, S., Stone, C., 1987. Introduction to Stochastic Processes, Waveland Press Inc.zbMATHGoogle Scholar
  7. Hunter, B., 2005. Gambler’s Ruin and the Three State Process, Master’s Thesis, California State Polytechnic University, Pomona.Google Scholar
  8. Krinik, A., Rubino, G., Marcus, D., Swift, R., Kasfy, H., Lam, H., 2005. Dual processes to solve single serve systems. Journal of Statistical Planning and Inference Special Issue on Lattice Path Combinatorics and Discrete Distributions 135, 121–147.zbMATHGoogle Scholar
  9. Marcus, D., 1998. Combinatorics: A Problem Oriented Approach, The Mathematical Association of America, Washington.zbMATHGoogle Scholar
  10. Mohanty, S. G., 1979. Lattice Path Counting and Applications, Academic Press.zbMATHGoogle Scholar
  11. Narayana, T. V., 1979. Lattice Path Combinatorics With Statistical Applications, University of Toronto Press, Toronto.zbMATHGoogle Scholar
  12. Takacs, L., 1969. On the Classical Ruin Problems. American Statistical Association Journal 64, 889–906.MathSciNetzbMATHGoogle Scholar

Copyright information

© Grace Scientific Publishing 2008

Authors and Affiliations

  • B. Hunter
    • 1
    Email author
  • A. C. Krinik
    • 2
  • C. Nguyen
    • 2
  • J. M. Switkes
    • 2
  • H. F. von Bremen
    • 2
  1. 1.Department of MathematicsUniversity of CaliforniaDavisUSA
  2. 2.Department of Mathematics and StatisticsCalifornia State Polytechnic UniversityPomonaUSA

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