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.
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References
Asmussen S, Albrecher H (2010) Ruin Probabilities
Asmussen S, Sigman K (2009) Monotone Stochastic Recursions and their Duals. Probab Eng Inf Sci 10(01):1
Bernoulli J (1713) Ars conjectandi
Błaszczyszyn B, Sigman K (1999) Risk and duality in multidimensions. Stochastic processes and their applications 83:331–356
Bremaud P (1999) Markov chains: Gibbs fields, Monte Carlo simulation, and queues
Diaconis P, Fill J A (1990) Strong stationary times via a new form of duality. Ann Probab 18(4):1483–1522
Dette H, Fill J A, Pitman J, Studden W J (1997) Wall and siegmund duality relations for birth and death chains with reflecting barrier. J Theor Probab 10 (2):349–374
Edwards A W (1983) Pascal’s problem: The ’Gambler’s ruin’. Int Stat Rev 51:73–79
El-Shehawey M A (2009) On the gambler’s ruin problem for a finite Markov chain. Statistics & Probability Letters 79(14):1590–1595
El-Shehawey M A (2000) Absorption probabilities for a random walk between two partially absorbing boundaries: I. J Phys A Math Gen 33(49):9005–9013
Harik G, Cantú-Paz E, Goldberg D E, Miller B L (1999) The gambler’s ruin problem, genetic algorithms, and the sizing of populations. Evol Comput 7(3):231–253
Huillet T (2010) Siegmund duality with applications to the neutral Moran model conditioned on never being absorbed. J Phys A Math Theor 43:37
Huillet T (2014) Servet Martínez On Möbius Duality and Coarse-Graining. Journal of Theoretical Probability
Isaac R (1995) The pleasures of probability. Undergraduate texts in mathematics. Springer new york, new york, NY
Kaigh W D (1979) An attrition problem of gambler’s ruin. Math Mag 52:22–25
Kmet A, Petkovšek M (2002) Gambler’s ruin problem in several dimensions. Adv Appl Math 28(2):107–118
Lefebvre M (2008) The gambler’s ruin problem for a Markov chain related to the Bessel process. Statistics & Probability Letters 78(15):2314–2320
Lengyel T (2009a) Gambler’s ruin and winning a series by m games. Ann Inst Stat Math 63(1):181–195
Lengyel T (2009b) The conditional gambler’s ruin problem with ties allowed. Appl Math Lett 22(3):351–355
Lindley D V (1952) The theory of queues with a single server. Math Proc Camb Philos Soc 48(02):277–289
Lorek P (2016) Siegmund duality for Markov chains on partially ordered state spaces. Submitted:1–14
Lorek P, Szekli R (2012) Strong stationary duality for Möbius monotone Markov chains. Queueing Systems 71(1-2):79–95
Rocha A L, Stern F (2004) The asymmetric n-player gambler’s ruin problem with equal initial fortunes. Adv Appl Math 33(3):512–530
Rolski T, Schmidli H, Schmidt V, Teugels J (2009) stochastic processes for insurance and finance
Rota G-C (1964) On the foundations of combinatorial theory I. Theory of Möbius functions. Probab Theory Relat Fields 368:340–368
Scott J (1981) The probability of bankruptcy. J Bank Financ 5(3):317–344
Siegmund D (1976) The equivalence of absorbing and reflecting barrier problems for stochastically monotone Markov processes The Annals of Probability
Snell J L (2009) Gambling, probability and martingales. The Mathematical Intelligencer 4(3):118–124
Theodore Cox J., Rösler U (1984) A duality relation for entrance and exit laws for Markov processes. Stochastic processes and their applications 16:141–156
Liggett TM (2004) Interacting particle systems
Tsai C W, Hsu Y, Lai K-C, Wu N-K (2014) Application of gambler’s ruin model to sediment transport problems. J Hydrol 510:197–207
Veitch V, Ferrie C, Gross D, Emerson J (2012) Negative quasi-probability as a resource for quantum computation. New J Phys 14(11):113011
Wigner E (1932) On the quantum correction for thermodynamic equilibrium. Phys Rev 40(5):749–759
Yamamoto K (2013) Solution and analysis of a One-Dimensional First-Passage problem with a nonzero halting probability. International Journal of Statistical Mechanics 2013:1–9
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Work supported by NCN Research Grant DEC-2013/10/E/ST1/00359
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Lorek, P. Generalized Gambler’s Ruin Problem: Explicit Formulas via Siegmund Duality. Methodol Comput Appl Probab 19, 603–613 (2017). https://doi.org/10.1007/s11009-016-9507-6
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DOI: https://doi.org/10.1007/s11009-016-9507-6
Keywords
- Generalized gambler’s ruin problem
- Markov chains
- Absorption probability
- Siegmund duality
- Möbius monotonicity
- Partial ordering