Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-30T14:37:19.278Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Mabinogion urn model

Part of: Markov processes Existence theories Stochastic processes

Published online by Cambridge University Press:  26 July 2018

David Stenlund
David Stenlund*
Affiliation:
Åbo Akademi University
*
* Postal address: Mathematics and Statistics, Faculty of Science and Engineering, Åbo Akademi University, Åbo, FIN-20500, Finland. Email address: david.stenlund@abo.fi
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we discuss the Mabinogion urn model introduced by Williams (1991). Therein he describes an optimal control problem where the objective is to maximize the expected final number of objects of one kind in the Mabinogion urn model. Our main contribution is formulae for the expected time to absorption and its asymptotic behaviour in the optimally controlled process. We also present results for the noncontrolled Mabinogion urn process and briefly analyze other strategies that become superior if a certain discount factor is included.

Keywords

Mabinogion sheep problemMarkov chaindifference equationextinction timerandom walkhypergeometric functionbinomial identityDoob's h-transform

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 49J21: Optimal control problems involving relations other than differential equations 60G50: Sums of independent random variables; random walks
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 2 , June 2018 , pp. 327 - 346
Copyright
Copyright © Applied Probability Trust 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Abramowitz, M. and Stegun, I. A. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. U.S. Government Printing Office, Washington, D.C.. Google Scholar
[2]Bharucha-Reid, A. T. (1960). Elements of the Theory of Markov Processes and Their Applications. McGraw-Hill, New York. Google Scholar
[3]Chan, T. (1996). Some diffusion models for the Mabinogion sheep problem of Williams. Adv. Appl. Prob. 28, 763–783. Google Scholar
[4]Davis, M. H. A. and Norman, A. R. (1990). Portfolio selection with transaction costs. Math. Operat. Res. 15, 676–713. Google Scholar
[5]Flajolet, P. and Huillet, T. (2008). Analytic combinatorics of the Mabinogion urn. In Fifth Colloquium on Mathematics and Computer Science, Association of Discrete Mathematics and Theoretical Computer Science, Nancy, pp. 549–571. Google Scholar
[6]Flajolet, P., Dumas, P. and Puyhaubert, V. (2006). Some exactly solvable models of urn process theory. In Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities, Association of Discrete Mathematics and Theoretical Computer Science, Nancy, pp. 59–118. Google Scholar
[7]Gould, H. W. (1972). Combinatorial Identities. Henry W. Gould, Morgantown, WV. Google Scholar
[8]Guest, C. E. (translated by) (1849). The Mabinogion. Available at https://ebooks.adelaide.edu.au/m/mabinogion/guest/. Google Scholar
[9]Kingman, J. F. C. (1999). Martingales in the OK Corral. Bull. London Math. Soc. 31, 601–606. Google Scholar
[10]Lin, Y.-S. (2016). A generalization of the Mabinogion sheep problem of D. Williams. J. Appl. Prob. 53, 1240–1256. Google Scholar
[11]Petkovšek, M., Wilf, H. S. and Zeilberger, D. (1996). A = B. A K Peters, Wellesley, MA. Google Scholar
[12]Stenlund, D. and Wan, J. G. (2018). Some double sums involving ratios of binomial coefficients. In preparation. Google Scholar
[13]Taylor, H. M. and Karlin, S. (1998). An Introduction to Stochastic Modeling, 3rd edn. Academic Press, San Diego, CA. Google Scholar
[14]Williams, D. (1991). Probability with Martingales. Cambridge University Press. Google Scholar
[15]Williams, D. and McIlroy, P. (1998). The OK Corral and the power of the law (a curious Poisson-kernel formula for a parabolic equation). Bull. London Math. Soc. 30, 166–170. Google Scholar