### Abstract

An (m,p) urn contains m balls of value − 1 and p balls of value +1. A player starts with fortune k and in each game draws a ball without replacement with the fortune increasing by one unit if the ball is positive and decreasing by one unit if the ball is negative, having to stop when k = 0 (risk aversion). Let V(m,p,k) be the expected value of the game. We are studying the question of the minimum k such that the net gain function of the game V(m,p,k) − k is positive, in both the discrete and the continuous (Brownian bridge) settings. Monotonicity in various parameters m, p, k is established for both the value and the net gain functions of the game. For the cut-off value k, since the case m − p

Original language | English (US) |
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Pages (from-to) | 702-722 |

Number of pages | 21 |

Journal | Stochastics |

Volume | 87 |

Issue number | 4 |

DOIs | |

State | Published - Jul 4 2015 |

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### Keywords

- Brownian bridge
- Chow and Robbins S<inf>n</inf>/n problem
- optimal stopping
- optimal strategy
- Shepp's urn

### ASJC Scopus subject areas

- Statistics and Probability
- Modeling and Simulation

### Cite this

*Stochastics*,

*87*(4), 702-722. https://doi.org/10.1080/17442508.2014.995660