### Abstract

A player starts at x in (0, 1) and tries to reach 1. The process (X//T, t greater than equivalent to 0) of his positions moves according to a diffusion process (or, more generally, an Ito process) whose infinitesimal parameters mu , sigma are chosen by the player at each instant of time from a set depending on his current position. To maximize the probability of reaching 1, the player should choose the parameters so as to maximize mu / sigma **2, at least when the maximum is achieved by bounded, measurable functions. This implies that bold (timid) play is optimal for subfair (superfair), continuous-time red-and-black. Furthermore, in superfair red-and-black, the strategy which maximizes the drift coefficient of left brace log X//t right brace minimizes the expected time to reach 1.

Original language | English (US) |
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Pages (from-to) | 599-611 |

Number of pages | 13 |

Journal | Mathematics of Operations Research |

Volume | 10 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 1985 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Computer Science Applications
- Management Science and Operations Research

### Cite this

*Mathematics of Operations Research*,

*10*(4), 599-611. https://doi.org/10.1287/moor.10.4.599