## Abstract

Motivated by a problem posed by Aldous [2, 1] our goal is to find the maximal-entropy win-martingale: In a sports game between two teams, the chance the home team wins is initially x
_{0} ∈ (0, 1) and finally 0 or 1. As an idealization we take a continuous time interval [0, 1] and let M
_{t} be the probability at time t that the home team wins. Mathematically, M = (M
_{t})
_{t}
_{∈[0,1]} is modelled as a continuous martingale. We consider the problem to find the most random martingale M of this type, where ‘most random’ is interpreted as a maximal entropy criterion. In discrete time this is equivalent to the minimization of relative entropy w.r.t. a Gaussian random walk. The continuous time analogue is that the max-entropy win-martingale M should minimize specific relative entropy with respect to Brownian motion in the sense of Gantert [20]. We use this to prove that M is characterized by the stochastic differential equation (Formula Present) To derive the form of the optimizer we use a scaling argument together with a new first order condition for martingale optimal transport, which may be of interest in its own right.

Original language | English |
---|---|

Article number | 6 |

Journal | Electronic Communications in Probability |

Volume | 29 |

DOIs | |

Publication status | Published - 2024 |

## Austrian Fields of Science 2012

- 101007 Financial mathematics
- 101019 Stochastics

## Keywords

- entropy
- martingale optimal transport
- max-entropy win-martingale
- prediction markets
- specific relative entropy