The most exciting game

Julio Backhoff-Veraguas; Mathias Beiglböck

doi:10.1214/24-ecp574

Titel

The most exciting game

Autor*in

Julio Backhoff-Veraguas

Universität Wien

Mathias Beiglböck

Universität Wien

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 x0∈(0,1) and finally 0 or 1. As an idealization we take a continuous time interval [0,1] and let Mt be the probability at time t that the home team wins. Mathematically, M=(Mt)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 dMt=sin(πMt)π√1−tdBt. 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.

Stichwort

EntropyMartingale optimal transportmax-entropy win-martingaleprediction marketsspecific relative entropy

Objekt-Typ

journal article

Sprache

Englisch [eng]

Persistent identifier

phaidra.univie.ac.at/o:2081755

DOI

10.1214/24-ecp574

Erschienen in

Titel