Robust contracting in general contract spaces

Abstract

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 consider the process M = (M t ) t∈[0,1] giving the probability at time t that the home team wins. This is a martingale which we idealize further to have continuous paths. We consider the problem to find the most random martingale M of this type, where 'most random' is interpreted as a maximal entropy criterion. We observe that this max-entropy win-martingale M also minimizes specific relative entropy with respect to Brownian motion in the sense of Gantert [18] and use this to prove that M is characterized by the stochastic differential equationTo 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.