Of 'most exciting' games and the specific relative entropy between martingales
Abstract
The laws of two continuous martingales will typically be singular to each other and hence have infinite relative entropy. But this does not need to happen in discrete time. This suggests defining a new object, the specific relative entropy, as a scaled limit of the relative entropy between the discretized laws of the martingales. This definition goes back to Nina Gantert's PhD thesis, and in recent time Hans Foellmer has rekindled the study of this object. Independently, this object has made sporadic appearances in finance over the years, for instance in works by Avellaneda et al. and more recently Dolinsky and Cohen. In this talk we will first discuss the existence of a closed-form formula for the specific relative entropy, depending on the quadratic variation of the involved martingales. Next we will describe an application of this object to prediction markets. Concretely, David Aldous asked in an open question to determine the 'most exciting' game, i.e. the prediction market with the highest entropy. With M. Beiglböck we give an answer to this question by solving a stochastic control problem whose cost criterion is the specific relative entropy. Finally we will discuss an application to the 'most exciting' game with multiple outcomes, based on joint work with Wang and Zhang, highlighting a novel connection to the field of Monge-Ampere equations.
Watch recording ↗About the speaker
Julio Backhoff obtained his PhD in Mathematics from Humboldt-Universität zu Berlin, in 2015, under the supervision of Prof. Ulrich Horst. From 2015 to 2019, he held various postdoc and university assistant positions in the group of Prof. Mathias Beiglböck at the University of Vienna and the Vienna University of Technology. In 2020, he became an assistant professor at the University of Twente. Since 2021, he has been an assistant professor at the University of Vienna (Mathematics Faculty, Mathematical Finance and Probability group). His research lies at the intersection of Mathematical Finance, Stochastic control / analysis, and optimal transport theory. For the most part, he is interested in problems coming from model-free finance and model calibration, with an emphasis on adapted and martingale transport.