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 all the way back to Nina Gantert's PhD thesis, and in recent time Hans Foellmer has rekindled the study of this object by for instance obtaining a novel transport-information inequality.

In this talk I will first discuss the existence of a closed formula for the specific relative entropy, depending on the quadratic variation of the involved martingales. Next I 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. Beiglbock we give a concise answer to this question. Finally, if time permits, I will give a glimpse to different extensions of this object, e.g. to higher dimensions or when we replace the role of the relative entropy by a power of the Wasserstein distance, as recently developed with X. Zhang and co-authors.