We conduct a systematic large-scale analysis of order book-driven predictability in high-frequency returns by leveraging deep learning techniques. First, we introduce a new and robust representation of the order book, the volume representation. Next, we carry out an extensive empirical experiment to address various questions regarding predictability. We investigate if and how far ahead there is predictability, the importance of a robust data representation, the advantages of multi-horizon modeling, and the presence of universal trading patterns. We use model confidence sets, which provide a formalized statistical inference framework particularly well suited to answer these questions. Our findings show that at high frequencies predictability in mid-price returns is not just present, but ubiquitous. The performance of the deep learning models is strongly dependent on the choice of order book representation, and in this respect, the volume representation appears to have multiple practical advantages.

Skeletal joints are often modelled as two adjacent layers of poroviscoelastic cartilage that are permitted to slide past each other.  The talk will begin by outlining a mathematical model that may be used, focusing on two unusual features of the model: (i) the solid component of the poroviscoelastic body has a charged surface that ionises the fluid within the pores, generating a swelling pressure; and (ii) appropriate conditions are required at the interface between the two adjacent layers of cartilage.  The remainder of the talk will then address various theoretical and practical issues in computing a finite element solution of the governing equations.