The Skorokhod embedding problem aims to represent a given probability measure on the real line as the distribution of Brownian motion stopped at a chosen stopping time. In this talk, we consider an extension of the weak formulation of the optimal Skorokhod embedding problem. Using the classical convex duality approach together with the optimal stopping theory, we establish some duality. Moreover, based on the duality, we provide an alternative proof of the monotonicity principle proved by Beiglbock, Cox and Huesmann.

We pursue robust approach to pricing and hedging in mathematical
finance. We develop a general discrete time setting in which some
underlying assets and options are available for dynamic trading and a
further set of European options, possibly with varying maturities, is
available for static trading. We include in our setup modelling beliefs by
allowing to specify a set of paths to be considered, e.g.
super-replication of a contingent claim is required only for paths falling
in the given set. Our framework thus interpolates between
model-independent and model-specific settings and allows to quantify the
impact of making assumptions. We establish suitable FTAP and
Pricing-Hedging duality results which include as special cases previous
results of Acciaio et al. (2013), Burzoni et al. (2016) as well the
Dalang-Morton-Willinger theorem. Finally, we explain how to treat further
problems, such as insider trading (information quantification) or American
options pricing.
Based on joint works with Burzoni, Frittelli, Hou, Maggis; Aksamit, Deng and Tan.
 

Roxana Pamfil
Analysis of consumer behaviour with annotated networks

Rachel Philip
Modelling droplet breakup in a turbulent jet

Asbjørn Riseth
Stochastic optimal control of a retail pricing problem
 

We introduce Learning with Errors and Ring Learning with Errors, two hard
lattice problems which are widely used for security of Homomorphic
Encryption schemes. Following a study we conducted comparing four such
schemes, the best scheme was the so-called BGV scheme, introduced by
Brakerski-Gentry-Vaikuntanathan in 2012. We present it as an example of a
ring-based homomorphic scheme, discussing its number theoretic
optimisations.

We present “Ouroboros,” the first blockchain protocol based on proof of stake with rigorous security guarantees. We establish security properties for the protocol comparable to those achieved by the bitcoin blockchain protocol. As the protocol provides a “proof of stake” blockchain discipline, it offers qualitative efficiency advantages over blockchains based on proof of physical resources (e.g., proof of work). We showcase the practicality of our protocol in real world settings by providing experimental results on transaction processing time obtained with a prototype implementation in the Amazon cloud. We also present a novel reward mechanism for incentivizing the protocol and we prove that given this mechanism, honest behavior is an approximate Nash equilibrium, thus neutralizing attacks such as selfish mining. 

Joint work with  Alexander Russell and Bernardo David and Roman Oliynykov