In the first part of the talk, we study risk-sharing equilibria where heterogenous agents trade subject to quadratic transaction costs. The corresponding equilibrium asset prices and trading strategies are characterised by a system of nonlinear, fully-coupled forward-backward stochastic differential equations. We show that a unique solution generally exists provided that the agents’ preferences are sufficiently similar. In a benchmark specification, the illiquidity discounts and liquidity premia observed empirically correspond to a positive relationship between transaction costs and volatility.
In the second part of the talk, we discuss how the model can be calibrated to time series of prices and the corresponding trading volume, and explain how extensions of the model with general transaction costs, for example, can be solved numerically using the deep learning approach of Han, Jentzen, and E (2018).
 (Based on joint works with Martin Herdegen and Dylan Possamai, as well as with Lukas Gonon and Xiaofei Shi)

Suppose that you have a long strip of paper, and draw the central line through it. You then glue it together so as to make a Möbius band. Can the drawn curve be contained in a plane?

We'll answer the question in this talk, as well as introduce the concepts from the Geometry of Surfaces course required to go through it; including Gauss' one and only Theorema Egregium! (we won't prove it though).

I will talk about recent work with Jack Thorne in which we find the average size of the Selmer group for a family of genus-2 curves by analyzing a graded Lie algebra of type E_8. I will focus on the role representation theory plays in our proofs.

When solving partial differential equations driven by additive spatial white noise, the efficient sampling of white noise realizations can be challenging. In this talk we focus on the efficient sampling of white noise using quasi-random points in a finite element method and multilevel Quasi Monte Carlo (MLQMC) setting. This work is an extension of previous research on white noise sampling for MLMC.

We express white noise as a wavelet series expansion that we divide in two parts. The first part is sampled using quasi-random points and contains a finite number of terms in order of decaying importance to ensure good QMC convergence. The second part is a correction term which is sampled using standard pseudo-random numbers.

We show how the sampling of both terms can be performed in linear time and memory complexity in the number of mesh cells via a supermesh construction. Furthermore, our technique can be used to enforce the MLQMC coupling even in the case of non-nested mesh hierarchies. We demonstrate the efficacy of our method with numerical experiments.

With growing population of humans being clustered in large cities and connected by fast routes more suitable environments for epidemics are being created. Topped by rapid mutation rate of viral and bacterial strains, epidemiological studies stay a relevant topic at all times. From the beginning of 2019, the World Health Organization publishes at least five disease outbreak news including Ebola virus disease, dengue fever and drug resistant gonococcal infection, the latter is registered in the United Kingdom.

To control the outbreaks it is necessary to gain information on mechanisms of appearance and evolution of pathogens. Close to all disease-causing virus and bacteria undergo a specialization towards a human host from the closest livestock or wild fauna of a shared habitat. Every strain (or subtype) of a pathogen has a set of characteristics (e.g. infection rate and burst size) responsible for its success in a new environment, a host cell in case of a virus, and with the right amount of skepticism that set can be framed as fitness of the pathogen. In our model, we consider a population of a mutating strain of a virus. The strain specialized towards a new host usually remains in the environment and does not switch until conditions get volatile. Two subtypes, wild and mutant, of the virus share a host. This talk will illustrate findings on an explicitly independent cycling coexistence of the two subtypes of the parasite population. A rare transcritical bifurcation of limit cycles is discussed. Moreover, we will find conditions when one of the strains can outnumber and eventually eliminate the other strain focusing on an infection rate as fitness of strains.