L6

I introduce Stochastic Portfolio Theory (SPT), which is an alternative approach to optimal investment, where the investor aims to beat an index instead of optimising a mean-variance or expected utility criterion. Portfolios which achieve this are called relative arbitrages, and simple and implementable types of such trading strategies have been shown to exist in very general classes of continuous semimartingale market models, with unspecified drift and volatility processes but realistic assumptions on the behaviour of stocks which come from empirical observation. I present some of my recent work on this, namely the so-called diversity-weighted portfolio with negative parameter. This portfolio outperforms the market quite significantly, for which I have found both theoretical and empirical evidence.

1. Minimising Regret in Portfolio Optimisation (Simões)

When looking for an "optimal" portfolio the traditional approach is to either try to minimise risk or maximise profit. While this approach is probably correct for someone investing their own wealth, usually traders and fund managers have other concerns. They are often assessed taking into account others' performance, and so their decisions are molded by that. We will present a model for this decision making process and try to find our own "optimal" portfolio.

2. Systemic risk in financial networks (Murevics)

Abstract: In this paper I present a framework for studying systemic risk and financial contagion in interbank networks. The current financial health of institutions is expressed through an abstract measure of robustness, and the evolution of robustness in time is described through a system of stochastic differential equations. Using this model I then study how the structure of the interbank lending network affects the spread of financial contagion through different contagion channels and compare the results for different network structures. Finally I outline the future directions for developing this model.

1. Calibration and Pricing of Financial Derivatives using Forward PDEs (Mariapragassam)

Nowadays, various calibration techniques are in use in the financial industry and the exact re-pricing of call options is a must-have standard. However, practitioners are increasingly interested in taking into account the quotes of other derivatives as well.
We describe our approach to the calibration of a specific Local-Stochastic volatility model proposed by the FX group at BNP Paribas. We believe that forward PDEs are powerful tools as they allow to achieve stable and fast best-fit routines. We will expose our current results on this matter.

Joint work with Prof. Christoph Reisinger

2. Infinite discrete-time investment and consumption problem (Li)

We study the investment and consumption problem in infinite discrete-time framework. In our problem setting, we do not need the wealth process to be positive at any time point. We first analyze the time-consistent case and give the convergence of value function for infinite-horizon problem by value functions of finite-horizon problems.

Then we discuss the time-consistent case, and hope the value functions of finite-horizon problems will still converge to the infinite-horizon problem.

We use a growth procedure for binary trees due to Luczak and Winkler, a bijection between binary trees and irreducible quadrangulations of the hexagon due to Fusy, Poulalhon and Schaeffer, and the classical angular mapping between quadrangulations and maps, to define a growth procedure for maps. The growth procedure is local, in that every map is obtained from its predecessor by an operation that only modifies vertices lying on a common face with some fixed vertex. The sequence of maps has an almost sure limit G; we show that G is the distributional local limit of large, uniformly random 3-connected graphs.
A classical result of Brooks, Smith, Stone and Tutte associates squarings of rectangles to edge-rooted planar graphs. Our map growth procedure induces a growing sequence of squarings, which we show has an almost sure limit: an infinite squaring of a finite rectangle, which almost surely has a unique point of accumulation. We know almost nothing about the limit, but it should be in some way related to "Liouville quantum gravity".
Parts joint with Nicholas Leavitt.

Fluid-structure interaction (FSI) problems arise in many applications. The widely known examples are aeroelasticity and biofluids.

In biofluidic applications, such as, e.g., the study of interaction between blood flow and cardiovascular tissue, the coupling between the fluid and the

relatively light structure is {highly nonlinear} because the density of the structure and the density of the fluid are roughly the same.

In such problems, the geometric nonlinearities of the fluid-structure interface

and the significant exchange in the energy between a moving fluid and a structure

require sophisticated ideas for the study of their solutions.

In the blood flow application, the problems are further exacerbated by the fact that the walls of major arteries are composed of several layers, each with

different mechanical characteristics.

No results exist so far that analyze solutions to fluid-structure interaction problems in which the structure is composed of several different layers.

In this talk we make a first step in this direction by presenting a program to study the {\bf existence and numerical simulation} of solutions

for a class of problems

describing the interaction between a multi-layered, composite structure, and the flow of an incompressible, viscous fluid,

giving rise to a fully coupled, {\bf nonlinear moving boundary, fluid-multi-structure interaction problem.}

A stable, modular, loosely coupled scheme will be presented, and an existence proof

showing the convergence of the numerical scheme to a weak solution to the fully nonlinear FSI problem will be discussed.

Our results reveal a new physical regularizing mechanism in

FSI problems: the inertia of the fluid-structure interface regularizes the evolution of the FSI solution.

All theoretical results will be illustrated with numerical examples.

This is a joint work with Boris Muha (University of Zagreb, Croatia, and with Martina Bukac, University of Pittsburgh and Notre Dame University).

In the Erdös-Rényi random graph process, starting from an empty graph, in each step a new random edge is added to the evolving graph. One of its most interesting features is the `percolation phase transition': as the ratio of the number of edges to vertices increases past a certain critical density, the global structure changes radically, from only small components to a single giant component plus small ones.


In this talk we consider Achlioptas processes, which have become a key example for random graph processes with dependencies between the edges. Starting from an empty graph these proceed as follows: in each step two potential edges are chosen uniformly at random, and using some rule one of them is selected and added to the evolving graph. We discuss why, for a large class of rules, the percolation phase transition is qualitatively comparable to the classical Erdös-Rényi process.


Based on joint work with Oliver Riordan.

A system of linear equations is called partition regular if, whenever the naturals are finitely coloured, there is a monochromatic solution of the equations. Many of the classical theorems of Ramsey Theory may be phrased as assertions that certain systems are partition regular.

What happens if we are colouring not the naturals but the (non-zero) integers, rationals, or reals instead? After some background, we will give various recent results.

This talk is based on a joint work with B. Rémy (Lyon) in which we study some subgroups of topological Kac–Moody groups and the implications of this study on the subgroup structure of the ambient Kac–Moody group.

In this talk we will discuss when one right-angled Artin group is a subgroup of another one and explain how this basic algebraic problem may provide answers to questions in geometric group theory and model theory such as classification of right-angled Artin groups up to quasi-isometries and universal equivalence.