L2

The risk of a financial position is usually summarized by a risk measure.

As this risk measure has to be estimated from historical data, it is important to be able to verify and compare competing estimation procedures. In

statistical decision theory, risk measures for which such verification and comparison is possible, are called elicitable. It is known that quantile based risk

measures such as value-at-risk are elicitable. However, the coherent risk measure expected shortfall is not elicitable. Hence, it is unclear how to perform

forecast verification or comparison. We address the question whether coherent and elicitable risk measures exist (other than minus the expected value).

We show that one positive answer are expectiles, and that they play a special role amongst all elicitable law-invariant coherent risk measures.

We construct and study market models admitting optimal arbitrage. We say that a model admits optimal arbitrage if it is possible, in a zero-interest rate setting, starting with an initial wealth of 1 and using only positive portfolios, to superreplicate a constant c>1. The optimal arbitrage strategy is the strategy for which this constant has the highest possible value. Our definition of optimal arbitrage is similar to the one in Fenrholz and Karatzas (2010), where optimal relative arbitrage with respect to the market portfolio is studied. In this work we present a systematic method to construct market models where the optimal arbitrage strategy exists and is known explicitly. We then develop several new examples of market models with arbitrage, which are based on economic agents' views concerning the impossibility of certain events rather than ad hoc constructions. We also explore the concept of fragility of arbitrage introduced in Guasoni and Rasonyi (2012), and provide new examples of arbitrage models which are not fragile in this sense.

References:

Fernholz, D. and Karatzas, I. (2010). On optimal arbitrage. The Annals of Applied Probability, 20(4):1179–1204.

Guasoni, P. and Rasonyi, M. (2012). Fragility of arbitrage and bubbles in diffusion models. preprint.

There are many financial models used in practice (CIR/Heston, Vasicek,

Stein-Stein, quadratic normal) whose popularity is due, in part, to their

analytically tractable asset pricing. In this talk we will show that it is

possible to generalise these models in various ways while maintaining

tractability. Conversely, we will also characterise the family of models

which admit this type of tractability, in the spirit of the classification

of polynomial term structure models.

An investor trades a safe and several risky assets with linear price impact to maximize expected utility from terminal wealth.

In the limit for small impact costs, we explicitly determine the optimal policy and welfare, in a general Markovian setting allowing for stochastic market,

cost, and preference parameters. These results shed light on the general structure of the problem at hand, and also unveil close connections to

optimal execution problems and to other market frictions such as proportional and fixed transaction costs.

Question: Is it a realistic proposition for a mathematician to use his/her skills to make a living from sports betting? The introduction of betting exchanges have fundamentally changed the potential profitability of gambling, and a professional mathematician's arsenal of numerical and theoretical weapons ought to give them a huge natural advantage over most "punters", so what might be realistically possible and what potential risks are involved? This talk will give some idea of the sort of plan that might be required to realise this ambition, and what might be further required to attain the aim of sustainable gambling profitability.

We are interested in large systems of agents collectively looking for a

consensus (about e.g. their direction of motion, like in bird flocks). In

spite of the local character of the interactions (only a few neighbours are

involved), these systems often exhibit large scale coordinated structures.

The understanding of how this self-organization emerges at the large scale

is still poorly understood and offer fascinating challenges to the modelling

science. We will discuss a few of these issues on a selection of specific

examples.

A perturbative method is introduced to analyze shear bands formation and

development in ductile solids subject to large strain.

Experiments on discrete systems made up of highly-deformable elements [1]

confirm the validity of the method and suggest that an elastic structure

can be realized buckling for dead, tensile loads. This structure has been

calculated, realized and tested and provides the first example of an

elastic structure buckling without elements subject to compression [2].

The perturbative method introduced for the analysis of shear bands can be

successfuly employed to investigate other material instabilities, such as

for instance flutter in a frictional, continuum medium [3]. In this

context, an experiment on an elastic structure subject to a frictional

contact shows for the first time that a follower load can be generated

using dry friction and that this load can induce flutter instability [4].

The perturbative approach may be used to investigate the strain state near

a dislocation nucleated in a metal subject to a high stress level [5].

Eshelby forces, similar to those driving dislocations in solids, are

analyzed on elastic structures designed to produce an energy release and

therefore to evidence configurational forces. These structures have been

realized and they have shown unexpected behaviours, which opens new

perspectives in the design of flexible mechanisms, like for instance, the

realization of an elastic deformable scale [6].

[1] D. Bigoni, Nonlinear Solid Mechanics Bifurcation Theory and Material

Instability. Cambridge Univ. Press, 2012, ISBN:9781107025417.

[2] D. Zaccaria, D. Bigoni, G. Noselli and D. Misseroni Structures

buckling under tensile dead load. Proc. Roy. Soc. A, 2011, 467, 1686.

[3] A. Piccolroaz, D. Bigoni, and J.R. Willis, A dynamical interpretation

of flutter instability in a continuous medium. J. Mech. Phys. Solids,

2006, 54, 2391.

[4] D. Bigoni and G. Noselli Experimental evidence of flutter and

divergence instabilities induced by dry friction. J. Mech. Phys.

Solids,2011,59,2208.

[5] L. Argani, D. Bigoni, G. Mishuris Dislocations and inclusions in

prestressed metals. Proc. Roy. Soc. A, 2013, 469, 2154 20120752.

[6] D. Bigoni, F. Bosi, F. Dal Corso and D. Misseroni, Instability of a

penetrating blade. J. Mech. Phys. Solids, 2014, in press.

We prove a generalization of Helly's theorem concerning intersections of convex sets that has an interesting voting theory interpretation. We then
consider various extensions in which compelling mathematical problems are motivated from very natural questions in the voting context.

The Ramsey number $R(K_s, Q_n)$ is the smallest integer $N$ such that every red-blue colouring of the edges of the complete graph $K_N$ contains either a red $n$-dimensional hypercube, or a blue clique on $s$ vertices. Note that $N=(s-1)(2^n -1)$ is not large enough, since we may colour in red $(s-1)$ disjoint cliques of cardinality $2^N -1$ and colour the remaining edges blue. In 1983, Burr and Erdos conjectured that this example was the best possible, i.e., that $R(K_s, Q_n) = (s-1)(2^n -1) + 1$ for every positive integer $s$ and sufficiently large $n$. In a recent breakthrough, Conlon, Fox, Lee and Sudakov proved the conjecture up to a multiplicative constant for each $s$. In this talk we shall sketch the proof of the conjecture and discuss some related problems.

(Based on joint work with Gonzalo Fiz Pontiveros, Robert Morris, David Saxton and Jozef Skokan)