Tue, 28 May 2013

13:00 - 14:00
DH 1st floor SR

### Community detection in spatially embedded epidemic networks

Marta Sarzynska
Abstract

We detect communities on time-dependent correlation networks to study the geographical spread of disease. Using data on country-wide dengue fever, rubella, and H1N1 influenza occurrences spanning several years, we create multilayer similarity networks, with the provinces of a country as nodes and the correlations between the time series of case numbers giving weights to the edges.

We perform community detection on these temporal networks of disease outbreaks, looking for groups of provinces in which disease patterns change in similar ways. Optimizing multilayer modularity with a Newman-Girvan null model over a wide parameter range, we observe several partitions that corresponding roughly to relevant historical time points, such as large epidemics and introduction of new disease strains, as well as many strongly spatial partitions.

We develop a novel null model for community detection that takes into account spatial information, thereby allows to uncover additional structure that might otherwise be obscured by spatial proximity. The null model is based on a radiation model that was proposed recently for modelling human mobility, and we believe that it might be better at capturing disease spread than existing spatial null models based on gravity models for interaction between nodes.

The radiation null model performs better than the Newman-Girvan null model and similarly to the gravity model on benchmark spatial networks with distance-dependent links and a known community structure (both static and multislice networks), and it strongly outperforms both on flux-based benchmarks. When applied to the disease networks, the radiation null model uncovers novel, clear temporal partitions, that might shed light on disease patterns, the introduction of new strains, and provide epidemic warning signals.

Fri, 24 May 2013

10:00 - 11:15
DH 1st floor SR

### Flash Sintering

Richard Todd
(Dept. of Materials)
Abstract
“Flash sintering” is a process reported by R Raj and co-workers in which very rapid densification of a ceramic powder compact is achieved by the passage of an electrical current through the specimen. Full density can be achieved in a few seconds (sintering normally takes several hours) and at furnace temperatures several hundred Kelvin below the temperatures required with conventional sintering. The name of the process comes from a runaway power spike that is observed at the point of sintering. Although it is acknowledged by Raj that Joule heating plays a role in the process, he and his co-authors claim that this is of minor importance and that entirely new physical effects must also be involved. However, the existence and possible relevance of these other effects of the electric field/current remains controversial. The aim of this workshop is to introduce the subject and to stimulate discussion of how mathematics could shed light on some the factors that are difficult to measure and understand experimentally.
Fri, 07 Jun 2013

10:00 - 11:00
DH 1st floor SR

### Microelectromechanical Systems, Inverse Eigenvalue Analysis and Nonlinear Lattices

(Department of Engineering Science, University of Oxford)
Abstract
Collective behaviours of coupled linear or nonlinear resonators have been of interest to engineers as well as mathematician for a long time. In this presentation, using the example of coupled resonant nano-sensors (which leads to a Linear pencil with a Jacobian matrix), I will show how previously feared and often avoided coupling between nano-devices along with their weak nonlinear behaviour can be used with inverse eigenvalue analysis to design multiple-input-single-output nano-sensors. We are using these matrices in designing micro/Nano electromechanical systems, particularly resonant sensors capable for measuring very small mass for use as environmental as well as biomedical monitors. With improvement in fabrication technology, we can design and build several such sensors on one substrate. However, this leads to challenges in interfacing them as well as introduces undesired parasitic coupling. More importantly, increased nonlinearity is being observed as these sensors reduce in size. However, this also presents an opportunity to experimentally study chains or matrices of coupled linear and/or nonlinear structures to develop new sensing modalities as well as to experimentally verify theoretically or numerically predicted results. The challenge for us is now to identify sensing modalities with chain of linear or nonlinear resonators coupled either linearly or nonlinearly. We are currently exploring chains of Duffing resonators, van der Pol oscillators as well as FPU type lattices.
Fri, 31 May 2013

10:00 - 11:15
DH 1st floor SR

### Understanding Composite Hydrophones' Sensitivity at Low Frequency

Mike Clifton
(Thales UK (Underwater Systems))
Abstract

In order to reduce cost, the MOD are attempting to reduce the number of array types fitted to their assets. There is also a requirement for the arrays to increase their frequency coverage. A wide bandwidth capability is thus needed from a single array. The need for high sensitivity and comparatively high frequencies of operation has led to the view that 1 3 composites are suitable hydrophones for this purpose. These hydrophones are used widely in ultra-sonics, but are not generally used down to the frequency of the new arrays.

Experimental work using a single hydrophone (small in terms of wavelengths) has shown that the sensitivity drops significantly as the frequency approaches the bottom of the required band, and then recovers as the frequency reduces further. Complex computer modelling appears to suggest the loss in sensitivity is due to a "lateral mode" where the hydrophone "breathes" in and out. In order to engineer a solution, the mechanics of the cause of this problem and the associated parameters of the materials need to be identified (e.g. is changing the 1 3 filler material the best option?). In order to achieve this understanding, a mathematical model of the 1 3 composite hydrophone (ceramic pegs and filler) is required that can be used to explain why the hydrophone changes from the simple compression and expansion in the direction of travel of the wave front to a lateral "breathing" mode.

More details available from @email

Fri, 14 Jun 2013

16:00 - 17:00
DH 1st floor SR

### Weak solutions of the Kolmogorov backward equations for option pricing in Lévy models

Kathrin Glau
(Technical University Munich)
Abstract
Advanced models such as Lévy models require advanced numerical methods for developing efficient pricing algorithms. Here we focus on PIDE based methods. There is a large arsenal of numerical methods for solving parabolic equations that arise in this context. Especially Galerkin and Galerkin inspired methods have an impressive potential. In order to apply these methods, what is required is a formulation of the equation in the weak sense.

We therefore classify Lévy processes according to the solution spaces of the associated parabolic PIDEs. We define the Sobolev index of a Lévy process by a certain growth condition on the symbol. It follows that for Lévy processes with a certain Sobolev index b the corresponding evolution problem has a unique weak solution in the Sobolev-Slobodeckii space with index b/2. We show that this classification applies to a wide range of processes. Examples are the Brownian motion with or without drift, generalised hyperbolic (GH), CGMY and (semi) stable Lévy processes.

A comparison of the Sobolev index with the Blumenthal-Getoor index sheds light on the structural implication of the classification. More precisely, we discuss the Sobolev index as an indicator of the smoothness of the distribution and of the variation of the paths of the process.

An application to financial models requires in particular to admit pure jump processes as well as unbounded domains of the equation. In order to deal at the same time with the typical payoffs which can arise, the weak formulation of the equation has to be based on exponentially weighted Sobolev-Slobodeckii spaces. We provide a number of examples of models that are covered by this general framework. Examples of options for which such an analysis is required are calls, puts, digital and power options as well as basket options.

The talk is based on joint work with Ernst Eberlein.

Fri, 07 Jun 2013

16:00 - 17:00
DH 1st floor SR

### Martingale Optimal Transport and Robust Hedging

Nizar Touzi
(Ecole Polytechnique (ParisTech))
Abstract
The martingale optimal transportation problem is motivated by

model-independent bounds for the pricing and hedging exotic options in

financial mathematics.

In the simplest one-period model, the dual formulation of the robust

superhedging cost differs from the standard optimal transport problem by

the presence of a martingale constraint on the set of coupling measures.

The one-dimensional Brenier theorem has a natural extension. However, in

the present martingale version, the optimal coupling measure is

concentrated on a pair of graphs which can be obtained in explicit form.

These explicit extremal probability measures are also characterized as

the unique left and right monotone martingale transference plans, and

induce an optimal solution of the kantorovitch dual, which coincides

with our original robust hedging problem.

By iterating the above construction over n steps, we define a Markov

process whose distribution is optimal for the n-periods martingale

transport problem corresponding to a convenient class of cost functions.

Similarly, the optimal solution of the corresponding robust hedging

problem is deduced in explicit form. Finally, by sending the time step

to zero, this leads to a continuous-time version of the one-dimensional

Brenier theorem in the present martingale context, thus providing a new

remarkable example of Peacock, i.e. Processus Croissant pour l'Ordre

Convexe. Here again, the corresponding robust hedging strategy is

obtained in explicit form.

Fri, 31 May 2013

16:00 - 17:00
DH 1st floor SR

### CANCELLED

Ioannis Karatzas
(Columbia)
Abstract
In an equity market with stable capital distribution, a capitalization-weighted index of small stocks tends to outperform a capitalization-weighted index of large stocks.} This is a somewhat careful statement of the so-called "size effect", which has been documented empirically and for which several explanations have been advanced over the years. We review the analysis of this phenomenon by Fernholz (2001) who showed that, in the presence of (a suitably defined) stability for the capital structure, this phenomenon can be attributed entirely to portfolio rebalancing effects, and will occur regardless of whether or not small stocks are riskier than their larger brethren. Collision local times play a critical role in this analysis, as they capture the turnover at the various ranks on the capitalization ladder.

We shall provide a rather complete study of this phenomenon in the context of a simple model with stable capital distribution, the so-called Atlas model" studied in Banner et al.(2005).

This is a Joint work with Adrian Banner, Robert Fernholz, Vasileios Papathanakos and Phillip Whitman.

Fri, 24 May 2013

16:00 - 17:00
DH 1st floor SR

### Markov Modulated Weak Stochastic Maximum Principle

Harry Zheng
(London)
Abstract
In this paper we prove a weak necessary and sufficient maximum principle for Markov regime switching stochastic optimal control problems. Instead of insisting on the maximum condition of the Hamiltonian, we show that 0 belongs to the sum of Clarke's generalized gradient of the Hamiltonian and Clarke's normal cone of the control constraint set at the optimal control. Under a joint concavity condition on the Hamiltonian and a convexity condition on the terminal objective function, the necessary condition becomes sufficient. We give four examples to demonstrate the weak stochastic maximum principle.
Fri, 17 May 2013

16:00 - 17:00
DH 1st floor SR

### Superhedging under Model Uncertainty

Michael Kupper
(Institut fut Mathematik (Humboldt))
Abstract
We discuss the superhedging problem under model uncertainty based on existence

and duality results for minimal supersolutions of backward stochastic differential equations.

The talk is based on joint works with Samuel Drapeau, Gregor Heyne and Reinhard Schmidt.

Fri, 10 May 2013

16:00 - 17:00
DH 1st floor SR

### Option pricing, fake Brownian motion, and minimal variation

David Hobson
(Warwick)
Abstract
Suppose we are given a double continuum (in time and strike) of discounted

option prices, or equivalently a set of measures which is increasing in

convex order. Given sufficient regularity, Dupire showed how to construct

a time-inhomogeneous martingale diffusion which is consistent with those

prices. But are there other martingales with the same 1-marginals? (In the

case of Gaussian marginals this is the fake Brownian motion problem.)

In this talk we show that the answer to the question above is yes.

Amongst the class of martingales with a given set of marginals we

construct the process with smallest possible expected total variation.

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