L4

Multidimensional single molecule microscopy (MSMM) generates image time series of biomolecules in a cellular environment that have been tagged with fluorescent labels. Initial analysis steps of such images consist of image registration of multiple channels, feature detection and single particle tracking. Further analysis may involve the estimation of diffusion rates, the measurement of separations between molecules that are not optically resolved and more. The analysis is done under the condition of poor signal to noise ratios, high density of features and other adverse conditions. Pushing the boundary of what is measurable, we are facing among others the following challenges. Firstly the correct assessment of the uncertainties and the significance of the results, secondly the fast and reliable identification of those features and tracks that fulfil the assumptions of the models used. Simpler models require more rigid preconditions and therefore limiting the usable data, complexer models are theoretically and especially computationally challenging.

 About 10 years ago Minahan and Zarembo made a remarkable discovery: the one-loop Dilatation Operator in the SO(6) sector of planar N=4 SYM can be identified with the Hamiltonian of an integrable spin chain. This one-loop Dilatation operator was obtained by computing a two-point correlation function at one loop, which is a completely off-shell quantity. Around the same time, Witten proposed a duality between N=4 SYM and twistor string theory, which initiated a revolution in the field of on-shell objects like scattering amplitudes. In this talk we illustrate that these techniques that have been sucessfully used for on-shell quantities can also be employed for the computation of off-shell quantities by computing the one-loop Dilatation Operator in the SO(6) sector. The first half of the talk will be dedicated to doing this calculation using MHV diagrams and the second half of the talk shows the computation in twistor space. 

These two short talks will be followed by an informal afternoon session for those interested in further details of these approaches, and in form factors in Class Room C2 from 2-4.30 pm then from 4.30pm in N3.12.  All are welcome.

The cube attack of Dinur and Shamir and the AIDA attack of Vielhaber have been used successfully on 

reduced round versions of the Trivium stream cipher and a few other ciphers. 

These attacks can be viewed in the framework of higher order differentiation, as introduced by Lai in 

the cryptographic context. We generalise these attacks from the binary case to general finite fields, 

showing that we would need to differentiate several times with respect to each variable in order to have

a reasonable chance of a successful attack.

We also investigate the notion of “fast points” for a binary polynomial function f  

(i.e. vectors such that the derivative of f with respect to this vector has a lower 

than expected degree). These were  introduced by Duan and Lai, motivated by the fact that higher order 

differential attacks are usually more efficient if they use such points. The number of functions which 

admit fast points were computed by Duan et al in a few particular cases; we give explicit formulae for 

all remaining cases and discuss the cryptographic significance of these results.

This is joint work with Richard A. Davis (Columbia Statistics) and Johannes Heiny (Copenhagen). In recent years the sample covariance matrix of high-dimensional vectors with iid entries has attracted a lot of attention. A deep theory exists if the entries of the vectors are iid light-tailed; the Tracy-Widom distribution typically appears as weak limit of the largest eigenvalue of the sample covariance matrix. In the heavy-tailed case (assuming infinite 4th moments) the situation changes dramatically. Work by Soshnikov, Auffinger, Ben Arous and Peche shows that the largest eigenvalues are approximated by the points of a suitable nonhomogeneous Poisson process. We follows this line of research. First, we consider a p-dimensional time series with iid heavy-tailed entries where p is any power of the sample size n. The point process of the scaled eigenvalues of the sample covariance matrix converges weakly to a Poisson process. Next, we consider p-dimensional heavy-tailed time series with dependence through time and across the rows. In particular, we consider entries with a linear dependence or a stochastic volatility structure. In this case, the limiting point process is typically a Poisson cluster process. We discuss the suitability of the aforementioned models for large portfolios of return series. 

We describe the possible influence of stochastic 
dependence on the evaluation of
the risk of joint portfolios and establish relevant risk bounds.Some 
basic tools for this purpose are  the distributional transform,the 
rearrangement method and extensions of the classical Hoeffding -Frechet 
bounds based on duality theory.On the other hand these tools find also 
essential applications to various problems of optimal investments,to the 
construction of cost-efficient payoffs as well as to various optimal 
hedging problems.We
discuss in detail the case of optimal payoffs in Levy market models as 
well as utility optimal payoffs and hedgings
with state dependent utilities.

The problem of optimal stopping with finite horizon in discrete time
is considered in view of maximizing the expected gain. The algorithm
presented in this talk is completely nonparametric in the sense that it
uses observed data from the past of the process up to time -n+1 (n being
a natural number), not relying on any specific model assumption. Kernel
regression estimation of conditional expectations and prediction theory
of individual sequences are used as tools.
The main result is that the algorithm is universally consistent: the
achieved expected gain converges to the optimal value for n tending to
infinity, whenever the underlying process is stationary and ergodic.
An application to exercising American options is given.

In this talk, we present a pathwise method to construct confidence 
intervals on the value of some discrete time stochastic dynamic 
programming equations, which arise, e.g., in nonlinear option pricing 
problems such as credit value adjustment and pricing under model 
uncertainty. Our method generalizes the primal-dual approach, which is 
popular and well-studied for Bermudan option pricing problems. In a 
nutshell, the idea is to derive a maximization problem and a 
minimization problem such that the value processes of both problems 
coincide with the solution of the dynamic program and such that 
optimizers can be represented in terms of the solution of the dynamic 
program. Applying an approximate solution to the dynamic program, which 
can be precomputed by any algorithm, then leads to `close-to-optimal' 
controls for these optimization problems and to `tight' lower and upper 
bounds for the value of the dynamic program, provided that the algorithm 
for constructing the approximate solution was `successful'. We 
illustrate the method numerically in the context of credit value 
adjustment and pricing under uncertain volatility.
The talk is based on joint work with C. Gärtner, N. Schweizer, and J. 
Zhuo.

With few exceptions, optimal stopping assumes that the underlying system is stopped immediately after the decision is made. 
In fact, most stoppings take time. This has been variously referred to as "time-to-build", "investment lag" and "gestation period", 
which is often non negligible. 
In this talk, we consider a class of optimal stopping/switching problems with delivery lags, or equivalently, delayed information, 
by using reflected BSDE method. As an example, we study American put option with delayed exercise, and show that it can be decomposed 
as a European put option and a premium, the latter of which involves a new optimal stopping problem where the investor decides when to stop
to collect the Greek theta of such a European option. We also give a complete characterization of the optimal exercise boundary by resorting to free boundary analysis.  

Joint work with Zhou Yang and Mihail Zervos. 

The large majority of risk-sharing transactions involve few agents, each of whom can heavily influence the structure and the prices of securities. This paper proposes a game where agents' strategic sets consist of all possible sharing securities and pricing kernels that are consistent with Arrow-Debreu sharing rules. First, it is shown that agents' best response problems have unique solutions, even when the underlying probability space is infinite. The risk-sharing Nash equilibrium admits a finite-dimensional characterisation and it is proved to exist for general number of agents and be unique in the two-agent game. In equilibrium, agents choose to declare beliefs on future random outcomes different from their actual probability assessments, and the risk-sharing securities are endogenously bounded, implying (amongst other things) loss of efficiency. In addition, an analysis regarding extremely risk tolerant agents indicates that they profit more from the Nash risk-sharing equilibrium as compared to the Arrow-Debreu one.
(Joint work with Michail Anthropelos)

Motivated by the European sovereign debt crisis, we propose a hybrid sovereign default model which combines an accessible part which takes into account the movement of the sovereign solvency and the impact of critical political events, and a totally inaccessible part for the idiosyncratic credit risk. We obtain closed-form formulas for the probability that the default occurs at political critical dates in a Markovian CEV process setting. Moreover, we introduce a generalized density framework for the hybrid default times and deduce the compensator process of default. Finally we apply the hybrid model and the generalized density to the valuation of sovereign bond and explain the significant jumps in the long-term government bond yield during the sovereign crisis.