In this talk we aim at emphasizing the role of information in financial markets (public information versus insider information). In particular, if the information about a particular event (as for instance the default event of a company) is incorporated into a pricing model, then by a change of the underlying filtration, one can compute risk premiums attached to particular events. We also show that modeling of the information leads eventually to modeling of dependencies.

We start with showing how rearrangement inequalities may be used in probabilistic contexts such as e.g. for obtaining bounds on survival probabilities in trapping models. This naturally motivates the need for a new rearrangement inequality which can be interpreted as involving symmetric rearrangements around infinity. After outlining the proof of this inequality we proceed to give some further applications to the volume of Lévy sausages as well as to capacities for Lévy processes.
(Joint work with P. Sousi and R. Sun)

Suppose we take a sample of size n from a population and follow the ancestral lines backwards in time.  Under standard assumptions, this process can be modeled by Kingman's coalescent, in which each pair of lineages merges at rate one.  However, if some individuals have large numbers of offspring or if the population is affected by selection, then many ancestral lineages may merge at one time.  In this talk, we will introduce the family of coalescent processes with multiple mergers and discuss some circumstances under which populations can be modeled by these coalescent processes.  We will also describe how genetic data, such as the number of segregating sites and the site frequency spectrum, would be affected by multiple mergers of ancestral lines, and we will discuss the implications for statistical inference.

 We study the path properties of the random polymer attracted to a defect line by a potential with disorder, and we prove that in the delocalized regime, at any temperature, the number of contacts with the defect line remains in a certain sense ``tight in probability'' as the polymer length varies. On the other hand we show that at sufficiently low temperature, there exists a.s. a subsequence where the number of contacts grows like the log of the length of the polymer.

A two-dimensional discrete Gaussian Free Field (DGFF) is a centered Gaussian process over a finite subset (say, a square) of the square lattice with covariance given by the Green function of the simple random walk killed upon exit from this set. Recently, much effort has gone to the study of the concentration properties and tail estimates for the maximum of DGFF. In my talk I will address the limiting extreme-order statistics of DGFF as the square-size tends to infinity. In particular, I will show that for any sequence of squares along which the centered maximum converges in law, the (centered) extreme process converges in law to a randomly-shifted Gumbel Poisson point process which is decorated, independently around each point, by a random collection of auxiliary points. If there is any time left, I will review what we know and/or believe about the law of the random shift. This talk is based on joint work with Oren Louidor (UCLA).