ETH Zurich

We show how to solve optimization problems in the presence of proportional transaction costs by determining a shadow price, which is a solution to the dual problem. Put differently, this is a fictitious frictionless market evolving within the bid-ask spread, that leads to the same optimization problem as in the original market with transaction costs. In addition, we also discuss how to obtain asymptotic expansions of arbitrary order for small transaction costs. This is joint work with Stefan Gerhold, Paolo Guasoni, and Walter Schachermayer.

We present theory and numerics of affine processes and several of their applications in finance. The theory is appealing due to methods from probability theory, analysis and geometry. Applications are diverse since affine processes combine analytical tractability with a high flexibility to model stylized facts like heavy tails or stochastic volatility.

Jointly with Number Theory

Consider a family of abelian varieties whose base is an algebraic variety. The union of all torsion groups over all fibers of the family will be called the set of torsion points of the family. If the base variety is a point then the family is just an abelian variety.

In this case the Manin-Mumford Conjecture, a theorem of Raynaud, implies that a subvariety of the abelian variety contains a Zariski dense set of torsion points if and only if it is itself essentially an abelian subvariety. This talk is on possible extensions to certain families where the base is a curve. Conjectures of André and Pink suggest considering "special points": these are torsion points whose corresponding fibers satisfy an additional arithmetic property. One possible property is for the fiber to have complex multiplication; another is for the fiber to be isogenous to an abelian variety fixed in advance.

We discuss some new results on the distribution of such "special points"

on the subvarieties of certain families of abelian varieties. One important aspect of the proof is the interplay of two height functions.

I will give a brief introduction to the theory of heights in the talk.

In the past decades the $G_{n,p}$ model of random graphs has led to numerous beautiful and deep theorems. A key feature that is used in basically all proofs is that edges in $G_{n,p}$ appear independently.

The independence of the edges allows, for example, to obtain extremely tight bounds on the number of edges of $G_{n,p}$ and its degree sequence by straightforward applications of Chernoff bounds. This situation changes dramatically if one considers graph classes with structural side constraints. In this talk we show how recent progress in the construction of so-called Boltzmann samplers by Duchon, Flajolet, Louchard, and Schaeffer can be used to reduce the study of degree sequences and subgraph counts to properties of sequences of independent and identically distributed random variables -- to which we can then again apply Chernoff bounds to obtain extremely tight results. As proof of concept we study properties of random graphs that are drawn uniformly at random from the class consisting of the dissections of large convex polygons. We obtain very sharp concentration results for the number of vertices of any given degree, and for the number of induced copies of a given fixed graph.