Random walks on Bratteli diagrams

In the eighties, A. Connes and E. J. Woods made a connection between hyperfinite von Neumann algebras and Poisson boundaries of time dependent random walks. The present paper explains this connection and gives a detailed proof of two theorems quoted there: the construction of a large class of states on a hyperfinite von Neumann algebra (due to A. Connes) and the ergodic decomposition of a Markov measure via harmonic functions (a classical result in probability theory). The crux of the first theorem is a model for conditional expectations on finite dimensional C*-algebras. The proof of the second theorem hinges on the notion of cotransition probability.Comment: 18 pages, written version of a talk given at the Operator Theory 26th Conference, Timisoara 201

AF-equivalence relations and their cocycles

After a review of some of the main results about hyperfinite equivalence relations and their cocycles in the measured setting, we give a definition of a topological AF-equivalence relation. We show that every cocycle is cohomologous to a quasi-product cocycle. We then study the problem of determining the quasi-invariant probability measures admitting a given cocycle as their Radon-Nikodym derivative.Comment: 15 pages, talk at 4th International Conference on Operator Algebras, July 2-7 2001, Constanza, Romani

Relative equilibria with holes for the surface quasi-geostrophic equations

We study the existence of doubly connected rotating patches for the inviscid surface quasi- geostrophic equation left open in \cite{HHH}. By using the approach proposed by \cite{CCGS} we also prove that close to the annulus the boundaries are actually analytic curves

Concentration risk and the optimal number of central counterparties for a single asset

We model the central counterparty (CCP) clearing of a single asset traded over-the-counter by two groups of banks in two currencies. We compare a variety of different clearing set-ups involving one or two CCPs according to their ability to withstand a combined market and banking crisis. Using stress testing, the model shows that the question of the optimal clearing set-up for a specifi c asset is complex and depends on many parameters such as the level of funding available to the CCP(s), the degree of integration between the different groups of participants and the particular risk profiles of these different groups. On the whole, however, a single CCP solution appears less resilient than a two-CCP arrangement when the magnitude of the crisis is large and only more resilient when the magnitude of the crisis is small in relation to the clearing fund of the CCP(s). Another interesting outcome is that the two-CCP set-ups perform better than the single CCP set-up for low levels of participation.