Multivariate risks and depth-trimmed regions

We describe a general framework for measuring risks, where the risk measure takes values in an abstract cone. It is shown that this approach naturally includes the classical risk measures and set-valued risk measures and yields a natural definition of vector-valued risk measures. Several main constructions of risk measures are described in this abstract axiomatic framework. It is shown that the concept of depth-trimmed (or central) regions from the multivariate statistics is closely related to the definition of risk measures. In particular, the halfspace trimming corresponds to the Value-at-Risk, while the zonoid trimming yields the expected shortfall. In the abstract framework, it is shown how to establish a both-ways correspondence between risk measures and depth-trimmed regions. It is also demonstrated how the lattice structure of the space of risk values influences this relationship.Comment: 26 pages. Substantially revised version with a number of new results adde

Role of Ectonucleotidases in the Synapse Formation During Brain Development: Physiological and Pathological Implications

Extracellular adenine nucleotides and nucleosides, such as ATP and adenosine, are among the most recently identified and least investigated diffusible signaling factors that contribute to the structural and functional remodeling of the brain, both during embryonic and postnatal development. Their levels in the extracellular milieu are tightly controlled by various ectonucleotidases: ectonucleotide pyrophosphatase/phosphodiesterases (E-NPP), alkaline phosphatases (AP), ectonucleoside triphosphate diphosphohydrolases (E-NTPDases) and ecto-5'-nucleotidase (eN). During central nervous system development and in adulthood all ectonucleotidases have diverse expression pattern, cell specific localization and function. Formation, maturation, and refinement of synaptic contacts are influenced by neurotransmitters and neuromodulators, and control of extracellular adenine nucleotide levels by ectonucleotidases are important for understanding the role of purinergic signaling in developing tissues and potential targets in developmental disorders such as autism