In the first part of this work, we consider second order supersymmetric differential operators in the semiclassical limit, including the Kramers-Fokker-Planck operator, such that the exponent of the associated Maxwellian $\phi$ is a Morse function with two local minima and one saddle point. Under suitable additional assumptions of dynamical nature, we establish the long time convergence to the equilibrium for the associated heat semigroup, with the rate given by the first non-vanishing, exponentially small, eigenvalue. In the second part of the paper, we consider the case when the function $\phi$ has precisely one local minimum and one saddle point. We also discuss further examples of supersymmetric operators, including the Witten Laplacian and the infinitesimal generator for the time evolution of a chain of classical anharmonic oscillators.

In this paper, we generalize the parametric Delta-VaR methods from portfolios with elliptic distributed risk factors to portfolios with mixture of elliptically distributed ones. We treat both the Expected Shortfall and the Value-at-Risk of such portfolios. Special attention is given to the particular case of the mixture of Student-t distributions.

In this note, we initiate the systematic study of the Lie algebra structure of the necklace Lie algebra n of a free algebra in 2d variables. We begin by giving a description of n as an sp(2d)-module. Specializing to d = 1, we decompose n into a direct sum of highest weight modules for sl_2, the coefficients of which are given by a closed formula. Next, we observe that n has a nontrivial center, which we link through the center C of the trace ring of couples of generic 2x2 matrices to the Poisson center of S(sl_2). The Lie algebra structure of n induces a Poisson structure on C, the symplectic leaves of which we are able to describe as coadjoint orbits for the Lie group of the semidirect product sl_2\rtimes h of sl_2 with the Heisenberg Lie algebra h. Finally, we provide a link between double Poisson algebras on one hand and Poisson orders on the other hand, showing that all trace rings of a double Poisson algebra are Poisson orders over their center.