On explicit free field realization of current algebras

We construct the explicit free field representations of the current algebras so(2n)_k, so(2n+1)_k and sp(2n)_k for a generic positive integer n and an arbitrary level k. The corresponding energy–momentum tensors and screening currents of the first kind are also given in terms of free fields.     

Linearized filtering of affine processes using stochastic Riccati equations

We consider an affine process $X$ which is only observed up to an additive white noise, and we ask for the law of $X_t$, for some $t>0$, conditional on all observations up to time $t$. This is a general, possibly high dimensional filtering problem which is not even locally approximately Gaussian, whence essentially only particle filtering methods remain as solution techniques. In this work we present an efficient numerical solution by introducing an approximate filter for which conditional characteristic functions can be calculated by solving a system of generalized Riccati differential equations depending on the observation and the process characteristics of $X$. The quality of the approximation can be controlled by easily observable quantities in terms of a macro location of the signal in state space. Asymptotic techniques as well as maximization techniques can be directly applied to the solutions of the Riccati equations leading to novel very tractable filtering formulas. The efficiency of the method is illustrated with numerical experiments for Cox–Ingersoll–Ross and Wishart processes, for which Gaussian approximations usually fail.     

The Fontaine–Ogus realization of Laumon 1-motives

Abstract

We construct the (filtered) Ogus realization of Laumon 1-motives over a number field. This realization extends the functor defined on Deligne 1-motives by Andreatta, Barbieri-Viale and Bertapelle.     

Integral quantizations with two basic examples

The paper concerns integral quantization, a procedure based on operator-valued measure and resolution of the identity. We insist on covariance properties in the important case where group representation theory is involved. We also insist on the inherent probabilistic aspects of this classical–quantum map. The approach includes and generalizes coherent state quantization. Two applications based on group representation are carried out. The first one concerns the Weyl–Heisenberg group and the euclidean plane viewed as the corresponding phase space. We show that a world of quantizations exist, which yield the canonical commutation rule and the usual quantum spectrum of the harmonic oscillator. The second one concerns the affine group of the real line and gives rise to an interesting regularization of the dilation origin in the half-plane viewed as the corresponding phase space.