A Fourier transform method for nonparametric estimation of multivariate volatility

We provide a nonparametric method for the computation of instantaneous multivariate volatility for continuous semi-martingales, which is based on Fourier analysis. The co-volatility is reconstructed as a stochastic function of time by establishing a connection between the Fourier transform of the prices process and the Fourier transform of the co-volatility process. A nonparametric estimator is derived given a discrete unevenly spaced and asynchronously sampled observations of the asset price processes. The asymptotic properties of the random estimator are studied: namely, consistency in probability uniformly in time and convergence in law to a mixture of Gaussian distributions.Comment: Published in at http://dx.doi.org/10.1214/08-AOS633 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Cohomologie locale de certains anneaux Auslander-Gorenstein

We give axiornatic conditions in order to calculate the local collomology of some idempotent kernel functors . The results lie on some new dimension introduced by T. Levasseur for Auslander-Gorenstein rings . Under some hypothesis, we generalize previous resul

Unitarizing probability measures for representations of Virasoro algebra

AbstractDetermination of a formula of integration by part insuring the unitarity

Separating tubular series

Ringel CM. Separating tubular series. In: Malliavin M-P, ed. Séminaire d' Algèbre Paul Dubreil et Marie-Paule Malliavin. Proceedings Paris 1982 (35ème Année). Lecture Notes in Mathematics. Vol 1029. Berlin, Heidelberg: Springer; 1983: 134-158

Frame Bundle of Riemannian Path Space and Ricci Tensor in Adapted Differential Geometry

AbstractThe vanishing of the renormalized Ricci tensor of the path space above a Ricci flat Riemannian manifold is discussed

The Dirichlet problem for superdegenerate differential operators

Let $L$ be an infinitely degenerate second-order linear operator defined on a bounded smooth Euclidean domain. Under weaker conditions than those of H\"ormander, we show that the Dirichlet problem associated with $L$ has a unique smooth classical solution. The proof uses the Malliavin calculus. At present, there appears to be no proof of this result using classical analytic techniques

Stochastic differential equations with coefficients in Sobolev spaces

We consider It\^o SDE \d X_t=\sum_{j=1}^m A_j(X_t) \d w_t^j + A_0(X_t) \d t on $\R^d$. The diffusion coefficients $A_1,..., A_m$ are supposed to be in the Sobolev space $W_\text{loc}^{1,p} (\R^d)$ with $p>d$, and to have linear growth; for the drift coefficient $A_0$, we consider two cases: (i) $A_0$ is continuous whose distributional divergence $\delta(A_0)$ w.r.t. the Gaussian measure $\gamma_d$ exists, (ii) $A_0$ has the Sobolev regularity $W_\text{loc}^{1,p'}$ for some $p'>1$. Assume \int_{\R^d} \exp\big[\lambda_0\bigl(|\delta(A_0)| + \sum_{j=1}^m (|\delta(A_j)|^2 +|\nabla A_j|^2)\bigr)\big] \d\gamma_d0, in the case (i), if the pathwise uniqueness of solutions holds, then the push-forward (X_t)_# \gamma_d admits a density with respect to $\gamma_d$. In particular, if the coefficients are bounded Lipschitz continuous, then $X_t$ leaves the Lebesgue measure \Leb_d quasi-invariant. In the case (ii), we develop a method used by G. Crippa and C. De Lellis for ODE and implemented by X. Zhang for SDE, to establish the existence and uniqueness of stochastic flow of maps.Comment: 31 page

How Euler would compute the Euler-Poincar\'e characteristic of a Lie superalgebra

The Euler-Poincar\'e characteristic of a finite-dimensional Lie algebra vanishes. If we want to extend this result to Lie superalgebras, we should deal with infinite sums. We observe that a suitable method of summation, which goes back to Euler, allows to do that, to a certain degree. The mathematics behind it is simple, we just glue the pieces of elementary homological algebra, first-year calculus and pedestrian combinatorics together, and present them in a (hopefully) coherent manner.Comment: v3: minor English correction