On non-local ergodic Jacobi semigroups: spectral theory, convergence-to-equilibrium and contractivity

In this paper, we introduce and study non-local Jacobi operators, which generalize the classical (local) Jacobi operators. We show that these operators extend to generators of ergodic Markov semigroups with unique invariant probability measures and study their spectral and convergence properties. In particular, we derive a series expansion of the semigroup in terms of explicitly defined polynomials, which generalize the classical Jacobi orthogonal polynomials. In addition, we give a complete characterization of the spectrum of the non-self-adjoint generator and semigroup. We show that the variance decay of the semigroup is hypocoercive with explicit constants, which provides a natural generalization of the spectral gap estimate. After a random warm-up time, the semigroup also decays exponentially in entropy and is both hypercontractive and ultracontractive. Our proofs hinge on the development of commutation identities, known as intertwining relations, between local and non-local Jacobi operators and semigroups, with the local objects serving as reference points for transferring properties from the local to the non-local case

On the use of the l(2)-norm for texture analysis of polarimetric SAR data

In this paper, the use of the l2-norm, or Span, of the scattering vectors is suggested for texture analysis of polarimetric synthetic aperture radar (SAR) data, with the benefits that we need neither an analysis of the polarimetric channels separately nor a filtering of the data to analyze the statistics. Based on the product model, the distribution of the l2-norm is studied. Closed expressions of the probability density functions under the assumptions of several texture distributions are provided. To utilize the statistical properties of the l2-norm, quantities including normalized moments and log-cumulants are derived, along with corresponding estimators and estimation variances. Results on both simulated and real SAR data show that the use of statistics based on the l2-norm brings advantages in several aspects with respect to the normalized intensity moments and matrix variate log-cumulants.Peer ReviewedPostprint (published version

Distribution Functions for Random Variables for Ensembles of positive Hermitian Matrices

Distribution functions for random variables that depend on a parameter are computed asymptotically for ensembles of positive Hermitian matrices. The inverse Fourier transform of the distribution is shown to be a Fredholm determinant of a certain operator that is an analogue of a Wiener-Hopf operator. The asymptotic formula shows that up to the terms of order $o(1)$, the distributions are Gaussian

Perturbation Expansion for Option Pricing with Stochastic Volatility

We fit the volatility fluctuations of the S&P 500 index well by a Chi distribution, and the distribution of log-returns by a corresponding superposition of Gaussian distributions. The Fourier transform of this is, remarkably, of the Tsallis type. An option pricing formula is derived from the same superposition of Black-Scholes expressions. An explicit analytic formula is deduced from a perturbation expansion around a Black-Scholes formula with the mean volatility. The expansion has two parts. The first takes into account the non-Gaussian character of the stock-fluctuations and is organized by powers of the excess kurtosis, the second is contract based, and is organized by the moments of moneyness of the option. With this expansion we show that for the Dow Jones Euro Stoxx 50 option data, a Delta-hedging strategy is close to being optimal.Comment: 33 pages, 13 figures, LaTeX