The Heston Riemannian distance function

The Heston model is a popular stock price model with stochastic volatility that has found numerous applications in practice. In the present paper, we study the Riemannian distance function associated with the Heston model and obtain explicit formulas for this function using geometrical and analytical methods. Geometrical approach is based on the study of the Heston geodesics, while the analytical approach exploits the links between the Heston distance function and the sub-Riemannian distance function in the Grushin plane. For the Grushin plane, we establish an explicit formula for the Legendre-Fenchel transform of the limiting cumulant generating function and prove a partial large deviation principle that is true only inside a special set

Nonstationary Stochastic Simulation of Strong Ground-Motion Time Histories : Application to the Japanese Database

For earthquake-resistant design, engineering seismologists employ time-history analysis for nonlinear simulations. The nonstationary stochastic method previously developed by Pousse et al. (2006) has been updated. This method has the advantage of being both simple, fast and taking into account the basic concepts of seismology (Brune's source, realistic time envelope function, nonstationarity and ground-motion variability). Time-domain simulations are derived from the signal spectrogram and depend on few ground-motion parameters: Arias intensity, significant relative duration and central frequency. These indicators are obtained from empirical attenuation equations that relate them to the magnitude of the event, the source-receiver distance, and the site conditions. We improve the nonstationary stochastic method by using new functional forms (new surface rock dataset, analysis of both intra-event and inter-event residuals, consideration of the scaling relations and VS30), by assessing the central frequency with S-transform and by better considering the stress drop variability.Comment: 10 pages; 15th World Conference on Earthquake Engineering, Lisbon : Portugal (2012

Bayes and maximum likelihood for L1L^1-Wasserstein deconvolution of Laplace mixtures

We consider the problem of recovering a distribution function on the real line from observations additively contaminated with errors following the standard Laplace distribution. Assuming that the latent distribution is completely unknown leads to a nonparametric deconvolution problem. We begin by studying the rates of convergence relative to the $L^2$-norm and the Hellinger metric for the direct problem of estimating the sampling density, which is a mixture of Laplace densities with a possibly unbounded set of locations: the rate of convergence for the Bayes' density estimator corresponding to a Dirichlet process prior over the space of all mixing distributions on the real line matches, up to a logarithmic factor, with the $n^{-3/8}\log^{1/8}n$ rate for the maximum likelihood estimator. Then, appealing to an inversion inequality translating the $L^2$-norm and the Hellinger distance between general kernel mixtures, with a kernel density having polynomially decaying Fourier transform, into any $L^p$-Wasserstein distance, $p\geq1$, between the corresponding mixing distributions, provided their Laplace transforms are finite in some neighborhood of zero, we derive the rates of convergence in the $L^1$-Wasserstein metric for the Bayes' and maximum likelihood estimators of the mixing distribution. Merging in the $L^1$-Wasserstein distance between Bayes and maximum likelihood follows as a by-product, along with an assessment on the stochastic order of the discrepancy between the two estimation procedures