Comparing Fr\'echet and positive stable laws

Let ${\bf L}$ be the unit exponential random variable and ${\bf Z}_\alpha$ the standard positive $\alpha$-stable random variable. We prove that $\{(1-\alpha) \alpha^{\gamma_\alpha} {\bf Z}_\alpha^{-\gamma_\alpha}, 0< \alpha <1\}$ is decreasing for the optimal stochastic order and that $\{(1-\alpha){\bf Z}_\alpha^{-\gamma_\alpha}, 0< \alpha < 1\}$ is increasing for the convex order, with $\gamma_\alpha = \alpha/(1-\alpha).$ We also show that $\{\Gamma(1+\alpha) {\bf Z}_\alpha^{-\alpha}, 1/2\le \alpha \le 1\}$ is decreasing for the convex order, that {\bf Z}_\alpha^{-\alpha}\,\prec_{st}\, \Gamma(1-\alpha) \L and that \Gamma(1+\a){\bf Z}_\alpha^{-\alpha} \,\prec_{cx}\,{\bf L}. This allows to compare ${\bf Z}_\alpha$ with the two extremal Fr\'echet distributions corresponding to the behaviour of its density at zero and at infinity. We also discuss the applications of these bounds to the strange behaviour of the median of ${\bf Z}_\alpha$ and ${\bf Z}_\alpha^{-\alpha}$ and to some uniform estimates on the classical Mittag-Leffler function. Along the way, we obtain a canonical factorization of ${\bf Z}_\alpha$ for $\alpha$ rational in terms of Beta random variables. The latter extends to the one-sided branches of real strictly stable densities.Comment: To appear in Electronic Journal of Probabilit

Hitting densities for spectrally positive stable processes

A multiplicative identity in law connecting the hitting times of completely asymmetric $\alpha-$stable L\'evy processes in duality is established. In the spectrally positive case, this identity allows with an elementary argument to compute fractional moments and to get series representations for the density. We also prove that the hitting times are unimodal as soon as $\alpha\le 3/2.$ Analogous results are obtained, in a much simplified manner, for the first passage time across a positive level

Diffusion hitting times and the Bell-shape

Consider a generalized diffusion on R with speed measure m, in the natural scale. It is known that the conditional hitting times have a unimodal density function. We show that these hitting densities are bell-shaped if and only if m has infinitely many points of increase between the starting point and the hit point. This result can be viewed as a visual corollary to Yamazato's general factorization for diffusion hitting times.Comment: To appea

Correcting Newton--C\^{o}tes integrals by L\'{e}vy areas

In this note we introduce the notion of Newton--C\^{o}tes functionals corrected by L\'{e}vy areas, which enables us to consider integrals of the type $\int f(y) \mathrm{d}x,$ where $f$ is a ${\mathscr{C}}^{2m}$ function and $x,y$ are real H\"{o}lderian functions with index $\alpha>1/(2m+1)$ for all $m\in {\mathbb{N}}^*.$ We show that this concept extends the Newton--C\^{o}tes functional introduced in Gradinaru et al., to a larger class of integrands. Then we give a theorem of existence and uniqueness for differential equations driven by $x$, interpreted using the symmetric Russo--Vallois integral.Comment: Published at http://dx.doi.org/10.3150/07-BEJ6015 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm