The scaling limit of Poisson-driven order statistics with applications in geometric probability

Let _ηt${\eta }_t$ be a Poisson point process of intensity t≥1$t\ge 1$ on some state space Y$\mathbb{Y}$ and let f$f$ be a non-negative symmetric function on Y^k${\mathbb{Y}}^k$ for some k≥1$k\ge 1$. Applying f$f$ to all k$k$-tuples of distinct points of _ηt${\eta }_t$ generates a point process _ξt${\xi }_t$ on the positive real half-axis. The scaling limit of _ξt${\xi }_t$ as t$t$ tends to infinity is shown to be a Poisson point process with explicitly known intensity measure. From this, a limit theorem for the m$m$-th smallest point of _ξt${\xi }_t$ is concluded. This is strengthened by providing a rate of convergence. The technical background includes Wiener–Itô chaos decompositions and the Malliavin calculus of variations on the Poisson space as well as the Chen–Stein method for Poisson approximation. The general result is accompanied by a number of examples from geometric probability and stochastic geometry, such as k$k$-flats, random polytopes, random geometric graphs and random simplices. They are obtained by combining the general limit theorem with tools from convex and integral geometry.     

Adaptive estimation of an ergodic diffusion process based on sampled data

We consider adaptive maximum likelihood type estimation of both drift and diffusion coefficient parameters for an ergodic diffusion process based on discrete observations. Two kinds of adaptive maximum likelihood type estimators are proposed and asymptotic properties of the adaptive estimators, including convergence of moments, are obtained.     

Markovian quadratic and superquadratic BSDEs with an unbounded terminal condition

This article deals with the existence and the uniqueness of solutions to quadratic and superquadratic Markovian backward stochastic differential equations (BSDEs) with an unbounded terminal condition. Our results are deeply linked with a strong a priori estimate on Z$Z$ that takes advantage of the Markovian framework. This estimate allows us to prove the existence of a viscosity solution to a semilinear parabolic partial differential equation with nonlinearity having quadratic or superquadratic growth in the gradient of the solution. This estimate also allows us to give explicit convergence rates for time approximation of quadratic or superquadratic Markovian BSDEs.     

The exact packing measure of Lévy trees

We study fine properties of Lévy trees that are random compact metric spaces introduced by Le Gall and Le Jan in 1998 as the genealogy of continuous state branching processes. Lévy trees are the scaling limits of Galton-Watson trees and they generalize the Aldous continuum random tree which corresponds to the Brownian case. In this paper, we prove that Lévy trees always have an exact packing measure: we explicitly compute the packing gauge function and we prove that the corresponding packing measure coincides with the mass measure up to a multiplicative constant.     

Convex regularization of local volatility models from option prices: Convergence analysis and rates

We study a convex regularization of the local volatility surface identification problem for the Black-Scholes partial differential equation from prices of European call options. This is a highly nonlinear ill-posed problem which in practice is subject to different noise levels associated to bid-ask spreads and sampling errors. We analyze, in appropriate function spaces, different properties of the parameter-to-solution map that assigns to a given volatility surface the corresponding option prices. Using such properties, we show stability and convergence of the regularized solutions in terms of the Bregman distance with respect to a class of convex regularization functionals when the noise level goes to zero.We improve convergence rates available in the literature for the volatility identification problem. Furthermore, in the present context, we relate convex regularization with the notion of exponential families in Statistics. Finally, we connect convex regularization functionals with convex risk measures through Fenchel conjugation. We do this by showing that if the source condition for the regularization functional is satisfied, then convex risk measures can be constructed.     

Periodic solutions of asymptotically linear autonomous Newtonian systems with resonance

In this paper, we study the existence, continuation and bifurcation from infinity of2π$2\pi$-periodic solutions of autonomous Newtonian systems. We underline that the resonant case is considered. To prove the results, we apply the degree for S¹$S^1$-equivariant gradient maps defined by Rybicki (1994) in [15] and the angle condition introduced by Bartsch and Li (1997) in [16].