Estimation for stochastic differential equations with a small diffusion coefficient

We consider a multidimensional diffusion X$X$ with drift coefficient b(_Xt,α)$b(X_t,\alpha )$ and diffusion coefficient εa(_Xt,β)$\epsilon a(X_t,\beta )$ where α$\alpha$ and β$\beta$ are two unknown parameters, while ε$\epsilon$ is known. For a high frequency sample of observations of the diffusion at the time points k/n$k/n$, k=1,…,n$k=1,\dots ,n$, we propose a class of contrast functions and thus obtain estimators of (α,β)$(\alpha ,\beta )$. The estimators are shown to be consistent and asymptotically normal when n→∞$n\to \infty$ and ε→0$\epsilon \to 0$ in such a way that ε⁻¹n^−ρ${\epsilon }^{-1}{n}^{-\rho }$ remains bounded for some ρ>0$\rho >0$. The main focus is on the construction of explicit contrast functions, but it is noted that the theory covers quadratic martingale estimating functions as a special case. In a simulation study we consider the finite sample behaviour and the applicability to a financial model of an estimator obtained from a simple explicit contrast function.     

Spectral problems of a class of non-self-adjoint one-dimensional Schrodinger operators

In this paper we investigate the one-dimensional Schrodinger operator L(q)$L(q)$ with complex-valued periodic potential q   when q∈_L1[0,1]$q\in L_1[0,1]$ and _qn=0$q_n=0$ for n=0,−1,−2,...$n=0,-1,-2,...$, where _qn$q_n$ are the Fourier coefficients of q   with respect to the system {e^i2πnx}$\{e^{i2\pi nx}\}$. We prove that the Bloch eigenvalues are (2πn+t)²${(2\pi n+t)}^2$ for n∈Z$n\in \mathbb{Z}$, t∈C$t\in \mathbb{C}$ and find explicit formulas for the Bloch functions. Then we consider the inverse problem for this operator.     

On the length of an external branch in the Beta-coalescent

In this paper, we consider Beta(2−α,α)$(2-\alpha ,\alpha )$ (with 1<α<2$1<\alpha <2$) and related Λ$\Lambda$-coalescents. If T⁽ⁿ⁾$T^{\left(n\right)}$ denotes the length of a randomly chosen external branch of the n$n$-coalescent, we prove the convergence of n^α−1T⁽ⁿ⁾$n^{\alpha -1}{T}^{\left(n\right)}$ when n$n$ tends to ∞$\infty$, and give the limit. To this aim, we give asymptotics for the number σ⁽ⁿ⁾${\sigma }^{\left(n\right)}$ of collisions which occur in the n$n$-coalescent until the end of the chosen external branch, and for the block counting process associated with the n$n$-coalescent.     

Regularity for the fractional Gelfand problem up to dimension 7

We study the problem (−Δ)^su=λe^u${(-\mathrm{\Delta })}^{s}u=\lambda e^u$ in a bounded domain Ω⊂Rⁿ$\Omega \subset {\mathbb{R}}^n$, where λ   is a positive parameter. More precisely, we study the regularity of the extremal solution to this problem. Our main result yields the boundedness of the extremal solution in dimensions n≤7$n\le 7$ for all s∈(0,1)$s\in (0,1)$ whenever Ω   is, for every i=1,...,n$i=1,...,n$, convex in the _xi$x_i$-direction and symmetric with respect to {_xi=0}$\{x_i=0\}$. The same holds if n=8$n=8$ and s?0.28206...$s?0.28206...$, or if n=9$n=9$ and s?0.63237...$s?0.63237...$. These results are new even in the unit ball Ω=_B1$\Omega =B_1$.     

Sur la répartition des puissances modulo 1

For almost all x>1$x>1$, (xⁿ)$(x^n)$(n=1,2,…)$(n=1,2,\dots )$ is equidistributed modulo 1, a classical result. What can be said on the exceptional set? It has Hausdorff dimension one. Much more: given an (b_n)$(b_n)$ in [0,1[$[0,1[$ and ε>0$\epsilon >0$, the x  -set such that |xⁿ−b_n|<ε$|x^n-b_n|<\epsilon$ modulo 1 for n   large enough has dimension 1. However, its intersection with an interval [1,X]$[1,X]$ has a dimension <1, depending on ε and X. Some results are given and a question is proposed.     

Which finite simple groups are unit groups?

We prove that if G$G$ is a finite simple group which is the unit group of a ring, then G$G$ is isomorphic to: (a) a cyclic group of order 2; or (b) a cyclic group of prime order 2^k−1$2^k-1$ for some k$k$; or (c) a projective special linear group _PSLn(_F2)${PSL}_n\left({\mathbb{F}}_2\right)$ for some n≥3$n\ge 3$. Moreover, these groups do all occur as unit groups. We deduce this classification from a more general result, which holds for groups G$G$ with no non-trivial normal 2-subgroup.     

Concentration and multiplicity of solutions for a fourth-order equation with critical nonlinearity

In this paper, we consider the problem (_Pε)$(P_{\epsilon })$ : Δ²u=u^n+4/n-4+εu,u>0${\mathrm{\Delta }}^{2}u=u^{n+4/n-4}+\epsilon u,u>0$ in Ω,u=Δu=0$\Omega ,u=\mathrm{\Delta }u=0$ on ∂Ω$\mathrm{\partial }\Omega$, where Ω$\Omega$ is a bounded and smooth domain in Rⁿ,n>8${\mathbb{R}}^n,n>8$ and ε>0$\epsilon >0$. We analyze the asymptotic behavior of solutions of (_Pε)$(P_{\epsilon })$ which are minimizing for the Sobolev inequality as ε→0$\epsilon \to 0$ and we prove existence of solutions to (_Pε)$(P_{\epsilon })$ which blow up and concentrate around a critical point of the Robin's function. Finally, we show that for ε$\epsilon$ small, (_Pε)$(P_{\epsilon })$ has at least as many solutions as the Ljusternik–Schnirelman category of Ω$\Omega$.     

Blow-up and propagation of disturbances for fast diffusion equations

This paper is concerned with the Cauchy problem for the fast diffusion equation _ut−Δu^m=αu^_p1$u_t-\Delta u^m=\alpha u^{p_1}$ in R^N${\mathbb{R}}^N$ (N≥1$N\ge 1$), where m∈(0,1)$m\in (0,1)$, _p1>1$p_1>1$ and α>0$\alpha >0$. The initial condition _u0$u_0$ is assumed to be continuous, nonnegative and bounded. Using a technique of subsolutions, we set up sufficient conditions on the initial value _u0$u_0$ so that u(t,x)$u(t,x)$ blows up in finite time, and we show how to get estimates on the profile of u(t,x)$u(t,x)$ for small enough values of t>0$t>0$.