Characterizations of Ellipsoid and Sphere

The purpose of this paper is to characterize the ellipsoids in the unimodularaffine space of dimension 3 by the affine principal curvatures and the spheres inthe space of constant curvature of dimension 3 by the principal curvatures. Let A~3 be the unimodular affine space of dimension 3,x:M→A~3 be a closed,locally strongly convex C~5 surface.Denote the equiaffine principal curvatures of Mby λ_1,λ_2 and let     

Cumulant operators and moments of the Itô and Skorohod integrals

We propose a formula for the computation of the moments of all orders of Itô and Skorohod stochastic integrals with respect to Brownian motion, based on cumulant operators defined by the Malliavin calculus. Some characterizations of Gaussian distributions for stochastic integrals are recovered as a consequence.     

Excluded–Minor Characterizations of Antimatroids arisen from Posets and Graph Searches

An antimatroid is a family of sets such that it contains an empty set, and it is accessible and closed under union of sets. An antimatroid is a ’dual’ or ‘antipodal’ concept of matroid.We shall show that an antimatroid is derived from shelling of a poset if and only if it does not contain a minor isomorphic to S₇ where S₇ is the smallest semimodular lattice that is not modular (See Fig. 1). It is also shown that an antimatroid is a node-search antimatroid of a digraph if and only if it does not contain a minor isomorphic to D₅ where D₅ is a lattice consisting of five elements Ø {x},{y}, {x, y} and {x, y, z}. Furthermore, an antimatroid is shown to be a node-search antimatroid of an undirected graph if and only if it does not contain D₅ nor S₁₀ as a minor: S₁₀ is a locally free lattice consisting of ten elements shown in Fig. 2.     

The Classical Average-Cost Inventory Models of Iglehart and Veinott–Wagner Revisited

This paper revisits the classical papers of Iglehart (Ref. 1) and Veinott and Wagner (Ref. 2) devoted to stochastic inventory problems with the criterion of long-run average cost minimization. We indicate some of the assumptions that are used implicitly without verification in their stationary distribution approach to the problems and provide the missing (nontrivial) verification. In addition to completing their analysis, we examine the relationship between the stationary distribution approach and the dynamic programming approach to the average-cost stochastic inventory problems.     

Characterizations and Extensions of Lipschitz-α Operators

In this work, we prove that a map F from a compact metric space K into a Banach space X over F is a Lipschitz-α operator if and only if for each σ in X^＊ the map σoF is a Lipschitz-α function on K. In the case that K = [a, b], we show that a map f from [a, b] into X is a Lipschitz-1 operator if and only if it is absolutely continuous and the map σ→ （σ o f）＇ is a bounded linear operator from X^＊ into L^∞（[a, b]）. When K is a compact subset of a finite interval （a, b） and 0 〈 α ≤ 1, we show that every Lipschitz-α operator f from K into X can be extended as a Lipschitz-α operator F from [a, b] into X with Lα（f） ≤ Lα（F） ≤ 3^1-α Lα（f）. A similar extension theorem for a little Lipschitz-α operator is also obtained.     

Operator Characterizations,Rigidity and Constructions of (Ω,μ)-Frames

In this paper, first, we give some operator characterizations of (Ω,μ)-frames. We obtain that normalized tight (Ω,μ)-frames are precisely the (Ω,μ)-frames which are unitary equivalent to normalized tight (Ω,μ)-frames for some closed subspace ? of L²(Ω,μ) and (Ω,μ)-frames are precisely the (Ω,μ)-frames which are similar to normalized tight (Ω,μ)-frames for some closed subspace ? of L²(Ω,μ). We also characterize the alternate dual (Ω,μ)-frames through an operator equation. Then we establish some rigidity in the pairs of dual (super) (Ω,μ)-frames related with disjointness. Finally, we consider the constructions of (Ω,μ)-frames, including the constructions of new (Ω,μ)-frames or new pair of dual (Ω,μ)-frames from known ones and the constructions of the canonical dual of a (Ω,μ)-frame under certain conditions, which generalize the corresponding results on discrete frames.     

Characterizations of Bochner–Krall Orthogonal Polynomials of Jacobi Type

Let $\{P_n(x) \}_{n=0}^\infty$ be an orthogonal polynomial system relative to a compactly supported measure. We find characterizations for $\{P_n(x) \}_{n=0}^\infty$ to be a Bochner--Krall orthogonal polynomial system, that is, $\{P_n(x) \}_{n=0}^\infty$ are polynomial eigenfunctions of a linear differential operator of finite order. In particular, we show that $\{P_n(x) \}_{n=0}^\infty$ must be generalized Jacobi polynomials which are orthogonal relative to a Jacobi weight plus two point masses.     

A Computational Approach for Fluid Queues Driven by Truncated Birth–Death Processes

In this paper, we analyze fluid queues driven by truncated birth-death processes with general birth and death rates. We compute the equilibrium distribution of the content of the fluid buffer by providing efficient numerical procedures to compute the eigenvalues and the eigenvectors of the associated real sign-asymmetric tridiagonal matrix. We illustrate the effectiveness of the procedures through tables and graphs.     

Moment identities for Poisson–Skorohod integrals and application to measure invariance

We present a moment identity on the Poisson space that extends the Skorohod isometry to arbitrary powers of the Skorohod integral. Applications of this identity are given to the invariance of Poisson measures under intensity preserving random transformations. To cite this article: N. Privault, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Characterizations of K-smooth Spaces and K-strongly Smooth Spaces

Some necessary and sufficient conditions for a Banach space to be K-smooth or Kstrongly smooth are obtained.     

Fluctuations of Stable Processes and Exponential Functionals of Hypergeometric Lévy Processes

We study the distribution and various properties of exponential functionals of hypergeometric Lévy processes. We derive an explicit formula for the Mellin transform of the exponential functional and give both convergent and asymptotic series expansions of its probability density function. As applications we present a new proof of some of the results on the density of the supremum of a stable process, which were recently obtained in Hubalek and Kuznetsov (Electron. Commun. Probab. 16:84–95, 2011) and Kuznetsov (Ann. Probab. 39(3):1027–1060, 2011). We also derive several new results related to (i) the entrance law of a stable process conditioned to stay positive, (ii) the entrance law of the excursion measure of a stable process reflected at its past infimum, (iii) the distribution of the lifetime of a stable process conditioned to hit zero continuously and (iv) the entrance law and the last passage time of the radial part of a multidimensional symmetric stable process.     

Method of Moments Estimators and Multi–step MLE for Poisson Processes

We introduce two types of estimators of the finite–dimensional parameters in the case of observations of inhomogeneous Poisson processes. These are the estimators of the method of moments and Multi–step MLE. It is shown that the estimators of the method of moments are consistent and asymptotically normal and the Multi–step MLE are consistent and asymptotically efficient. The construction of Multi–step MLE–process is done in two steps. First we construct a consistent estimator by the observations on some learning interval and then this estimator is used for construction of One–step and Two–step MLEs. The main advantage of the proposed approach is its computational simplicity.     

Local Characterizations of Left Noetherian Rings and Left Ideals

Let R be a ring with identity, M a maximal ideal of R.Y. Lequain proved the following results in [1]: (a) R is a Noetherian ring every ideal contained in M is finitely generated. (b) R is a Dedekind domain every nonzero ideal contained in M isinvertible.     

Surjectivity of Generaized Expanding Maps and Some Characterizations of Monotone Operators

In [1] L. Nirenberg stated the following problem: Suppose T is a continuous map H→H (H is a Hilbert space), which is expanding,i.e. and T(0)=0.Suppose T map a neighborhood of the origin onto a neighborhood of the origin. Does T maps H onto H? In [2] Kung-Ching Chang and Li Shujie answered positively the problem when T is differentiable.     

The Class of Distributions of Periodic Ornstein–Uhlenbeck Processes Driven by Lévy Processes

The class I(c) of stationary distributions of periodic Ornstein–Uhlenbeck processes with parameter c driven by Lévy processes is analyzed. A characterization of I(c) analogous to a well-known characterization of the selfdecomposable distributions is given. The relations between I(c) for varying values of c and the relations with the class of selfdecomposable distributions and with the nested classes L_m are discussed.     

Recurrency and Transiency of Markov Processes Governed by Coupled Systems of Second–Order Elliptic Operators

Markov processes corresponding to coupled systems of the form k=1,2,...,m, are considered. Here L_k are second–order linear elliptic operators and d_k,j are non–negative functions. We prove sufficient conditions for the recurrency and transiency of the Markov processes corresponding to such systems and alos provide some examples which show that it is possible to have a recurrent system even if some or all its components are transient; where by components, we mean the Markov processes engendered by the uncoupled equations. We also provide all example which shows that a system composed of recurrent processes may itself be transient. Finally, we consider necessary and sufficient conditions for the exterior Dirichlet problem of the coupled system to have a bounded solution as.     

The Entrance Laws of Self-Similar Markov Processes and Exponential Functionals of Lévy Processes

We consider the asymptotic behavior of semi-stable Markov processes valued in ]0,[ when the starting point tends to 0. The entrance distribution is expressed in terms of the exponential functional of the underlying Lévy process which appears in Lamperti's representation of a semi-stable Markov process.     

The Distribution of Discounted Compound PH–Renewal Processes

We continue and extend the work in Léveillé et al. (2010) Scand Actuar J 3:165–184 that gives analytical formulas for the moment generating function (mgf) of some discounted compound renewal processes. Here these mgf’s are used to derive and study the distribution of discounted compound Phase-type (PH) renewal sums. The approach consists in first deriving a differential equation, in the time variable, for the mgf of discounted compound sums when the inter–arrival times are PH–distributed. Then the corresponding distribution of the discounted compound PH sum is obtained by inversion of its mgf (or the equivalent Laplace transform). Analytical expressions for the asymptotic distribution of these discounted compound renewal sums are also given, to test general expressions in some limiting cases. Despite the technical difficulty, several new examples are provided where the inversion is possible, through symbolic computation in MAPLE. When the inversion is too complex, a truncated series solution method is proposed, which is not new but turns out to be very fast and accurate here, much more than other methods (such as rational approximations). The article concludes with some applications where the mgf and the distribution are used to calculate risk measures such as VaR, CTE, Esscher’s, Wang’s Proportional Hazard transforms and evaluate their evolution in time.