Integration with Respect to Hilbert Space‐Valued Semimartingales via Jacod‐Grigelionis Characteristics

Abstract

The main aim of this paper is to describe the space of operator‐valued predictable processes, which are integrable with respect to a Hilbert space valued quasi‐left continuous semimartingale. The space of integrable processes is a randomized Musielak‐Orlicz space with a modular explicitely expressed in terms of Jacod‐Grigelionis characteristics.     

向量值Nevanlinna理论初探

本文主要讨论定义在复平面取值为无穷维希尔伯特空间上的向量值亚纯函数的Ｎｅｖａｎｌｉｎｎａ理论．给出了推广的Ｐｏｉｓｓｏｎ－Ｊｅｎｓｅｎ－Ｎｅｖａｎｌｉｎｎａ公式以及推广的第一基本定理的证明，并对向量值特征函数的性质予以讨论     

Darboux integrating factors: Inverse problems

We discuss planar polynomial vector fields with prescribed Darboux integrating factors, in a nondegenerate affine geometric setting. We establish a reduction principle which transfers the problem to polynomial solutions of certain meromorphic linear systems, and show that the space of vector fields with a given integrating factor, modulo a subspace of explicitly known “standard” vector fields, has finite dimension. For several classes of examples we determine this space explicitly.     

Imprecise set and fuzzy valued probability

Set valued probability and fuzzy valued probability theory is used for analyzing and modeling highly uncertain probability systems. In this paper the set valued probability and fuzzy valued probability are defined over the measurable space. They are derived from a set and fuzzy valued measure using restricted arithmetics. The range of set valued probability is the set of subsets of the unit interval and the range of fuzzy valued probability is the set of fuzzy sets of the unit interval. The expectation with respect to set valued and fuzzy valued probability is defined and some properties are discussed. Also, the fuzzy model is applied to binomial model for the price of a risky security.     

集值上鞅的轨道正则性

本文证明了在有限维Ｂａｎａｃｈ空间中，任一闭凸集值鞅（在ｋ－收敛意义下）存在右连左极修正．     

Well‐posedness of fractional degenerate differential equations with finite delay on vector‐valued functional spaces

We study the well‐posedness of the fractional degenerate differential equations with finite delay on Lebesgue–Bochner spaces , periodic Besov spaces and periodic Triebel–Lizorkin spaces , where A and M are closed linear operators on a Banach space X satisfying , F is a bounded linear operator from (resp. and ) into X, where is given by when and . Using known operator‐valued Fourier multiplier theorems, we give necessary or sufficient conditions for the well‐posedness of in the above three function spaces.     

数值函数关于集值测度的积分的极限定理

本文给出了实值函数关于集值测度的积分序列的极限定理．     

On the theory of Banach space valued multifunctions. 2. Set valued martingales and set valued measures

This second part of the work on Banach space valued multifunctions begins with a detailed study of set valued martingales, which have their values in a Banach space. Several new convergence theorems are established for different modes of convergence. The profile of a multifunction in connection with set valued martingales is also studied. The notion of weak convergence of multifunctions is introduced and used to obtain additional convergence theorems for set valued martingales. In the last two sections of the paper set valued measures dealt with and an integral with respect to a set valued measure is introduced.     

A note on operator‐valued Fourier multipliers on Besov spaces

Let X be a Banach space. We show that each m : ? \ {0} → L (X ) satisfying the Mikhlin condition sup_{x ≠0}(‖m (x )‖ + ‖xm ′(x )‖) < ∞ defines a Fourier multiplier on B ^s _p,q (?; X ) if and only if 1 < p < ∞ and X is isomorphic to a Hilbert space; each bounded measurable function m : ? → L (X ) having a uniformly bounded variation on dyadic intervals defines a Fourier multiplier on B ^s _p,q (?; X ) if and only if 1 < p < ∞ and X is a UMD space. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Well‐posedness of fractional integro‐differential equations in vector‐valued functional spaces

We study the well‐posedness of the fractional differential equations with infinite delay on Lebesgue–Bochner spaces and Besov spaces , where A and B are closed linear operators on a Banach space X satisfying ,  and . Under suitable assumptions on the kernels a and b, we completely characterize the well‐posedness of in the above vector‐valued function spaces on by using known operator‐valued Fourier multiplier theorems. We also give concrete examples where our abstract results may be applied.     

Linearly independent homoclinic bifurcations parameterized by a small function

In this paper we discuss a small nonautonomous perturbation of an autonomous system on Rn which has a homoclinic solution. Regarding the small perturbation as a parameter in an appropriate space of functions we discuss various situations of co-existence of homoclinic orbits. Those conditions of various co-existence actually define bifurcation manifolds in the space of functions for linearly independent homoclinic bifurcations.     

Representations of MV‐algebras by sheaves

In this paper, inspired by methods of Bigard, Keimel, and Wolfenstein ([2]), we develop an approach to sheaf representations of MV‐algebras which combines two techniques for the representation of MV‐algebras devised by Filipoiu and Georgescu ([18]) and by Dubuc and Poveda ([16]). Following Davey approach ([12]), we use a subdirect representation of MV‐algebras that is based on local MV‐algebras. This allowed us to obtain: (a) a representation of any MV‐algebras as MV‐algebra of all global sections of a sheaf of local MV‐algebras on the spectruum of its prime ideals; (b) a representation of MV‐algebras, having the space of minimal prime ideals compact, as MV‐algebra of all global sections of a Hausdorff sheaf of MV‐chains on the space of minimal prime ideals, which is a Stone space; (c) an adjunction between the category of all MV‐algebras and the category of MV‐algebraic spaces, where an MV‐algebraic space is a pair (X, F), where X is a compact topological space and F is a sheaf of MV‐algebras with stalks that are local (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On then-th order iterative ordinary linear differential equations

Summary It is shown that a change of the basis in the solution space of a second order linear differential equation induces a covariant change in the solution space of the corresponding iterative equation. Also studied is the problem to what extent a solution of an iterative equation determines the equation.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.     

Multipoint boundary value problems for nonlinear ordinary differential equations

In this paper we provide sufficient conditions for the existence of solutions to multipoint boundary value problems for nonlinear ordinary differential equations. We consider the case where the solution space of the associated linear homogeneous boundary value problem is less than 2. When this solution space is trivial, we establish existence results via the Schauder Fixed Point Theorem. In the resonance case, we use a projection scheme to provide criteria for the solvability of our nonlinear boundary value problem. We accomplish this by analyzing a link between the behavior of the nonlinearity and the solution set of the associated linear homogeneous boundary value problem.     

Causal functional differential equations in Banach spaces

The aim of this paper is to establish the existence of solutions and some properties of set solutions for a Cauchy problem with causal operator in a separable Banach space.     

Menger概率n-度量空间中的不动点定理

引进了概率ｎ－度量空间,并将文「１」、「２」的某些主要结果拓广至Ｍｅｎｇｅｒ概率ｎ－度量空间。     

概率n-度量空间中的不动点定理

本文引进了概率 n-度量空间 ,并将 [1 ]、[2 ]的某些主要结果拓广至 Menger概率 n-度量空间     

Characterization of set‐valued conditional expectation

For an essentially bounded closed‐convex‐nonempty set‐valued random variable, it is shown that the conditional expectation is characterized by its integrals over the sets of the associated σ ‐field. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A class of vector‐valued dilation‐and‐modulation frames on the half real line

Vector‐valued frames were first introduced under the name of superframes by Balan in the context of signal multiplexing and by Han and Larson from the mathematical aspect. Since then, the wavelet and Gabor frames in have interested many mathematicians. The space models vector‐valued causal signal spaces because of the time variable being nonnegative. But it admits no nontrivial shift‐invariant system and thus no wavelet or Gabor frame since is not a group by addition (not as ). Observing that is a group by multiplication, we, in this paper, introduce a class of multiplication‐based dilation‐and‐modulation ( ) systems, and investigate the theory of frames in . Since is not closed under the Fourier transform, the Fourier transform does not fit . We introduce the notion of Θ_a transform in , and using Θ_a‐transform matrix method, we characterize frames, Riesz bases, and dual frames in and obtain an explicit expression of duals for an arbitrary given frame. An example theorem is also presented.