Wiener Integrals with Respect to the Hermite Process and a Non-Central Limit Theorem

Abstract

We introduce Wiener integrals with respect to the Hermite process and we prove a non-central limit theorem in which this integral appears as limit. As an example, we study a generalization of the fractional Ornstein–Uhlenbeck process.     

ANTICIPATING QUADRANT AND SYMMETRIC INTEGRALS IN THE PLANE WITH APPLICATION TO WIENER SPACE

1IntroductionItiswellknowntattheadaptednessconditionontheintegrandisnecessarytoestablishthetheoryofIt6'sstochasticintegrals.Butinmanysitustions,theprocessesmaynotsatisfytheadaptedness.TherehavebeenmanyattemptstoweakentheadaptednessconditionfortheintegrandofIt6'stochasticintegral.ThemostsignificantcontributionisSkorohod'sintegralwhichwasintroducedbySkorohodin1975.Skorohodintegralgeneralizesboththeits'sforwardandbackwardintegralsandmakesnorestrictionontheintegrand.ButSkorohod'sintegralhasalso…     

Stochastic integral with respect to set-valued square integrable martingales

In this paper, we shall firstly illustrate why we should consider integral of a stochastic process with respect to a set-valued square integrable martingale. Secondly, we shall prove the representation theorem of set-valued square integrable martingale. Thirdly, we shall give the definition of stochastic integral of a stochastic process with respect to a set-valued square integrable martingale and the representation theorem of this kind of integrals. Finally, we shall prove that the stochastic integral is a set-valued sub-martingale.     

函数粗集生成的粗积分(英文)

The concepts of the lower approximation integral,the upper approximation integral and rough integrals are given on the basis of function rough sets.Based on these concepts,the relation of the lower approximation integrals,the relation of the upper approximation integrals,the relation of rough integrals,and the double median theorem of rough integrals are discussed.Rough integrals have finite contraction characteristic and finite extension characteristic.     

Henstock-Kurzweil and McShane product integration; Descriptive definitions

The Henstock-Kurzweil and McShane product integrals generalize the notion of the Riemann product integral. We study properties of the corresponding indefinite integrals (i.e. product integrals considered as functions of the upper bound of integration). It is shown that the indefinite McShane product integral of a matrix-valued function A is absolutely continuous. As a consequence we obtain that the McShane product integral of A over [a, b] exists and is invertible if and only if A is Bochner integrable on [a, b]. Supported by grant No. 201/04/0690 of the Grant Agency of the Czech Republic.     

ON A HYPER HILBERT TRANSFORM＊＊＊＊

The authors define the directional hyper Hilbert transform and give ita mixed norm estimate. The similar conclusions for the directional fractional integral of one dimension are also obtained in this paper. As an application of the above results, the authors give the Lp-boundedness for a class of the hyper singular integrals and the fractional integrals with variable kernel. Moreover, as another application of the above results, the authors prove the dimension free estimate for the hyper Riesz transform. This is an extension of the related result obtained by Stein.     

分数Brown 运动驱动的非 Lipschitz随机微分方程

讨论了一类带分数Brown 运动的非Lipschitz 增长的随机微分方程适应解的存在唯一性。关于分数 Brown 运动的随机积分有多种定义,本文使用一种广义 Stieltjes积分定义方法,利用这种积分的性质,建立了一类由标准 Brown 运动和一个 Hurst 指数H ∈(1/2,1)的分数Brown 运动共同驱动的、系数为非Lipschitz 增长的随机微分方程适应解的存在唯一性定理。     

Integral Kernel Operators on Fock Space – Generalizations and Applications to Quantum Dynamics

A general theory of operators on Boson Fock space is discussed in terms of the white noise distribution theory on Gaussian space (white noise calculus). An integral kernel operator is generalized from two aspects: (i) The use of an operator-valued distribution as an integral kernel leads us to the Fubini type theorem which allows an iterated integration in an integral kernel operator. As an application a white noise approach to quantum stochastic integrals is discussed and a quantum Hitsuda–Skorokhod integral is introduced. (ii) The use of pointwise derivatives of annihilation and creation operators assures the partial integration in an integral kernel operator. In particular, the particle flux density becomes a distribution with values in continuous operators on white noise functions and yields a representation of a Lie algebra of vector fields by means of such operators.     

Multiple Integral Inequalities for Schur Convex Functions on Symmetric and Convex Bodies

In this paper, by making use of Divergence theorem for multiple integrals, we establish some integral inequalities for Schur convex functions defined on bodies $B⊂\mathbb{R}^n$ that are symmetric, convex and have nonempty interiors. Examples for three dimensional balls are also provided.     

Evaluations of some classes of the trigonometric moment integrals

Four classes of the trigonometric moment integrals are evaluated in closed form in a simple and unified manner by making use of the contour integration in conjunction with the Cauchy integral theorem. In all cases, the closed contour of the same shape is used and it is shown that the integrals are expressible only in terms of the Hurwitz zeta function and elementary functions. A number of interesting (known or new) special cases and consequences of the main results are also considered.     

Some inequalities and convergence theorems for Choquet integrals

Fuzzy measure (or non-additive measure), which has been comprehensively investigated, is a generalization of additive probability measure. Several important kinds of non-additive integrals have been built on it. Integral inequalities play important roles in classical probability and measure theory. In this paper, we discuss some of these inequalities for one kind of non-additive integrals—Choquet integral, including Markov type inequality, Jensen type inequality, Hölder type inequality and Minkowski type inequality. As applications of these inequalities, we also present several convergence concepts and convergence theorems as complements to Choquet integral theory.     

Linear and bilinear estimates for oscillatory integral operators related to restriction to hypersurfaces

We obtain bilinear estimates for oscillatory integral operators which are variable coefficient generalizations of bilinear restriction estimates for hypersurfaces. As applications, we improve the known estimates for oscillatory integrals.     

Multiplication formulas for Kubert functions

The aim of this paper is two folds. First, we shall prove a general reduction theorem to the Spannenintegral of products of （generalized） Kubert functions. Second, we apply the special case of Carlitz＇s theorem to the elaboration of earlier results on the mean values of the product of Dirichlet L-functions at integer arguments. Carlitz＇s theorem is a generalization of a classical result of Nielsen in 1923. Regarding the reduction theorem, we shall unify both the results of Carlitz （for sums） and Mordell （for integrals）, both of which are generalizations of preceding results by Frasnel, Landau, Mikolas, and Romanoff et al. These not only generalize earlier results but also cover some recent results. For example, Beck＇s lamma is the same as Carlitz＇s result, while some results of Maier may be deduced from those of Romanoff. To this end, we shall consider the Stiletjes integral which incorporates both sums and integrals. Now, we have an expansion of the sum of products of Bernoulli polynomials that we may apply it to elaborate on the results of afore-mentioned papers and can supplement them by related results.     

反常积分号下取极限的两个定理及应用

针对n→∞时相应反常积分的极限介绍两个有用的定理,特别是积分形式的Tannery定理,这在现有的文献中很少见到.这些结果有助于完善教材相关内容.最后给出了几个应用例子.     

Feynman path integrals as analysis on path space by time slicing approximation

Using the time slicing approximation, we give a mathematically rigorous definition of Feynman path integrals for a general class of functionals on the path space. As an application, we prove the interchange with Riemann-Stieltjes integrals, the interchange with a limit, the perturbation expansion formula, the semiclassical approximation, and the fundamental theorem of calculus in Feynman path integral.     

The widest continuous integral

An approach to a definition of an integral, which differs from definitions of Lebesgue and Henstock-Kurzweil integrals, is considered. We use trigonometrical polynomials instead of simple functions. Let V be the space of all complex trigonometrical polynomials without the free term. The definition of a continuous integral on the space V is introduced. All continuous integrals are described in terms of norms on V. The existence of the widest continuous integral is proved, the explicit form of its norm is obtained and it is proved that this norm is equivalent to the Alexiewicz norm. It is shown that the widest continuous integral is wider than the Lebesgue integral. An analog of the fundamental theorem of calculus for the widest continuous integral is given.     

Representation of a multiple integral of special form by a series

Multiple integrals generalizing the iterated kernels of linear integral equations are expressed by a series each of whose terms is proportional to the product of two orthogonal functions in the case of a similar representation of the kernel. Besides integral equations, these integrals have applications in the theory of Markov processes. The results obtained are illustrated by several examples.     

The s-Perron,sap-Perron and ap-McShane Integrals

In this paper, we study the s-Perron, sap-Perron and ap-McShane integrals. In particular, we show that the s-Perron integral is equivalent to the McShane integral and that the sap-Perron integral is equivalent to the ap-McShane integral.     

Homological integral of Hopf algebras

The left and right homological integrals are introduced for a large class of infinite dimensional Hopf algebras. Using the homological integrals we prove a version of Maschke's theorem for infinite dimensional Hopf algebras. The generalization of Maschke's theorem and homological integrals are the keys to studying noetherian regular Hopf algebras of Gelfand-Kirillov dimension one.

     

On Double Stratonovich Fractional Integrals and Some Strong and Weak Approximations

Abstract

Double Stratonovich integrals with respect to the odd part and even part of the fractional Brownian motion are constructed. The first and the second moments of such integrals are explicitly identified. As application of double Stratonovich integrals a strong law of large numbers for efBm and ofBm is derived.

Riemann–Stieltjes integral approximations to double Stratonovich fractional integrals are also considered. The strong convergence (almost surely and mean square) is obtained for approximations based on explicit series expansions of the fractional Brownian processes. The weak convergence is derived for approximations by processes with absolutely continuous paths which converge weakly to the considered fractional Brownian processes. The above-mentioned convergences are obtained for deterministic integrands which are given by bimeasures.