Regularity Properties of Some Stochastic Volterra Integrals with Singular Kernel

We prove the Hölder continuity of some stochastic Volterra integrals, with singular kernels, under integrability assumptions on the integrand. Some applications to processes arising in the analysis of the fractional Brownian motion are given. The main tool is the embedding of some Besov spaces into some sets of Hölder continuous functions.     

Two-Parameter Fuzzy-Valued Stochastic Integrals and Equations

This article is concerned with notions of fuzzy-valued stochastic integrals driven by two-parameter martingales and increasing processes. We present their main properties and formulate next two-parameter fuzzy-valued stochastic integral equations. We establish the existence and uniqueness of solutions to such equations as well as their additional properties.     

Boundedness of Fractional Integrals with a Rough Kernel on the Pro duct Triebel-Lizorkin Spaces

By using the Littlewood-Paley decomposition and the interpolation the-ory, we prove the boundedness of fractional integral on the product Triebel-Lizorkin spaces with a rough kernel related to the product block spaces.     

On Set-Valued Stochastic Integrals

We define a set-valued stochastic integral with respect to a 1-dimensional Brownian motion. The paper develops multivalued analogs to the theory of singlevalued stochastic integrals. It is expected that these results will be useful to study set-valued and fuzzy stochastic analysis.     

Multilinear Fractional Integrals and Commutators on Generalized Herz Spaces

Suppose b =(b_1, ···, b_m) ∈(BMO)~m, I_(α,m)~(Πb) is the iterated commutator of b and the m-linear multilinear fractional integral operator I_(α,m). The purpose of this paper is to discuss the boundedness properties of I_(α,m) and I_(α,m)~(Πb) on generalized Herz spaces with general Muckenhoupt weights.     

广义分数次积分多线性交换子的端点估计

设函数b=（b1,b2,…,bm）和广义分数次积分L-a／2（0〈α〈n）,它们生成多线性算子定义如下 Lb -a/2 f = [bm …, [b2[b1, L-a/2]],…, ]f,其中m ∈ Z＋ , bi ∈ Lipβi （0 〈βi 〈 1）,其中（1≤i≤m）．将讨论Lb -1a/2。从Mp^q（Rn）到Lip（α＋β-n/ q） （ Rn ）和q^q （ Rn ）到BMO（Rn）的有界性．     

Fractional Integrals on Product Manifolds

In this paper, the authors give a mixed norm estimate for the multi-parameter fractional integrals on product measurable spaces. This estimate is applied to obtain the boundedness for the fractional integrals of Nagel-Stein type on product manifolds, the fractional integral of Folland-Stein type with rough convolution kernels on product homogeneous groups, and the discrete fractional integrals of Stein-Wainger type. The research was supported by NSF of China (Grant: 10571015) and SRFDP of China (Grant: 20050027025).     

An Isometric Approach to Generalized Stochastic Integrals

The possibility to extend the classical Ito's construction of stochastic integrals is studied. This construction can be applied to fractional Brownian motions with Hurst index H(0, 1/2). A change of variables formula for fractional Brownian motions in terms of the stochastic integrals is given.     

Rational Approximations of Riemann--Liouville and Weyl Fractional Integrals

We obtain exact rational approximation orders for functions expressible as Riemann--Liouville and Weyl fractional integrals. New results and the strengthening and generalization of theorems due to Popov, Petrusheva, Pekarskii, Rusak, and the author, which are well known in the theory of rational approximation of differentiable functions, are obtained as consequences of theorems due to Pekarskii related to rational approximation of functions from the Hardy--Sobolev classes in the unit disk.     

变量核奇异积分和分数次微分加权范不等式

用T和D^γ（0 ≤ γ ≤ 1）分别表示变量核奇异积分和分数次微分算子.T^*和T^#分别为T的共轭算子及拟共轭算子.利用球调和多项式展式,本文得到了TD^γ-D^γT和（T^*-T^#）D^γ在?^q,λ_ω（Rⁿ）上的有界性.同时也得到了变量核奇异积分的积T₁T₂和拟积T₁°T₂的加权范不等式.     

Besov Regularity of Stochastic Integrals with Respect to the Fractional Brownian Motion with Parameter H > 1/2

Let {B _t ,t[0,1]} be a fractional Brownian motion with Hurst parameter H > 1/2. Using the techniques of the Malliavin calculus we show that the trajectories of the indefinite divergence integral ^t ₀ u _s B _s belong to the Besov space _p,q for all , provided the integrand u belongs to the space . Moreover, if u is bounded and belongs to for some even integer p2 and for some large enough, then the trajectories of the indefinite divergence integral ^t ₀ u _s B _s belong to the Besov space _p, ^H .     

多线性分数次积分的双权弱型不等式

本文给出了双权函数的一个如型条件使得多线性分数次积分满足双权弱型（p，q）不等式．     

Some Processes Associated with Fractional Bessel Processes

Let be a d-dimensional fractional Brownian motion with Hurst parameter H and let be the fractional Bessel process. Itôs formula for the fractional Brownian motion leads to the equation . In the Brownian motion case is a Brownian motion. In this paper it is shown that X_t is not an -fractional Brownian motion if H 1/2. We will study some other properties of this stochastic process as well.     

Weighted Weak-type Estimates for Multilinear Commutators of Fractional Integrals on Spaces of Homogeneous Type

We obtain weighted distributional inequalities for multilinear commutators of the fractional integral on spaces of homogeneous type, The techniques developed in this work involve the behavior of some fractional maximal functions. In relation to these operators, as a main tool, we prove a weighted weak type boundedness result, which is interesting in itself.     

Fractional Bilinear Stochastic Equations with the Drift in the First Fractional Chaos

Abstract

In the paper we compute the explicit form of the fractional chaos decomposition of the solution of a fractional stochastic bilinear equation with the drift in the fractional chaos of order one and initial condition in a finite fractional chaos. The large deviations principle is also obtained for the one-dimensional distributions of the solution of the equation perturbed by a small noise.     

The Fractional Calculus for Some Stochastic Processes

Abstract

The integration and differentiation of fractional orders are well known concepts for deterministic functions (see Miller, K.S.; Ross, B. An Introduction to Fractional Calculus and Fractional Differential Equations; John Wiley: New York, 1993; I. Podlubny and Ahmed M.A. El-Sayed, On two definitions of fractional calculus Slovak Academy of Sciences Institute of experimental Phys. UEF-03-96 ISBN 80-7099-252-2, 1996; Podlubny, I. Fractional Differential Equations; Acad. Press: San Diego – New York, London etc. 1999; Samko, S.G.; Kilbas, A.A.; Marichev, O. Integral and derivatives of the fractional orders and some of their applications. Nauka i Teknika Minisk 1983). In earlier work, we have studied the fractional calculus for mean square continuous stochastic processes. In this work, we shall study the mean square (m.s.) fractional calculus for stochastic processes which are m.s. Riemann-integrable and prove some its properties.     

分数次积分算子的交换子在齐型空间上的弱型估计

本文研究分数次积分交换子,其中Kα(x,y)=d(x,y)α-1,m∈N且b(x)∈BMO(X,μ),证明了Iα,bm是从Orlicz空间L(log L)m(X)到弱Lq(X)空间的映照．同时还证明了分数次极大算子交换子Mα,bm也有类似性质．     

分数次积分在加权Herz型Hardy空间的有界性

讨论了具有齐性核的分数次积分算子TΩ,μ在加权Herz型Hardy空间的有界性,证明TΩ,μ是从HKq1α,p1(w1,w2q1)到Kq2 α,p2(w1,w2 q2)或HKq1α,p1(1,w2q2)到HKq2α,p2(1,w2q2)有界的.     

Stochastic Integrals and Stochastic Differential Equations over the Field of p-Adic Numbers

We construct, for various classes of p-adic-valued functions, stochastic integrals with respect to the Poisson random measure. This leads to the construction of Markov processes over the field of p-adic numbers by means of stochastic differential equations.     

Boundedness for the Singular Integral with Variable Kernel and Fractional Differentiation on Weighted Morrey Spaces

Let T be the singular integral operator with variable kernel, T*be the adjoint of T and T~#be the pseudo-adjoint of T. Let T_1T_2 be the product of T_1 and T_2, T_1? T_2 be the pseudo product of T_1 and T_2. In this paper, we establish the boundedness for commutators of these operators and the fractional differentiation operator Dγon the weighted Morrey spaces.