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.     

多重分数斯特拉托诺维奇积分的一个注记

本文研究了多重分数斯特拉托诺维奇积分,通过卷积逼近技巧和分数布朗运动的随机积分的性质,构造了当Hurst参数小于二分之一时的多重随机积分.这种方法是新的不同于文[8]中的构造方法.     

多重分数Stratonovich积分的另一种构造(英文)

本文研究了当Hurst参数日小于1/2时关于分数布朗运动的随机积分问题.利用分数布朗运动的性质和卷积逼近的方法,获得了多重分数Stratonovich积分的另一种构造.     

On a Multiple Stratonovich-type Integral for Some Gaussian Processes

We construct a multiple Stratonovich-type integral with respect to Gaussian processes with covariance function of bounded variation. This construction is based on the previous definition of the multiple Itô-type integral given by Huang and Cambanis [Ann. Propab. 6(4), 585–614] and on a Hu–Meyer formula (that is, an expression of the multiple Stratonovich integral as a sum of Itô-type integrals of inferior or equal order) for the elementary functions. We also apply our results to the fractional Brownian motion with Hurst parameter $H > \frac{1}{2}We construct a multiple Stratonovich-type integral with respect to Gaussian processes with covariance function of bounded variation. This construction is based on the previous definition of the multiple It?-type integral given by Huang and Cambanis [Ann. Propab. 6(4), 585–614] and on a Hu–Meyer formula (that is, an expression of the multiple Stratonovich integral as a sum of It?-type integrals of inferior or equal order) for the elementary functions. We also apply our results to the fractional Brownian motion with Hurst parameter .     

On the Expectations of Multiple Stratonovich Integrals

Abstract

In the construction of numerical methods for solving stochastic differential equations it becomes necessary to calculate the expectations of products of multiple stochastic integrals. In the Itô case, explicit formulae for the expectation of a multiple integral with integrand identically equal to 1 and for the product of two such integrals are known. In this paper formulae for the expectation of any multiple Stratonovich integral as well as for the product of a broad class of two Stratonovich integrals have been derived.     

The hu-meyer formula for non deterministic kernels

In this paper we prove the analogue of the Hu-Meyer formula for random kernels. More precisely, using a suitable notion of trace we give the relation between the multiple Stratonovich integral of a non adapted process and the multiple Skorohod integral     

The Feynman-Stratonovich semigroup and stratonovich integral expansions in nonlinear filtering

Representations for the solution of the Zakai equation in terms of multiple Stratonovich integrals are derived. A new semigroup (the Feynman-Stratonovich semigroup) associated with the Zakai equation is introduced and using the relationship between multiple Stratonovich integrals and iterated Stratonovich integrals, a representation for the unnormalized conditional density,u(t,x), solely in terms of the initial density and the semigroup, is obtained. In addition, a Fourier seriestype representation foru(t,x) is given, where the coefficients in this representation uniquely solve an infinite system of partial differential equations. This representation is then used to obtain approximations foru(t,x). An explicit error bound for this approximation, which is of the same order as for the case of multiple Wiener integral representations, is obtained. Research supported by the National Science Foundation and the Air Force Office of Scientific Research Grant No. F49620 92 J 0154 and the Army Research Office Grant No. DAAL03-92-G0008.     

Convergence rate of multiple fractional Stratonovich type integral for Hurst parameter less than 1/2

In this paper, we have investigated the problem of the convergence rate of the multiple integralwhere f ∈ Cn+1([0, T ]n) is a given function, π is a partition of the interval [0, T ] and {BtHi ,π} is a family of interpolation approximation of fractional Brownian motion BtH with Hurst parameter H < 1/2. The limit process is the multiple Stratonovich integral of the function f . In view of known results, the convergence rate is different for different multiplicity n. Under some mild conditions, we obtain that the uniform convergence rate is 2H in the mean square sense, where is the norm of the partition generating the approximations.     

A strong approximation for double subfractional integrals

A Riemann–Stieltjes integral strong approximation to double Stratonovich integrals with respect to odd and even fractional Brownian motions is considered. We prove the convergence in quadratic mean, uniformly on compact time intervals, of the ordinary double integral process obtained by linear interpolation of the odd and even fractional Brownian motions, to the double Stratonovich integral. The deterministic integrands are continuous or are given by bimeasures.     

Stratonovich Calculus with Respect to Fractional Brownian Sheet

Abstract

We introduce two types of Stratonovich stochastic integrals for two-parameter process. The relationship of Stratonovich integrals to Skorohod integrals will be investigated. By using this relationship, we prove that a differentiation formula for fractional Brownian sheet in Stratonovich form can be expressed as the sum of Stratonovich integrals of two types introduced in this article.     

Differentiation formula in Stratonovich version for fractional Brownian sheet

We introduce two types of the Stratonovich stochastic integrals for two-parameter processes, and investigate the relationship of these Stratonovich integrals and various types of Skorohod integrals with respect to a fractional Brownian sheet. By using this relationship, we derive a differentiation formula in the Stratonovich sense for fractional Brownian sheet through Itô formula. Also the relationship between the two types of the Stratonovich integrals will be obtained and used to derive a differentiation formula in the Stratonovich sense. In this case, our proof is based on the repeated applications of differentiation formulas in the Stratonovich form for one-parameter Gaussian processes.     

Stability Analysis of nth Order Nonlinear Impulsive Differential Equations in Quasi–Banach Space

Abstract

This article is about Ulam’s type stability of n^th order nonlinear differential equations with fractional integrable impulses. It is a best procession to the stability of higher order fractional integrable impulsive differential equations in quasi–normed Banach space. Different Ulam’s type stability results are obtained by using the definitions of Riemann–Liouville fractional integral, Hölder’s inequality and the beta integral inequality.     

The generalized hu-meyer formula for random kernels

A theory of Hilbert-space-valued traces and multiple integration is developed for kernels inL ²([0, 1]^p × Θ). The multiple Ogawa and the multiple Stratonovich integrals for such kernels are introduced and sufficient conditions for their existence are obtained. The derivation of the Hu-Meyer formula connecting the multiple Ogawa and the multiple Stratonovich integrals requires the introduction of traces of random kernels. Such a derivation is obtained under appropriate conditions. This research was supported by the National Science Foundation and the Air Force Office of Scientific Research Grant No. F49620 92 J 0154 and the Army Research Office Grant No. DAAL03-92-G-0008.     

CONVERGENCE RATE OF AN APPROXIMATION TO MULTIPLE INTEGRAL OF FBM

In this article, we study the rate of convergence of the polygonal approximation to multiple stochastic integral Sp （f） of fractional Brownian motion of Hurst parameter H 〈 1/2 when the fractional Brownian motion is replaced by its polygonal approximation. Under different conditions on f and for different p, we obtain different rates.     

On the Weighted Local Time and the Tanaka Formula for the Multidimensional Fractional Brownian Motion

Abstract

We determine the weighted local time for the multidimensional fractional Brownian motion from the occupation time formula. We also discuss on the Itô and Tanaka formula for the multidimensional fractional Brownian motion. In these formulas the Skorohod integral is applicable if the Hurst parameter of fractional Brownian motion is greater than 1/2. If the Hurst parameter is less than 1/2, then we use the Skorohod type integral introduced by Nualart and Zakai for the stochastic integral and establish the Itô and Tanaka formulas.     

变量核分数次积分在Hardy空间的有界性

§ 1　 Introduction and main resultsLet Sn- 1 be the unitsphere in Rn(n≥ 2 ) equipped with normalized Lebesgue measure dσ= dσ(z′) .We say that a functionΩ(x,z) defined on Rn× Rnbelongs to L∞ (Rn)× Lr(Sn- 1 )(r≥ 1 ) ,ifΩ(x,z) satisfies the following two conditions,(i) for any x,z∈Rnandλ>0 ,there hasΩ(x,λz) =Ω(x,z) ;(ii)‖Ω‖L∞(Rn)× Lr(Sn- 1) :=supx∈ Rn∫Sn- 1|Ω(x,z′) | rdσ(z′) 1 / r<∞ .For 0 <α相似文献   

Hadamard-type fractional differential equations for the system of integral inequalities on time scales

ABSTRACT

This paper investigates some system of integral inequalities of one independent variable on time scales. The conclusion can be obtained by using Hadamard-type fractional differential equations and Greene's method which bring together and expand some integral inequalities on time scales. The established inequalities give explicit bounds on unknown functions which can be utilized as a key in examining the properties of certain classes of partial dynamic equations and difference equations on time scales. As an application, a system of fractional differential equations is considered to explain the value of our results.     

Approximation of a fractional power of an elliptic operator

Some mathematical models of applied problems lead to the need of solving boundary value problems with a fractional power of an elliptic operator. In a number of works, approximations of such a nonlocal operator are constructed on the basis of an integral representation with a singular integrand. In the present article, new integral representations are proposed for operators with fractional powers. Approximations are based on the classical quadrature formulas. The results of numerical experiments on the accuracy of quadrature formulas are presented. The proposed approximations are used for numerical solving a model two‐dimensional boundary value problem for fractional diffusion.     

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.     

The Wong–Zakai approximations of invariant manifolds and foliations for stochastic evolution equations

In this paper, we study the Wong–Zakai approximations given by a stationary process via the Wiener shift and their associated dynamics of a class of stochastic evolution equations with a multiplicative white noise. We prove that the solutions of Wong–Zakai approximations almost surely converge to the solutions of the Stratonovich stochastic evolution equation. We also show that the invariant manifolds and stable foliations of the Wong–Zakai approximations converge to the invariant manifolds and stable foliations of the Stratonovich stochastic evolution equation, respectively.