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.     

Approximation of Multiple Stratonovich Fractional Integrals

Abstract

We study multiple Riemann-Stieltjes integral approximations to multiple Stratonovich fractional integrals. Two standard approximations (Wong-Zakai and Mollifier approximations) are considered and we show the convergence in the mean square sense and uniformly on compact time intervals of these approximations to the multiple Stratonovich fractional integral.     

Expansion of Multiple Integrals in Linear Forms

We prove general theorems on expansions of multiple integrals in linear forms in generalized polylogarithms with coefficients that are rational functions.__________Translated from Matematicheskie Zametki, vol. 77, no. 5, 2005, pp. 683–706.Original Russian Text Copyright ©2005 by S. A. Zlobin.     

New Representations of the Taylor–Stratonovich Expansion

The problem of the Taylor–Stratonovich expansion of the Itô random processes in a neighborhood of a point is considered. The usual form of the Taylor–Stratonovich expansion is transformed to a new representation, which includes the minimal quantity of different types of multiple Stratonovich stochastic integrals. Therefore, these representations are more convenient for constructing algorithms of numerical solution of stochastic differential Itô equations. Bibliography: 14 titles.     

Absolute Continuity of Joint Laws of Multiple Stable Stochastic Integrals

We are interested in the laws of multiple stable stochastic integrals defined by LePage series representation in references(3,10,11). We continue the study started in Ref. 3 and give conditions ensuring absolute continuity of joint laws of stable integrals. To this end, we apply a stratification method on the Skorohod space on which we first take back the problem.     

Anticipating Stochastic Differential Equations of Stratonovich Type

We prove the existence and uniqueness of Stratonovich stochastic differential equations where the coefficients and the initial condition may depend on the whole path of the driving Wiener process. Our main hypothesis is that the diffusion coefficient satisfies the Frobenius condition. The solution is given in terms of solutions of ordinary differential equations and the Wiener process. We use this representation to study properties of the solution. Accepted 3 April 1996     

The Stratonovich Interpretation of Quantum Stochastic Approximations

The Stratonovich version of non-commutative stochastic calculus is introduced and shown to be equivalent to the Itô version developed by Hudson and Parthasarathy [1]. The conversion from Stratonovich to Itô version is shown to be implemented by a stochastic form of Wick's theorem: that is, involving the normal ordering of time-dependent noise fields. It is shown for a model of a quantum mechanical system coupled to a Bosonic field in a Gaussian state that under suitable scaling limits, in particular the weak coupling limit (for linear interactions) and low density limit (for scattering interactions), the limit form of the dynamical equation of motion is most naturally described as a quantum stochastic differential equation of Stratonovich form. We then convert the limit dynamical equations from Stratonovich to Itô form. Thermal Stratonovich noises are also presented.     

A Distributional Approach to Multiple Stochastic Integrals and Transformations of the Poisson Measure

We study a class of ‘nonpoissonian’ transformations of the configuration space and the corresponding transformations of the Poisson measure. For some class of Poisson measures we find conditions which are sufficient for the transformed measure (which in general is nonpoissonian) to be absolutely continuous with respect to the initial Poisson measure and get the expression for the corresponding Radon–Nikodym derivative. To solve this problem we use a distributional approach to Poisson multiple stochastic integrals.     

Numerical Evaluation of Fourier Integrals

A method is developed for evaluating Fourier integrals of theform A() = ¹_–1f(x) e^fax dx, 0. The method consists of expanding the function f in a seriesof Chebyshev polynomials and expressing the integral A() asa series of the Bessel functionsJ_r+(), r= 0, 1, 2,.... A partialsum A_N() of the series provides an approximant to A(). The principalfeature of the method is that one set of N+1 evaluations off(x) suffices for the calculation of A_N() for all , and alsothe truncation error A()–A_N() is essentially independentof . Numerical tests show that the method is accurate, economicaland reliable. An application to the inversion of Fourier andLaplace transforms is briefly described.     

随机积分的拟必然逼近

该文证明了随机积分的拟必然逼近. 基于此,作者用更简化的方法得到并拓展了文献[8]的主要结果. 并且, 该方法可以应用到光滑两参数鞅情形.     

Multiple Stable Stochastic Integrals: Series Representation and Absolute Continuity of Their Law

We study multiple stable stochastic integrals. We begin with a definition of these integrals using a generalization of the LePage representation as Samorodnitsky et al. in Ref. 15 but with a simplified probabilistic background and for general stable measure when 0<<1 and for symmetric stable measure when 1<2. This representation allows to study the law of multiple stable integrals. We prove they are absolutely continuous using a stratification method on the Skorokhod space on which we have previously taken back the problem.     

A Distributional Approach to Multiple Stochastic Integrals and Transformations of the Poisson Measure, II

We study a class of “nonpoissonian” transformations of the configuration space (over a space of the form G=S×?, where S is a complete separable metric space) and the corresponding transformations of the Poisson measure. For the Poisson measures of the Lévy-Khinchin type we find conditions which are sufficient to ensure that the transformed measure (which in general is nonpoissonian) is absolutely continuous with respect to the initial Poisson measure and derive an expression for the corresponding Radon-Nikodym derivative. To this end we use a distributional approach to Poisson multiple stochastic integrals. This is the second of a series of papers, as compared to the first part the space G is different and the intensity measure is more general, allowing a stronger singularity at the origin.     

Multiple Fourier series for functions from a Zygmund class

The paper considers necessary and sufficient conditions providing that a function of many variables with nonnegative trigonometric Fourier coefficients belongs to the big and small Zygmund classes. Earlier, such results were obtained by R.P. Boas in the one-dimensional case and by V. Fülöp in the two-dimensional case.     

Fractional Smoothness of Some Stochastic Integrals

We study the fractional smoothness in the sense of Malliavin calculus of stochastic integrals of the form ∫0^1Ф（Xs）dXs, where Xs is a semimartingale and Ф belongs to some fractional Sobolev space over R.     

多元函数积分的简化计算

利用多元函数积分区域的对称性,可通过对被积函数以及积分区域的变换来简化多元函数积分的计算.     

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.     

Remarks on Fourier series

We prove the following propositions. An even integrable function whose Fourier coefficients form a convex sequence is absolutely continuous if and only if its Fourier series converges absolutely. If the function f(t)is convex on [0, ],f(t)=f(—t), then for odd n while for even n, b₀=0.Translated from Matematicheskie Zametki, Vol. 3, No. 5, pp. 597–604, May, 1968.     

一种基于随机Taylor展开式的随机微分方程数值解法

给出了一种基于随机Taylor展开式的随机微分方程数值格式,证明了它的均方稳定性。此外,还证明了这种数值格式的均值意义下的局部收敛阶为2,均方意义下的局部收敛阶为1,均方强收敛阶为1.数值实验表明本文的方法比Euler法和Milstein方法具有更好的逼近效果.     

Fourier积分的数值计算法

本文建议采用样条函数插值作Fourier积分数值计算.同时,介绍了对几个一般函数采用各种不同的数值计算法的计算结果和构造样条函数的边界条件的简化处理方法.