On polygonal approximation of brownian motion in stochastic integral

For Brownian motion B denote by B^δ its polygonal approximation corresponding to a partition δ of [0,1]. It is proved that if E(f¹|X_t|^p dt<∞ for some p>2 then converges to in mean as the mesh |Δ|→0 provided the symmetric (Stratonovich) stochastic integral is determined (in the sense given in [4])     

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.     

The Wiener integral with respect to second order processes with stationary increments

We prove that for any second order stochastic process X with stationary increments with continuous paths and continuous variance function, there exists a tempered measure μ (for which we give an explicit expression) related with the domain of the Wiener integral with respect to X as follows: the space of tempered distributions f such that the Fourier transform of f is square integrable with respect to μ is always a dense subset of the domain of the Wiener integral. Moreover, we provide sufficient conditions on μ in order that the domain of the integral is exactly this space of distributions. We apply our results to the fractional Brownian motion. In particular, it is proved that the domain of the Wiener integral with respect to the fractional Brownian motion with Hurst parameter H>1/2 contains distributions that are not given by locally integrable functions, this fact was suggested by Pipiras and Taqqu (2000) in [5]. We have also considered the example of the process given by Ornstein and Uhlenbeck as a model for the position of a Brownian particle.     

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.     

Weighted BMO and discrete time hedging within the Black-Scholes model

The paper combines two objects rather different at first glance: spaces of stochastic processes having weighted bounded mean oscillation (weighted BMO) and the approximation of certain stochastic integrals, driven by the geometric Brownian motion, by integrals over piece-wise constant integrands. The consideration of the approximation error with respect to weighted BMO implies L_p and uniform distributional estimates for the approximation error by a John-Nirenberg type theorem. The general results about weighted BMO are given in the first part of the paper and applied to our approximation problem in the second one.     

Is the Approximation Rate for European Pay-offs in the Black–Scholes Model Always 1/ \sqrt{n}

For a Borel-function , we consider the approximation of a random variable f(W ₁) with by stochastic integrals with respect to the Brownian motion and the geometric Brownian motion, where the integrands are piecewise constant within certain deterministic time intervals. In earlier papers it has been shown that under certain regularity conditions the optimal approximation rate is 1/ , if one optimizes over deterministic time-nets of cardinality n. We will show the existence of random variables f(W ₁) such that the approximation error tends as slowly to zero as one wishes.     

Fractional Brownian motion approximation based on fractional integration of a white noise

We study simple approximations to fractional Gaussian noise and fractional Brownian motion. The approximations are based on spectral properties of the noise. They allow one to consider the noise as the result of fractional integration/differentiation of a white Gaussian noise. We consider correlation properties of the approximation to fractional Gaussian noise and point to the peculiarities of persistent and anti-persistent behaviors. We also investigate self-similarity properties of the approximation to fractional Brownian motion, namely, `τ^H laws' for the structure function and the range. We conclude that the models proposed serve as a convenient tool for modelling of natural processes and testing and improvement of methods aimed at analysis and interpretation of experimental data.     

<Emphasis Type="Italic">p</Emphasis>-Adic Brownian motion over ℚ<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

In this paper, we generalize the result of Bikulov and Volovich (1997) and construct a p-adic Brownian motion over ℚ _p . First, we construct directly a p-adic white noise over ℚ _p by using a specific complete orthonormal system of (ℚ _p ). A p-adic Brownian motion over ℚ _p is then constructed by the Paley-Wiener method. Finally, we introduce a p-adic random walk and prove a theorem on the approximation of a p-adic Brownian motion by a p-adic random walk.     

Controlled differential equations as Young integrals: A simple approach

The theory of rough paths allows one to define controlled differential equations driven by a path which is irregular. The most simple case is the one where the driving path has finite p-variations with 1?p<2, in which case the integrals are interpreted as Young integrals. The prototypal example is given by stochastic differential equations driven by fractional Brownian motion with Hurst index greater than 1/2. Using simple computations, we give the main results regarding this theory - existence, uniqueness, convergence of the Euler scheme, flow property … - which are spread out among several articles.     

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.     

Approximation of the finite dimensional distributions of multiple fractional integrals

We construct a family Inεt(f) of continuous stochastic processes that converges in the sense of finite dimensional distributions to a multiple Wiener-Itô integral with respect to the fractional Brownian motion. We assume that and we prove our approximation result for the integrands f in a rather general class.     

Fractional stochastic differential equation with discontinuous diffusion

In this article, we study a class of stochastic differential equations driven by a fractional Brownian motion with H > 1/2 and a discontinuous coefficient in the diffusion. We prove existence and uniqueness for the solution of these equations. This is a first step to define a fractional version of the skew Brownian motion.     

WEIGHTED Lp-APPROXIMATION OF DERIVATIVES BY THE METHOD OF BETA OPERATORS

The author consides Beta operators β_nf on suitable Sobolev type subspace of L_p[0, ∞) and characterizes the global rate of approximation of derivatives f^(τ) through corresponding derivatives (β_nf)^(τ) in an appropriate weighted L_p-metric by the rate of Ditzian and Totik's τ-th order weighted modulus of Smoothness.     

Brown运动在Hlder范数下的拟必然Strassen重对数律的收敛速率

本文研究了Brown运动的泛函极限问题.利用Brown运动在Hlder范数下关于容度的大偏差与小偏差,获得了Brown运动在Hlder范数下的Strassen型重对数律的拟必然收敛速率,推广了文[2]中的结果.     

Stock exchange fractional dynamics defined as fractional exponential growth driven by (usual) Gaussian white noise. Application to fractional Black-Scholes equations

Stock exchange dynamics of fractional order are usually modeled as a non-random exponential growth process driven by a fractional Brownian motion. Here we propose to use rather a non-random fractional growth driven by a (standard) Brownian motion. The key is the Taylor’s series of fractional order where E_α(.) denotes the Mittag-Leffler function, and is the so-called modified Riemann-Liouville fractional derivative which we introduced recently to remove the effects of the non-zero initial value of the function under consideration. Various models of fractional dynamics for stock exchange are proposed, and their solutions are obtained. Mainly, the Itô’s lemma of fractional order is illustrated in the special case of a fractional growth with white noise. Prospects for the Merton’s optimal portfolio are outlined, the path probability density of fractional stock exchange dynamics is obtained, and two fractional Black-Scholes equations are derived. This approach avoids using fractional Brownian motion and thus is of some help to circumvent the mathematical difficulties so involved.     

混合型随机微分方程的传输不等式

该文探讨一类由Wiener过程和Hurst参数1/2＜H＜1分数布朗运动驱动的混合型随机微分方程.通过使用一些变换技巧和逼近方法,这类方程的强解在d2度量和一致度量d∞下的二次传输不等式被建立.     

On central limit theorems,modulus of continuity and diophantine type for irrational rotations

LetR _α be a rotation on the circle by an irrational angle α. LetB(t) be a Brownian motion (for instance). Then (Lacey (1990), (1991)) there is anf ∈L ² so that $$m^{ - 1/2} (f + ... + f o R_\alpha ^{[m1] - 1} )\mathop \Rightarrow \limits^d B(t)$$ In this note, we show thatf can be taken to be continuous, and give a sharp estimate on the modulus of continuity off, in terms of number-theoretic properties of α. The same result is given for self-similar processes other than the Brownian motion.     

Hausdorff measures of the image, graph and level set of bifractional Brownian motion

Let BH,K = {BH,K(t), t ∈ R+} be a bifractional Brownian motion in Rd. This process is a selfsimilar Gaussian process depending on two parameters H and K and it constitutes a natural generalization of fractional Brownian motion (which is obtained for K = 1). The exact Hausdorff measures of the image, graph and the level set of BH,K are investigated. The results extend the corresponding results proved by Talagrand and Xiao for fractional Brownian motion.     

分数布朗单的幂函数随机积分逼近

本文研究了分数布朗单的逼近问题.利用Wiener积分,得到了分数布朗单的幂函数型随机积分逼近.     

Asymptotic Behavior of Weighted Power Variations of Fractional Brownian Motion in Brownian Time

We study the asymptotic behavior of weighted power variations of fractional Brownian motion in Brownian time \(Z_t:= X_{Y_t},t \geqslant 0\), where X is a fractional Brownian motion and Y is an independent Brownian motion.