分数布朗运动与随机积分

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"该资源是一本关于随机微积分与分数布朗运动的专业书籍，涵盖了分数布朗运动的基本属性、Wiener积分、发散型积分、分数Wick Itô Skorohod积分以及路径积分等多个主题。书中的内容深入探讨了分数布朗运动的特性，如长期依赖性、自相似性和Holder连续性，并介绍了适用于不同Hurst指数的积分理论。" 本书分为多个部分，首先介绍分数布朗运动的基础，包括它的内在性质，如它是如何定义的，以及它的随机积分表示。书中详细讨论了两个增量之间的相关性，展示了其具有长期依赖性，这意味着过去和未来的事件在某种程度上是相关的，这与传统的布朗运动不同。此外，分数布朗运动的自相似性意味着它的尺度变换保持某些统计特性不变，而Holder连续性则说明了其路径的平滑程度。对于H=1/2的情况，分数布朗运动不是半鞅，这是其与其他随机过程的一个关键区别。接着，第二部分深入到分数布朗运动的随机积分理论，分别讨论了Wiener积分和发散型积分。Wiener积分在H>1/2和H<1/2两种情况下有不同的处理方式。发散型积分同样分为H>1/2和H<1/2的两种情况，这些理论扩展了经典布朗运动的积分概念，使之适应分数布朗运动的非高斯特性。第三部分介绍了分数Wick Itô Skorohod积分，这部分涉及分数白噪声、分数Girsanov定理、分数随机梯度等概念。通过引入φ-导数，作者构建了一种新的积分形式，并给出了L2空间中的分数Wick Itô Skorohod积分，还发展了一个Itô公式。此外，还讨论了Lp估计、交错积分和混沌展开，以及分数Clark Hausmann Ocone定理。这部分内容对于理解分数布朗运动的复杂结构至关重要。第四部分介绍了Wick Itô Skorohod积分，这是另一种处理分数布朗运动的积分方法，书中阐述了M算子和WIS积分的定义，以及与标准Malliavin微积分的关系。第五部分则探讨了路径积分，特别是针对分数布朗运动的对称积分、前向积分和后向积分，以及它们与随机积分理论之间的联系。这本书是研究分数布朗运动及其在随机微积分应用中的理想资源，涵盖了广泛的理论和计算技术，适合对随机过程和现代概率理论有深入了解的读者。

Intrinsic properties of the fractional Brownian

motion

The aim of this book is to provide a comprehensive overview and system-

atization of stochastic calculus with respect to fractional Brownian motion.

However, for the reader’s convenience, in this chapter we review the main

properties that make fractional Brownian motion interesting for many appli-

cations in diﬀerent ﬁelds.

The main references for this chapter are [76], [156], [177], [195], [209], [215].

For further details concerning the theory and the applications of long-range

dependence from a more statistical point of view, we also refer to [81].

1.1 Fractional Brownian motion

The fractional Brownian motion was ﬁrst introduced within a Hilbert space

framework by Kolmogorov in 1940 in [141], where it was called Wiener Helix.

It was further studied by Yaglom in [230]. The name fractional Brownian

motion is due to Mandelbrot and Van Ness, who in 1968 provided in [156]

a stochastic integral representation of this process in terms of a standard

Brownian motion.

Deﬁnition 1.1.1. Let H be a constant belonging to (0, 1). A fractional Brown-

ian motion ( fBm) (B

(H)

(t))

t≥0

of Hurst index H is a continuous and centered

Gaussian process with covariance function



(H)

(t)B

(H)

(s)



=1/2(t

+ s

−|t − s|

For H =1/2, the fBm is then a standard Brownian motion. By Deﬁnition

1.1.1 we obtain that a standard fBm B

(H)

has the following properties:

1. B

(H)

(0) = 0 and E



(H)

(t)



= 0 for all t ≥ 0.

2. B

(H)

has homogeneous increments, i.e., B

(H)

(t + s) − B

(H)

(s) has the

same law of B

(H)

(t)fors, t ≥ 0.

6 1 Intrinsic properties of the fractional Brownian motion

3. B

(H)

is a Gaussian process and E



(H)

(t)



= t

, t ≥ 0, for all H ∈

(0, 1).

4. B

(H)

has continuous trajectories.

The existence of the fBm follows from the general existence theorem of cen-

tered Gaussian processes with given covariance functions (see [196]). We will

also give some constructions of the fBm through the white noise theory for

our special purposes in later chapters. The fBm is divided into three very

diﬀerent families corresponding to 0 <H<1/2, H =1/2, and 1/2 <H<1,

respectively, as we will see in the sequel. It was Mandelbrot that named the

parameter H of B

(H)

after the name of the hydrologist Hurst, who made

a statistical study of yearly water run-oﬀs of the Nile river (see [129]). He

considered the values δ

,...,δ

of n successive yearly run-oﬀs and their cor-

responding cumulative value ∆



i=1

over the period from the year

662 until 1469. He discovered that the behavior of the normalized values of

the amplitude of the deviation from the empirical mean was approximately

, where H =0.7. Moreover, the distribution of ∆



i=1

was ap-

proximately the same as n

, with H>1/2. Hence, this phenomenon could

not be modeled by using a process with independent increments, but rather

the δ

could be thought as the increments of a fBm. Because of this study,

Mandelbrot introduced the name Hurst index.

1.2 Stochastic integral representation

Here we discuss some of the integral representations for the fBm. In [156], it

is proved that the process

Z(t)=

Γ (H +1/2)





(t − s)

H−1/2

− (−s)

H−1/2



dB(s)

Γ (H +1/2)





−∞



(t − s)

H−1/2

− (−s)

H−1/2



dB(s)



(t − s)

H−1/2

dB(s)



(1.1)

where B(t) is a standard Brownian motion and Γ represents the gamma

function, is a fBm with Hurst index H ∈ (0, 1). If B(t) is replaced by a

complex-valued Brownian motion, the integral (1.1) gives the complex fBm.

By following [177] we sketch a proof for the representation (1.1). For further

detail we refer also to [207]. First we notice that Z(t) is a continuous centered

Gaussian process. Hence, we need only to compute the covariance functions.

In the following computations we drop the constant 1/Γ (H +1/2) for the sake

of simplicity. We obtain



(t)







(t − s)

H−1/2

− (−s)

H−1/2



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