Integration questions related to fractional Brownian motion
Integration questions related to fractional Brownian motion
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DOI：
10.1007/s440-000-8016-7
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发表时间：
2000
影响因子：
2
通讯作者：
V. Pipiras;M. Taqqu
中科院分区：
文献类型：
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作者：
V. Pipiras;M. Taqqu
令 {BH(u)}u∈ℝ 为分数布朗运动 (fBm)，索引 H∈(0, 1)，(BH) 为 fBmBH 增量的 spanSp(BH) 的闭包 inL2(Ω)。众所周知，当BH=B1/2为通常的布朗运动（Bm）时，元素X∈（B1/2）可以用唯一的函数fX∈L2（ℝ）来表征，此时将X写成积分形式为X=∫ℝfX（u）dB1/2（u）。从不同但等价的角度来看，空间 L2(ℝ) 形成了一类关于 BmB1/2 在实线上积分的被积函数。在这项工作中，我们探讨当 H ∈ (0, 1/2) 或 H ∈ (1/2, 1) 时是否可以获得 (BH) 元素的类似表征。由于很自然地可以通过 Σk=1nfk(BH(uk+1) −BH(uk)) 来定义初等函数 f= Σk=1nfk1[uk,uk+1) 的积分，因此我们希望被积函数的空间包含初等函数。这些类别的被积函数是内积空间。如果被积函数空间不完备，则它仅表征 (BH) 的严格子集。当0<H<1/2时，利用fBmBH的移动平均表示，构造一个完整的被积空间。然而，当 1/2<H<1 时，类似的构造会导致被积函数空间不完备。当 0<H<1/2 或 1/2<H<1 时，我们还考虑许多其他的被积函数空间。虽然较小且不完整，但它们形成了自然的选择并且易于使用。我们将这些被积函数空间与 fBm 的再现核希尔伯特空间进行比较。
Let {BH(u)}u∈ℝbe a fractional Brownian motion (fBm) with indexH∈(0, 1) and (BH) be the closure inL2(Ω) of the spanSp(BH) of the increments of fBmBH. It is well-known that, whenBH=B1/2is the usual Brownian motion (Bm), an elementX∈ (B1/2) can be characterized by a unique functionfX∈L2(ℝ), in which case one writesXin an integral form asX= ∫ℝfX(u)dB1/2(u). From a different, though equivalent, perspective, the spaceL2(ℝ) forms a class of integrands for the integral on the real line with respect to BmB1/2. In this work we explore whether a similar characterization of elements of (BH) can be obtained whenH∈ (0, 1/2) orH∈ (1/2, 1). Since it is natural to define the integral of an elementary functionf= ∑k=1nfk1[uk,uk+1)by ∑k=1nfk(BH(uk+1) −BH(uk)), we want the spaces of integrands to contain elementary functions. These classes of integrands are inner product spaces. If the space of integrands is not complete, then it characterizes only a strict subset of (BH). When 0<H<1/2, by using the moving average representation of fBmBH, we construct a complete space of integrands. When 1/2<H<1, however, an analogous construction leads to a space of integrands which is not complete. When 0<H<1/2 or 1/2<H<1, we also consider a number of other spaces of integrands. While smaller and henceincomplete, they form a natural choice and are convenient to workwith. We compare these spaces of integrands to the reproducing kernel Hilbert space of fBm.