Are classes of deterministic integrands for fractional Brownian motion on an interval complete
Are classes of deterministic integrands for fractional Brownian motion on an interval complete
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DOI：
10.2307/3318624
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发表时间：
2001
期刊：
影响因子：
1.5
通讯作者：
V. Pipiras;M. Taqqu
中科院分区：
文献类型：
--
作者：
V. Pipiras;M. Taqqu
设BH是一个具有自相似参数He（0，1）的分数布朗运动，a > 0是一个固定的真实的数.考虑积分fa f（u）dBH（u），其中f属于一类非随机被积函数AH，a。然后，积分将在L2（Q）意义下定义。我们希望AH，a是一个完全的内积空间。这对应于期望的情况，因为然后在AH，a和由BH（u）生成的跨度的闭包之间存在等距，0 < u < a。我们在这项工作中表明，当他（？），1），被积函数类AH，a通常被认为是不完备的内积空间，即使它们在文献中经常被假设是完备的。因此，它们不是等距于l--p{BH（u），0 u < a}，而是等距于一个真子空间。因此，在这个闭包中有一些（随机）元素不能用AH，a中的函数f表示。与HE（1，1）的情形不同，我们还证明了在完备内积空间[0，a]上存在一类具有HE（0，1）的分数布朗运动BH的被积函数.
Let BH be a fractional Brownian motion with self-similarity parameter H e (0, 1) and a > 0 be a fixed real number. Consider the integral fa f(u)dBH(u), where f belongs to a class of non-random integrands AH,a. The integral will then be defined in the L2(Q) sense. One would like AH,a to be a complete inner-product space. This corresponds to a desirable situation because then there is an isometry between AH,a and the closure of the span generated by BH(u), 0 < u < a. We show in this work that, when H e (?, 1), the classes of integrands AH,a one usually considers are not complete inner-product spaces even though they are often assumed in the literature to be complete. Thus, they are isometric not to l--p{BH(u), 0 u < a} but only to a proper subspace. Consequently, there are (random) elements in that closure which cannot be represented by functions f in AH,a. We also show, in contrast to the case H E (1, 1) that there is a class of integrands for fractional Brownian motion BH with H e (0, 1) on an interval [0, a] which is a complete inner-product space.