Estimating the p-Variation Index of a Sample Function: An Application to Financial Data Set

In this paper we apply a real analysis approach to test continuous time stochastic models of financial mathematics. Specifically, fractal dimension estimation methods are applied to statistical analysis of continuous time stochastic processes. To estimate a roughness of a sample function we modify a box-counting method typically used in estimating fractal dimension of a graph of a function. Here the roughness of a function f is defined as the infimum of numbers p > 0 such that f has bounded p-variation, which we call the p-variation index of f. The method is also tested on estimating the exponent [1, 2] of a simulated symmetric -stable process, and on estimating the Hurst exponent H (0, 1) of a simulated fractional Brownian motion.     

On Maps Of Bounded p-Variation With p>1

This paper addresses properties of maps of bounded p-variation (p>1) in the sense of N. Wiener, which are defined on a subset of the real line and take values in metric or normed spaces. We prove the structural theorem for these maps and study their continuity properties. We obtain the existence of a Hölder continuous path of minimal p-variation between two points and establish the compactness theorem relative to the p-variation, which is an analog of the well-known Helly selection principle in the theory of functions of bounded variation. We prove that the space of maps of bounded p-variation with values in a Banach space is also a Banach space. We give an example of a Hölder continuous of exponent 0<<1 set-valued map with no continuous selection. In the case p=1 we show that a compact absolutely continuous set-valued map from the compact interval into subsets of a Banach space admits an absolutely continuous selection.     

szasz-mirakyan算子对具有p次变差函数的收敛性

设f是定义于[0,∞)上的函数,则Szasz-Mirakyan算子S_n(f,x)定义如下 这里 Szaszr、Grof和Hermann等人研究过算子(1.1)的收敛性。最近,FuhuaCheng对在[0,∞)的每一有限子区间上具有有界变差的函数研究了算子(1.1)收敛于1/2[f(x+)+f(x-)]的速度,证明了下述的 定理A 设f是在[0,∞)的每一有限子区间上具有有界变差的函数且对某个α>0,f(t)=O(t~(at))(t→∞),若x∈(0,∞)是一无理数,则对于充分大的n,我们有     

Singularity of generalized grey Brownian motions with different parameters

In this note we prove that the probability measures generated by two generalized grey Brownian motions with different parameters are singular with respect to each other. This result can be interpreted as an extension of the Feldman–Hájek dichotomy of Gaussian measures to a family of non-Gaussian measures.