共查询到20条相似文献,搜索用时 31 毫秒
1.
Stochastic calculus with respect to fractional Brownian motion (fBm) has attracted a lot of interest in recent years, motivated in particular by applications in finance and Internet traffic modeling. Multifractional Brownian motion (mBm) is an extension of fBm enabling to control the local regularity of the process. It is obtained by replacing the constant Hurst parameter H of fBm by a function h(t), thus allowing for a finer modelling of various phenomena.In this work we extend to mBm the construction of the Wick–Itô stochastic integral with respect to fBm, as originally proposed in Bender (Stoch. Process. Appl. 104 (2003), pp. 81–106), Bender (Bernouilli 9(6) (2003), pp. 955–983), Biagini et al. (Proceedings of Royal Society, special issue on stochastic analysis and applications, 2004, pp. 347–372) and Elliott and Van der Hoek (Math. Finance 13(2) (2003), pp. 301–330). In that view, a multifractional white noise is defined and used to integrate with respect to mBm a large class of stochastic processes using Wick products. Itô formulas (both for tempered distributions and for functions with sub-exponential growth) are obtained, as well as a Tanaka Formula. 相似文献
2.
Let H be a Hilbert space and E a Banach space. We set up a theory of stochastic integration of ℒ(H,E)-valued functions with respect to H-cylindrical Liouville fractional Brownian motion with arbitrary Hurst parameter 0 < β < 1. For 0 < β < ? we show that a function Φ: (0, T) → ℒ(H,E) is stochastically integrable with respect to an H-cylindrical Liouville fractional Brownian motion if and only if it is stochastically integrable with respect to an H-cylindrical fractional Brownian motion. 相似文献
3.
Dong Sheng WU Yi Min XIAO 《数学学报(英文版)》2007,23(4):613-622
Let B^α = {B^α(t),t E R^N} be an (N,d)-fractional Brownian motion with Hurst index α∈ (0, 1). By applying the strong local nondeterminism of B^α, we prove certain forms of uniform Hausdorff dimension results for the images of B^α when N 〉 αd. Our results extend those of Kaufman for one-dimensional Brownian motion. 相似文献
4.
Jean-Christophe Breton Jean-Fran?ois Coeurjolly 《Statistical Inference for Stochastic Processes》2012,15(1):1-26
In this paper, we show how concentration inequalities for Gaussian quadratic form can be used to propose confidence intervals
of the Hurst index parametrizing a fractional Brownian motion. Both cases where the scaling parameter of the fractional Brownian
motion is known or unknown are investigated. These intervals are obtained by observing a single discretized sample path of
a fractional Brownian motion and without any assumption on the Hurst parameter H. 相似文献
5.
In this article, we study the existence of mild solutions to stochastic impulsive evolution equations with time delays, driven by fractional Brownian motion with the Hurst index H > 1/2 via a new fixed point analysis approach. 相似文献
6.
In this article, we study the rate of convergence of the polygonal approximation to multiple stochastic integral Sp (f) of fractional Brownian motion of Hurst parameter H 〈 1/2 when the fractional Brownian motion is replaced by its polygonal approximation. Under different conditions on f and for different p, we obtain different rates. 相似文献
7.
We give a result of stability in law of the local time of the fractional Brownian motion with respect to small perturbations
of the Hurst parameter. Concretely, we prove that the law (in the space of continuous functions) of the local time of the
fractional Brownian motion with Hurst parameter H converges weakly to that of the local time of , when H tends to H
0.
相似文献
8.
Michael Levine Soledad Torres Frederi Viens 《Statistical Inference for Stochastic Processes》2009,12(3):221-250
This paper investigates several strategies for consistently estimating the so-called Hurst parameter H responsible for the long-memory correlations in a linear class of ARCH time series, known as LARCH(∞) models, as well as
in the continuous-time Gaussian stochastic process known as fractional Brownian motion (fBm). A LARCH model’s parameter is
estimated using a conditional maximum likelihood method, which is proved to have good stability properties. A local Whittle
estimator is also discussed. The article further proposes a specially designed conditional maximum likelihood method for estimating
the H which is closer in spirit to one based on discrete observations of fBm. In keeping with the popular financial interpretation
of ARCH models, all estimators are based only on observation of the “returns” of the model, not on their “volatilities”. 相似文献
9.
10.
This paper deals with the problem of estimating the parameters for fractional Ornstein–Uhlenbeck processes from discrete observations when the Hurst parameter H is known. Both the drift and the diffusion coefficient estimators of discrete form are obtained based on approximating integrals via Riemann sums with Hurst parameter H ∈ (1/2, 3/4). By adapting the stochastic integral representation to the fractional Brownian motion, these two estimators can be efficiently computed by the use of computer software. Numerical examples are presented to examine the performance of our method. An application to real data is also presented to show how to apply this method in practice. 相似文献
11.
T. Sottinen 《Journal of Theoretical Probability》2004,17(2):309-325
We consider Gaussian processes that are equivalent in law to the fractional Brownian motion and their canonical representations. We prove a Hitsuda type representation theorem for the fractional Brownian motion with Hurst index H1/2. For the case H>1/2 we show that such a representation cannot hold. We also consider briefly the connection between Hitsuda and Girsanov representations. Using the Hitsuda representation we consider a certain special kind of Gaussian stochastic equation with fractional Brownian motion as noise. 相似文献
12.
Mario Abundo 《随机分析与应用》2018,36(1):181-187
For a time-homogenous one-dimensional diffusion process X(t), we investigate the distribution of the first instant, after a given time r, at which X(t) exceeds its maximum in the interval [0, r], generalizing a result of Papanicolaou, holding for Brownian motion. 相似文献
13.
We study a mixed financial market with risky asset governed by both the standard Brownian motion and the fractional Brownian
motion with Hurst index
H ? (\frac12, 1){H\in(\frac12, 1)}. We use representations of Hitsuda and Cheridito for the mixed Brownian and fractional Brownian process and present the solution
of the problem of efficient hedging for
H ? (\frac34, 1){H\in(\frac34, 1)}. To solve the problem for
H ? (\frac12, 1){H\in(\frac12, 1)} and to avoid some computational difficulties, we introduce the approximate incomplete semimartingale market, and the solution
of the approximate problem of efficient hedging is considered. Then we pass to the limit and observe the asymptotic behavior
of the solution of the efficient hedging problem. 相似文献
14.
Albert Benassi Pierre Bertrand Serge Cohen Jacques Istas 《Statistical Inference for Stochastic Processes》2000,3(1-2):101-111
We propose a semi-parametric estimator for a piece-wise constant Hurst coefficient of a step fractional Brownian motion (SFBM). For the applications, we want to detect abrupt changes of the Hurst index (which represents long-range correlation) for a Gaussian process with a.s. continuous paths. The previous model of multifractional Brownian motion give a.s. discontinuous paths at change times of the Hurst index. Thus, we first propose a new kind of Fractional Brownian Motion, the SFBM and prove some (Hölder) continuity results. After, we propose an estimator of the piecewise constant Hurst parameter and prove its consistency. 相似文献
15.
S. Albeverio P. E. T. Jorgensen A. M. Paolucci 《Complex Analysis and Operator Theory》2012,6(1):33-63
Given a fractional Brownian motion (fBm) with Hurst index H ? (0,1){H\in(0,1)} , we associate with this a special family of representations of Cuntz algebras related to frequency domains and wavelets.
Vice versa, we consider a pair of Haar wavelets satisfying some compatibility conditions, and we construct the covariance
functions of fBm with a fixed Hurst index. The Cuntz algebra representations enter the picture as filters of the associated
wavelets. Extensions to q-dependent covariance functions leading to a corresponding fBm process will also be discussed. 相似文献
16.
《随机分析与应用》2013,31(4):815-837
We find the chaos expansion of local time ? T (H)(x,·) of fractional Brownian motion with Hurst coefficient H∈(0,1) at a point x∈R d . As an application we show that when H 0 d<1 then ? T (H)(x,·)∈L 2(μ). Here μ denotes the probability law of B (H) and H 0=max{H 1,…,H d }. In particular, we show that when d=1 then ? T (H)(x,·)∈L 2(μ) for all H∈(0,1). 相似文献
17.
H. Uemura 《Journal of Theoretical Probability》2004,17(2):347-366
We study the Tanaka formula for multidimensional Brownian motions in the framework of generalized Wiener functionals. More precisely, we show that the submartingale U(B
t
–x) is decomposed in the sence of generalized Wiener functionals into the sum of a martingale and the Brownian local time, U being twice of the kernel of Newtonian potential and B
t
the multidimensional Brownian motion. We also discuss on an aspect of the Tanaka formula for multidimensional Brownian motions as the Doob–Meyer decomposition. 相似文献
18.
This paper provides a proof of the fact that asymptotically the R/S statistic and the self-similarity index of fractional Brownian motion agree in the expectation sense. In particular for fractional Gaussian noise time series, the R/S statistic is an estimator of the self-similarity index H. We also show that two other methods for estimating H yield consistent estimators. 相似文献
19.
In this paper, we introduce the linear fractional self-attracting diffusion driven by a fractional Brownian motion with Hurst
index 1/2<H<1, which is analogous to the linear self-attracting diffusion. For 1-dimensional process we study its convergence and the
corresponding weighted local time. For 2-dimensional process, as a related problem, we show that the renormalized self-intersection
local time exists in L
2 if 1/2<H<3/4.
The Project-sponsored by NSFC (10571025) and the Key Project of Chinese Ministry of Education (No.106076). 相似文献
20.
Richard F. Bass Nathalie Eisenbaum Zhan Shi 《Probability Theory and Related Fields》2000,116(3):391-404
Let X be a symmetric stable process of index α∈ (1,2] and let L
x
t
denote the local time at time t and position x. Let V(t) be such that L
t
V(t)
= sup
x∈
ℝ
L
t
x
. We call V(t) the most visited site of X up to time t. We prove the transience of V, that is, lim
t
→∞ |V(t)| = ∞ almost surely. An estimate is given concerning the rate of escape of V. The result extends a well-known theorem of Bass and Griffin for Brownian motion. Our approach is based upon an extension
of the Ray–Knight theorem for symmetric Markov processes, and relates stable local times to fractional Brownian motion and
further to the winding problem for planar Brownian motion.
Received: 14 October 1998 / Revised version: 8 June 1999 / Published online: 7 February 2000 相似文献