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1.
Let B
H
and
be two independent, d-dimensional fractional Brownian motions with Hurst parameter H∈(0,1). Assume d≥2. We prove that the intersection local time of B
H
and
exists in L
2 if and only if Hd<2.
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2.
The fractional Brownian density process is a continuous centered Gaussian
(
d
)-valued process which arises as a high-density fluctuation limit of a Poisson system of independent d-dimensional fractional Brownian motions with Hurst parameter H. (
(
d
) is the space of tempered distributions). The main result proved in the paper is that if the intensity measure of the (initial) Poisson random measure on
d
is either the Lebesgue measure or a finite measure, then the density process has self-intersection local time of order k 2 if and only if Hd < k/(k – 1). The latter is also the necessary and sufficient condition for existence of multiple points of order k for d-dimensional fractional Brownian motion, as proved by Talagrand12. This result extends to a non-Markovian case the relationship known for (Markovian) symmetric -stable Lévy processes and their corresponding density processes. New methods are used in order to overcome the lack of Markov property. Other properties of the fractional Brownian density process are also given, in particular the non-semimartingale property in the case H 1/2, which is obtained by a general criterion for the non-semimartingale property of real Gaussian processes that we also prove. 相似文献
3.
本文利用白噪声分析的方法,讨论了分式布朗运动的局部时,即将其看作一个Hida分布.进一步,给出分式布朗运动的局部时的混沌分解以及局部时平方可积性. 相似文献
4.
We define and prove the existence of a fractional Brownian motion indexed by a collection of closed subsets of a measure space.
This process is a generalization of the set-indexed Brownian motion, when the condition of independance is relaxed. Relations
with the Lévy fractional Brownian motion and with the fractional Brownian sheet are studied. We prove stationarity of the
increments and a property of self-similarity with respect to the action of solid motions. Moreover, we show that there no
“really nice” set indexed fractional Brownian motion other than set-indexed Brownian motion. Finally, behavior of the set-indexed
fractional Brownian motion along increasing paths is analysed.
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5.
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.
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6.
Charles El-Nouty 《Acta Appl Math》2003,78(1-3):103-114
We characterize the lower classes of the fractional integrated fractional Brownian motion by an integral test. 相似文献
7.
Fractional Brownian Motion and Sheet as White Noise Functionals 总被引:1,自引:0,他引:1
Zhi Yuan HUANG Chu Jin LI Jian Ping WAN Ying WU 《数学学报(英文版)》2006,22(4):1183-1188
In this short note, we show that it is more natural to look the fractional Brownian motion as functionals of the standard white noises, and the fractional white noise calculus developed by Hu and Фksendal follows directly from the classical white noise functional calculus. As examples we prove that the fractional Girsanov formula, the Ito type integrals and the fractional Black-Scholes formula are easy consequences of their classical counterparts. An extension to the fractional Brownian sheet is also briefly discussed. 相似文献
8.
Anatoliy Malyarenko 《Journal of Theoretical Probability》2008,21(2):459-475
We derive a series expansion for the multiparameter fractional Brownian motion. The derived expansion is proven to be rate optimal. This work is supported in part by the Foundation for Knowledge and Competence Development and Sparbanksstiftelsen Nya. 相似文献
9.
Vladimir I. Piterbarg 《Extremes》2001,4(2):147-164
We study probabilities of large extremes of the storage process Y(t) = sup
t
(X() - X(t) - c( - t)), where X(t) is the fractional Brownian motion. We derive asymptotic behavior of the maximum tail distribution for the process on fixed or slowly increased intervals by a reduction the problem to a large extremes problem for a Gaussian field. 相似文献
10.
Stochastic Analysis of the Fractional Brownian Motion 总被引:20,自引:0,他引:20
Since the fractional Brownian motion is not a semi-martingale, the usual Ito calculus cannot be used to define a full stochastic calculus. However, in this work, we obtain the Itô formula, the Itô–Clark representation formula and the Girsanov theorem for the functionals of a fractional Brownian motion using the stochastic calculus of variations. 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.
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
13.
14.
Abstract We introduce a class of continuous-time Gaussian processes with stationary increments via moving-average representation with good MA coefficient. The class includes fractional Brownian motion with Hurst index less than 1/2 as a typical example. It also includes processes which have different indices corresponding to the local and long-time properties, repsectively. We derive some basic properties of the processes, and, using the results, we establish a prediction formula for them. The prediction kernel in the formula is given explicitly in terms of MA and AR coefficients. 相似文献
15.
《随机分析与应用》2013,31(6):1577-1607
Abstract Linear and semilinear stochastic evolution equations with additive noise, where the forcing term is an infinite dimensional fractional Brownian motion are studied. Under usual dissipativity conditions the equations are shown to define random dynamical systems which have unique, exponentially attracting fixed points. The results are applied to stochastic parabolic PDE's. They are also applicable to standard finite-dimensional dissipative stochastic equation driven by fractional Brownian motion. 相似文献
16.
A fractional normal inverse Gaussian (FNIG) process is a fractional Brownian motion subordinated to an inverse Gaussian process. This paper shows how the FNIG process emerges naturally as the limit of a random walk with correlated jumps separated by i.i.d. waiting times. Similarly, we show that the NIG process, a Brownian motion subordinated to an inverse Gaussian process, is the limit of a random walk with uncorrelated jumps separated by i.i.d. waiting times. The FNIG process is also derived as the limit of a fractional ARIMA processes. Finally, the NIG densities are shown to solve the relativistic diffusion equation from statistical physics. 相似文献
17.
本文讨论两资产择好期权的定价问题。在风险中性假设下,建立了两资产价格过程遵循分数布朗运动和带非时齐Poisson跳跃—扩散过程的择好期权定价模型,应用期权的保险精算法,给出了相应的择好期权的定价公式。 相似文献
18.
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
19.
《随机分析与应用》2013,31(6):1487-1509
Abstract We apply Grenander's method of sieves to the problem of identification or estimation of the “drift” function for linear stochastic systems driven by a fractional Brownian motion (fBm). We use an increasing sequence of finite dimensional subspaces of the parameter space as the natural sieves on which we maximise the likelihood function. 相似文献
20.
The Generalized Multifractional Brownian Motion 总被引:1,自引:0,他引:1
It is well known that the fractional Brownian motion (FBM) is of great interest in modeling. However, its Hölder is the same all along its path and this restricts its field of application. Therefore, it would be useful to construct a Gaussian process extending the FBM and having a Hölder that is allowed to change. A partial answer to this problem is supplied by the multifractional Brownian motion (MBM); but the Hölder of the MBM must necessarily be continuous and this may be a drawback in some situations. In this paper we construct a Gaussian process generalizing the MBM and having a Hölder that can be a very irregular function. 相似文献