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Error estimates of a finite element method for stochastic time-fractional evolution equations with fractional brownian motion

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International Journal of Modeling, Simulation, and Scientific Computing 2022年第1期13卷 192-213页

作者： Jingyun Lv Key Laboratory of Systems and Control Institute of Systems Science Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 P.R.China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing 100049 P.R.China

The aim of this paper is to consider the convergence of the numerical methods for stochastic time-fractional evolution equations driven by fractional brownian *** spatial and temporal regularity of the mild solution is *** numerical scheme approximates the problem in space by the Galerkin finite element method and in time by the backward Euler convolution quadrature formula,and the noise by the L *** strong convergence error estimates for both semi-discrete and fully discrete schemes are established.A numerical example is presented to verify our theoretical analysis.

关键词： Stochastic time-fractional evolution equations finite element method error estimates fractional brownian motion

fractional brownian motion and Sheet as White Noise Functionals

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Acta Mathematica Sinica,English Series 2006年第4期22卷 1183-1188页

作者： Zhi Yuan HUANG Chu Jin LI Jian Ping WAN Ying WU Department of Mathematics Huazhong University of Science and Technology Wuhan 430074 P. R. China

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.

关键词： fractional brownian motion fractional white noise functionals fractional Girsanov formula Wick product

An Optimal Portfolio Problem Presented by fractional brownian motion and Its Applications

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Wuhan University Journal of Natural Sciences 2022年第1期27卷 53-56页

作者： YAN Li School of Mathematical Sciences Chongqing Normal UniversityChongqing 401331China

We use the dynamic programming principle method to obtain the Hamilton-Jacobi-Bellman(HJB)equation for the value function,and solve the optimal portfolio problem explicitly in a Black-Scholes type of market driven by fractional brownian motion with Hurst parameter H∈(0,1).The results are compared with the corresponding well-known results in the standard Black-Scholes market(H=1/2).As an application of our proposed model,two optimal problems are discussed and solved,analytically.

关键词： fractional brownian motion Merton’s optimal problem stochastic differential equation

Stochastic process-based degradation modeling and RUL prediction: from brownian motion to fractional brownian motion

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Science China(Information Sciences) 2021年第7期64卷 5-24页

作者： Hanwen ZHANG Maoyin CHEN Jun SHANG Chunjie YANG Youxian SUN School of Automation and Electrical Engineering University of Science and Technology Beijing State Key Laboratory of Industrial Control Technology College of Control Science and EngineeringZhejiang University Department of Automation Tsinghua University School of Automation and Electrical Engineering Linyi University Department of Electrical and Computer Engineering University of Alberta

brownian motion(BM) has been widely used for degradation modeling and remaining useful life(RUL) prediction, but it is essentially Markovian. This implies that the future state in a BM-based degradation process relies only on its current state, independent of the past states. However, some practical industrial devices such as Li-ion batteries, ball bearings, turbofans, and blast furnace walls show degradations with long-range dependence(LRD), where the future degradation states depend on both the current and past degradation states. This type of degradation naturally brings two interesting problems, that is, how to model the degradations and how to predict their RULs. Recently, in contrast to the work that uses only BM, fractional brownian motion(FBM) is introduced to model practical degradations. The most important feature of the FBM-based degradation models is the ability to characterize the non-Markovian degradations with LRD. Although FBM is an extension of BM, it is neither a Markovian process nor a semimartingale. Therefore, how to obtain the first passage time of an FBM-based degradation process has become a challenging task. In this paper, a review of the transition of RUL prediction from BM to FBM is provided. The peculiarities of FBM when addressing the LRD inherent in some practical degradations are discussed. We first review the BM-based degradation models of the past few decades and then give details regarding the evolution of FBM-based research. Interestingly, the existing BM-based models scarcely consider the effect of LRD on the prediction of RULs. Two practical cases illustrate that the newly developed FBMbased models are more generalized and suitable for predicting RULs than the BM-based models, especially for degradations with LRD. Along with the direction of FBM-based RUL prediction, we also introduce some important and interesting problems that require further study.

关键词： remaining useful life degradation model brownian motion fractional brownian motion long-range dependence

Dimensional Properties of fractional brownian motion

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Acta Mathematica Sinica,English Series 2007年第4期23卷 613-622页

作者： Dong Sheng WU Yi Min XIAO Department of Statistics and Probability A-413 Wells Hall Michigan State University East Lansing MI 48824 USA

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 result... 详细信息

关键词： fractional brownian motion Hausdorff dimension uniform dimension results strong local nondeterminism

Neutral stochastic delay partial functional integro-differential equations driven by a fractional brownian motion

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Frontiers of Mathematics in China 2013年第4期8卷 745-760页

作者： Tomas CARABALLO Mamadou Abdoul DIOP Dpto. Ecuaciones Diferenciales y Analisis Numerico Universidad de Sevilla Apdo. de Correos 1160 41080-Sevilla Spain Universite Gaston Berger de Saint-Louis UFR SAT Departement de Mathematiques 234 Saint-Louis Senegal

This paper deals with the existence and uniqueness of mild solutions to neutral stochastic delay functional integro-differential equations perturbed by a fractional brownian motion BH, with Hurst parameter H E （1/2, 1）. We use the theory of resolvent operators developed by R. Grimmer to show the existence of mild solutions. An example is provided to illustrate the results of this work.

关键词： Resolvent operator C0-semigroup Wiener process mild solution fractional brownian motion

CONTROLLABILITY OF NEUTRAL STOCHASTIC EVOLUTION EQUATIONS DRIVEN BY fractional brownian motion

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Acta Mathematica Scientia 2017年第1期37卷 108-118页

作者：崔静闫理坦 Department of Statistics Anhui Normal University Department of Mathematics Donghua University

In this paper,we investigate the controllability for neutral stochastic evolution equations driven by fractional brownian motion with Hurst parameter H ∈（1/2,1） in a Hilbert *** employ the α-norm in order to reflect the relationship between H and the fractional power α.Sufficient conditions are established by using stochastic analysis theory and operator *** example is provided to illustrate the effectiveness of the proposed result.

关键词： stochastic evolution equations fractional brownian motion controllability

Stochastic Volterra equations driven by fractional brownian motion

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Frontiers of Mathematics in China 2015年第3期10卷 595-620页

作者： Xiliang FAN Department of Statistics Anhui Normal University Wuhu 241003 China School of Mathematical Sciences Beijing Normal University Beijing 100875 China

This paper is devoted to study a class of stochastic Volterra equations driven by fractional brownian motion. We first prove the Driver type integration by parts formula and the shift Harnack type inequalities. As a direct application, we provide an alternative method to describe the regularities of the law of the solution. Secondly, by using the Malliavin calculus, the Bismut type derivative formula is established, which is then applied to the study of the gradient estimate and the strong Feller property. Finally, we establish the Talagrand type transportation cost inequalities for the law of the solution on the path space with respect to both the uniform metric and the L^2-metric.

关键词： fractional brownian motion derivative formula integration byparts formula stochastic Volterra equation Malliavin calculus

Some limit results on supremum of Shepp statistics for fractional brownian motion

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Applied Mathematics(A Journal of Chinese Universities) 2016年第3期31卷 269-282页

作者： TAN Zhong-quan CHEN Yang School of Mathematical Sciences Zhejiang University College of Mathematics Physics and Information Engineering Jiaxing University School of Mathematics and Physics Suzhou University of Science and Technology

Define the incremental fractional brownian field ZH（τ, s） = BH（s＋τ） -BH（s）,where BH（s） is a standard fractional brownian motion with Hurst parameter H ∈ （0, 1）. Inthis paper, we first derive an exact asymptotic of distribution of the maximum MH（Tu） =supτ∈[0,1],s∈[0,xτu] ZH（τ, s）, which holds uniformly for x ∈ [A, B] with A, B two positive con-stants. We apply the findings to analyse the tail asymptotic and limit theorem of MH （τ） witha random index τ. In the end, we also prove an almost sure limit theorem for the maximum M1/2（τ） with non-random index T.

关键词： Extremes Shepp statistics fractional brownian motion exact asymptotic almost sure limit theorem