Nguyen Tien Dung
In this note, we investigate the continuity in law with respect to the Hurst index of the exponential functional of the fractional Brownian motion. Based on the techniques of Malliavin's calculus, we provide an explicit bound on the Kolmogorov distance between two functionals with different Hurst indexes.
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On the Kolmogorov equation associated with Volterra equations and Fractional Brownian Motion
本文研究由Riemann-Liouville型分数布朗运动扰动的Volterra方程所对应的SPDE，通过扩展漂移算子及其Fréchet微分证明Markov随机流的二次可微性，并据此在无穷维微积分意义下求解路径依赖Kolmogorov方程，同时指出温和形式分析中的未解障碍。
We consider the persistence probability for the integrated fractional Brownian motion and the fractionally integrated Brownian motion with parameter $H,$ respectively. For the integrated fractional Brownian motion, we discuss a conjecture of Molchan and Khokhlov and determine the asymptotic behavior of the persistence exponent as $H\to 0$ and $H\to 1,$ which is in accordance with the conjecture. For the fractionally integrated Brownian motion, also called Riemann-Liouville process, we find the asymptotic behavior of the persistence exponent as $H\to 0$.
Fluctuations of the power variation of fractional Brownian motion in Brownian time
研究了分数阶布朗运动的幂变分在布朗时间中的涨落
Asymptotic behavior of weighted power variations of fractional Brownian motion in Brownian time
研究了分数阶布朗运动在布朗时间Z_t：= X_{Y_t}，t >= 0中加权幂变分的渐近性态，其中X是分数阶布朗运动，Y是独立的布朗运动.
Formalization of Brownian motion in Lean
本文在Lean中形式化了布朗运动，基于测度论基础，开发了扩展定理、高斯测度和路径连续性定理。
The Multiparameter Fractional Brownian Motion
定义并研究了多参数分数布朗运动。当放松独立性条件时，该过程是经典分数布朗运动和多参数布朗运动的推广。讨论了它与Lévy分数布朗运动和分数布朗单的关系。研究了多参数过程增量平稳性的不同概念，并将其应用于分式性质。利用自相似性，我们提出了这样的过程的表征。最后，分析了多参数分数布朗运动沿增路径的行为。
A strong uniform approximation of sub-fractional Brownian motion
亚分数阶布朗运动是一个类似于分数阶布朗运动但没有平稳增量的过程。在\cite{GGL1}中，我们利用迁移过程证明了分数阶布朗运动的一个强一致逼近的收敛速率。本文证明了亚分数布朗运动的一个类似的逼近。
We obtain a Berry-Esseen type bound for the distribution of the maximum likelihood estimator of the drift parameter for fractional Ornstein-uhlenbeck type process driven by sub-fractional Brownian motion.
Convergence to Weighted Fractional Brownian Sheets
定义了加权分数布朗单，这是一类含四个参数的高斯随机场，并给出了它的一些性质。我们表明，对于某些值的参数的加权分数布朗单的占有时间波动的随机粒子模型的法律得到的限制。与一些已知的分数布朗单的近似，其中使用的分数布朗运动的Volterra型整表示中的核相对于普通的布朗运动，我们的近似不使用核。
We prove a non-central limit theorem for the symmetric weighted odd-power variations of the fractional Brownian motion with Hurst parameter H< 1/2. As applications, we study the asymptotic behavior of the trapezoidal weighted odd-power variations of the fractional Brownian motion and the fractional Brownian motion in Brownian time Z_t:= X_{Y_t}, t >= 0, where X is a fractional Brownian motion and Y is an independent Brownian motion.
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 Levy 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, …
Stochastic delay fractional evolution equations driven by fractional Brownian motion
考虑了希尔伯特空间中一类由分数阶布朗运动驱动的随机时滞分数阶发展方程.得到了该方程的弱解存在唯一的充分条件。应用随机分数阶热方程来说明理论。
The $n$th order fractional Brownian motion was introduced by Perrin et al. It is the (upto a multiplicative constant) unique self-similar Gaussian process with Hurst index $H \in (n-1,n)$, having $n$th order stationary increments. We provide a transfer principle for the $n$th order fractional Brownian motion, i.e., we construct a Brownian motion from the $n$the order fractional Brownian motion and then represent the $n$the order fractional Brownian motion by using the Brownian motion in a non-anticipative way so that the filtrations of the $n$the order fractional Brownian motion and the associ…
Generalizations of tempered fractional Brownian from single index to two indices and variable index or tempered multifractional Brownian motion are studied. Tempered fractional Brownian motion and tempered multifractional Brownian motion with variable tempering parameter are considered.
Rate of convergence of Euler approximations of solution to mixed stochastic differential equation involving Brownian motion and fractional Brownian motion
考虑了一类同时含有标准布朗运动和分数阶布朗运动的混合随机微分方程，其参数为$H>1/2$ .得到了该方程解的欧拉近似的均方收敛速率。
We study a kinetic stochastic model with a non-linear time-inhomogeneous drag force and a Brownian-type random force. More precisely, the Kolmogorov type diffusion $(V,X)$ is considered: here $X$ is the position of the particle and $V$ is its velocity and is solution of a stochastic differential equation driven by a one-dimensional Brownian motion, with the drift of the form $t^{-β}F(v)$. The function $F$ satisfies some homogeneity condition and $β$ is positive. The behaviour of the process $(V,X)$ in large time is proved by using stochastic analysis tools.
A random walk approximation to fractional Brownian motion
我们提出了分数布朗运动的随机游走逼近，其中分数随机游动的增量被定义为伯努利随机游动过去增量的加权和。
Fractional Brownian motion in a nutshell
这是第七届Jagna国际研讨会的一门分数布朗运动迷你课程的讲义的延伸版。
In this paper, inspired by the fractional Brownian sheet of Riemann-Liouville type, we introduce the operator fractional Brownian sheet of Riemman-Liouville type, and study some properties of it. We also present an approximation in law to it based on the martingale differences.
On the sub-mixed fractional Brownian motion
设${S_t^H, t \geq 0} $是一个布朗运动和一个独立的次分数布朗运动的线性组合，其Hurst指数为$0 < H < 1$ .研究了它的主要性质，证明了$S^H $可以看作是介于亚分数布朗运动和混合分数布朗运动之间的中间过程.最后，我们确定了$H$的值，其中$S^H$不是半鞅。
We prove limit theorems for the weighted quadratic variation of trifractional Brownian motion and $n$-th order fractional Brownian motion. Furthermore, a sufficient condition for the $L^P$-convergence of the weighted quadratic variation for Gaussian processes is obtained as a byproduct. As an application, we give a statistical estimator for the self-similarity index of trifractional Brownian motion. These theorems extend results of Baxter, Gladyshev, and Norvaiša.
Stochastic Differential Equations Driven by Fractional Brownian Motion and Standard Brownian Motion
证明了由Hurst参数H> 1/2的多维分数布朗运动和多维标准布朗运动同时驱动的多维时变随机微分方程解的存在唯一性定理.证明依赖于一些先验估计，这是使用分数积分的方法，和经典的伊藤随机分析。存在性结果基于Yamada-Watanabe定理。
Flow properties of differential equations driven by fractional Brownian motion
证明了$H>1/2$的分数阶布朗运动驱动的随机微分方程的解定义了$\mathbb{R}^{d}$上的同胚流.
We consider the stochastic continuity equation perturbed by a fractional Brownian motion and the drift is allowed to be discontinuous. We show that for almost all paths of the fractional Brownian motion there exists a solution to the equation.
A Characterization of the Set-indexed Fractional Brownian Motion by Increasing Paths
证明了一个集指标过程是集指标分数布朗运动当且仅当它在所有增路上的投影是单参数时变分数布朗运动.作为应用，我们给出了这类过程的整表示.
Operator Fractional Brownian Motion and Martingale Differences
众所周知，鞅差序列在理论和应用上都是非常有用的。另一方面，分数阶布朗运动作为分数阶布朗运动的推广，在理论和应用上也有着重要的意义。在本文中，我们研究了它们之间的关系。我们将构造一个基于鞅差序列的算子分式布朗运动的逼近序列。
Large deviation for slow-fast McKean-Vlasov stochastic differential equations driven by fractional Brownian motions and Brownian motions
本文研究了由布朗运动和分数布朗运动驱动的慢-快McKean-Vlasov随机微分方程，提出乘积空间上的大偏差原理新定义，并在适当假设下利用弱收敛方法证明了该方程的LDP。
Hausdorff Dimension of the Record Set of a Fractional Brownian Motion
证明了具有Hurst参数$H$的分数阶布朗运动的记录集的豪斯多夫维数等于$H$ .
Approximation of fractional Brownian motion by martingales
研究了分数阶布朗运动的鞅最优逼近问题。证明了在特定意义下存在唯一的最接近分数布朗运动的鞅。证明了该鞅具有特定的形式。给出了有关逼近问题的数值结果。
Estimates for moments of supremum of reflected fractional Brownian motion
设$B_H(\cdot)$是具有Hurst参数$H\in(0,1]$的分数布朗运动.受分数布朗运动极大不等式应用的启发，本文给出了K_T（H，γ）：=E[ \sup_{t\in[0,T]}]的界|B_H（t）|]^γ，与$γ, T>0$。
We study the so-called multi-mixed fractional Brownian motions (mmfBm) and multi-mixed fractional Ornstein--Ulhenbeck (mmfOU) processes. These processes are constructed by mixing by superimposing (infinitely many) independent fractional Brownian motions (fBm) and fractional Ornstein--Uhlenbeck processes (fOU), respectively. We prove their existence as $L^2$ processes and study their path properties, viz. long-range and short-range dependence, Hölder continuity, $p$-variation, and conditional full support.
We study a modification of the fractional analogue of the Brownian meander, which is Brownian motion conditioned to be positive on the time interval ${[0,1]}$. More precisely, we determine the weak limit of a fractional Brownian motion which is penalized -- instead of being killed -- when leaving the positive half-axis. In the Brownian case, we give a representation of the limiting process in terms of an explicit SDE and compare it to the SDE fulfilled by the Brownian meander.
A smooth component of the fractional Brownian motion and optimal portfolio selection
考虑Hurst参数为（1/2，1）的分数阶布朗运动.我们发现分数阶布朗运动的增量可以表示为两个独立的高斯过程之和，其中一个过程是光滑的，因为它在均方上是可微的。我们考虑分数阶布朗运动和由Riemann和生成的随机积分。作为应用的一个例子，这一结果被用来找到一个最佳的预编程策略的均值-方差设置的Bachelier型市场模型驱动的分数布朗运动。
On the exponential integrability of the derivative of intersection and self-intersection local time for fractional Brownian motion and a limit theorem related to the self-intersection local time for fractional Brownian motion
该论文修正了分数布朗运动交局部时导数存在的条件，证明了其指数可积性，并给出了存在条件违反时自交局部时导数的极限定理。
Convergence in law to operator fractional Brownian motions
本文给出了算子分式布朗运动律的两个近似。一个是由Poisson过程构造的，另一个是Taqqu（1975）的一个结果的推广.
A fractional Brownian field indexed by $L^2$ and a varying Hurst parameter
利用抽象Wiener空间的结构，我们定义了一个分数布朗场，它由积空间$(0,1/2] \times L^2(T,m)$，$(T,m)$，一个可分测度空间所索引，其中第一个坐标对应于分数布朗运动的Hurst参数.这个领域包含了一个大类现有的分数布朗过程，如Lévy分数布朗运动和多参数分数布朗运动，并提供了一个新的设置。我们证明了它在两个坐标下都具有令人满意的增量方差，并得到了与度量熵有关的某些连续性和Hölder正则性。同时，给出了小球概率的精确估计，推广了Lévy分数布朗运动的一个结果.然后，我们将这些一般结果应用到多参数和集索引过程，证明了一般索引集合上具有指定局部Hölder正则性的过程的存在性。
A Random Matrix Approximation for the Non-commutative Fractional Brownian Motion
得到了矩阵分式布朗运动特征值的经验测度值过程的泛函极限定理。证明了极限测度值过程是Nourdin和Taqqu最近引入的非对易分数布朗运动. Young和Skorohod随机积分技术和分数阶微积分是使用的主要工具。
In this paper, we consider the drawdown and drawup of the fractional Brownian motion with trend, which corresponds to the logarithm of geometric fractional Brownian motion representing the stock price in financial market. We derive the asymptotics of tail probabilities of the maximum drawdown and maximum drawup as the threshold goes to infinity, respectively. It turns out that the extremes of drawdown leads to new scenarios of asymptotics depending on Hurst index of fractional Brownian motion.
Heavy-tailed fractional Pearson diffusions
我们定义了重尾分数倒数伽玛和Fisher-Snedecor扩散的非马尔可夫时变相应的皮尔逊扩散。皮尔逊扩散是由具有空间变化多项式系数的后向Kolmogorov方程控制的，并且在应用中被广泛使用。相应的分数倒数伽玛和Fisher-Snedecor扩散由分数向后Kolmogorov方程控制，并在稳态下具有重尾边缘分布。我们得到了分数阶倒向Kolmogorov方程的分数阶倒数伽玛扩散和Fisher-Snedecor扩散的转移密度的显式表达式以及相应的Cauchy问题的强解.
Second order limit laws for occupation times of the fractional Brownian motion
利用矩量法证明了具有Hurst指标$H=\frac{1}{d}$的$d$维分数布朗运动的（可加）泛函的二阶极限定律，推广了Kallianpur-Robbins定律.
Non-degeneracy of some Sobolev Pseudo-norms of fractional Brownian motion
利用分数布朗运动（fBm）的小$L^{2}$球概率的上界估计，证明了fBm的某些Sobolev伪范数的非退化性.
We introduce a generalized mixed fractional Brownian motion (gmfBm) as a linear combination of two independent fractional Brownian motions with possibly different Hurst indices and investigate conditions under which the time-changed gmfBm exhibit long range dependence when the time-change is induced by a tempered stable subordinator or a Gamma process.
Gradient Bounds for Solutions of Stochastic Differential Equations Driven by Fractional Brownian Motions
研究了由分数阶布朗运动驱动的随机微分方程解的分布所满足的函数不等式。在分数阶布朗运动的道路空间上，通过新的分部分部积分法公式得到了这类函数不等式。
Example of a Gaussian self-similar field with stationary rectangular increments that is not a fractional Brownian sheet
本文研究各向异性自相似随机场，特别是分数布朗单。这个高斯场是分数布朗运动的推广。本文证明了矩形增量自相似场的协方差函数的一些性质。利用Lamperti变换，我们得到了相应平稳场的协方差函数的性质。我们提出了一个例子，高斯自相似场与固定的矩形增量，这是不是一个分数布朗单。
Mixed Sub-Fractional Brownian Motion
引入了亚分数布朗运动的一个新扩展，从而也是布朗运动的一个新扩展。它是有限个次分数布朗运动的线性组合，我们称之为混合次分数布朗运动。本文研究了这一过程的一些基本性质，它的非马氏性和非平稳性，它是半鞅的条件，以及它的样本路径的主要特征。我们还表明，这个过程可以用来得到一个很好的模型，某些现象，不仅考虑的符号（如在亚分数布朗运动的情况下），但也考虑到这种现象的增量之间的依赖强度。
Stochastic Volterra equations driven by fractional Brownian motion with Hurst parameter H > 1/2
本文证明了Hurst参数H > 1/2的分数阶布朗运动驱动的随机Volterra积分方程解的存在唯一性，并证明了解具有有限矩.分数阶布朗运动的随机积分是一个路径方向的黎曼-斯蒂尔切斯积分。
Brownian and fractional Brownian stochastic currents via Malliavin calculus
利用Malliavin积分和多重Wiener-Itô积分，研究了定义为Skorohod（散度）积分的随机流关于布朗运动和分数阶布朗运动的存在性和正则性.我们还考虑了多维多参数的情况下，我们比较的规则性的电流分布在负索伯列夫空间与其规律性的渡边空间。
The Nyström method for functional quantization with an application to the fractional Brownian motion
在这篇文章中，所谓的“Nyström方法”被用来计算高斯过程的最佳量化器。特别是，我们通过逼近分数布朗运动的卡胡南-洛夫分解的首项来推导分数布朗运动的最佳量化。利用分数阶布朗运动对“功能分层”方差缩减算法进行了数值试验。
Cylindrical Fractional Brownian Motion in Banach Spaces
本文在巴拿赫空间中引入柱分式布朗运动，并发展了相关的随机积分理论。在这里，一个圆柱形分数布朗运动理解的经典框架内的圆柱形随机变量和柱测度。发展的确定性算子值被积函数的随机积分是基于圆柱分数布朗运动的一系列表示，这是类似于真正的随机过程的Karhunen-Loève展开。最后，我们将所得结果应用于巴拿赫空间中由柱分式布朗运动驱动的抽象随机Cauchy问题。
Bismut formulae and applications for stochastic (functional) differential equations driven by fractional Brownian motions
利用Malliavin演算，建立了一类由分数阶布朗运动驱动的随机（泛函）微分方程的Bismut导数公式.作为应用，给出了Harnack型不等式和强Feller性质.
Controllability of fractional stochastic neutral functional differential equations driven by fractional Brownian motion with infinite delay
研究了实可分希尔伯特空间中由分数阶布朗运动驱动的无穷时滞分数阶中立型随机泛函微分方程的能控性.利用随机分析和一个不动点策略得到了系统的能控性结果。最后通过一个算例验证了理论结果的有效性。
Central limit theorem for functionals of two independent fractional Brownian motions
利用矩量法证明了$(\frac{2}{d+1},\frac{2}{d})$中具有相同Hurst指标$H$的两个独立$d$维分数布朗运动泛函的中心极限定理.
Stochastic differential equation involving Wiener process and fractional Brownian motion with Hurst index $H> 1/2$
考虑由可能相依的分数阶布朗运动和布朗运动驱动的混合型随机微分方程。在对方程系数作较弱的正则性假设下，证明了该方程存在唯一解。
Harnack Inequalities and Applications for Stochastic Differential Equations Driven by Fractional Brownian Motion
本文建立了分数阶布朗运动驱动的带Hurst参数$H<1/2$的随机微分方程的Harnack不等式.作为应用，给出了强Feller性质、log-Harnack不等式和熵-代价不等式.
Smoothing effect of rough differential equations driven by fractional Brownian motions
本文研究了含参数$H>1/4$的分数布朗运动驱动的粗糙微分方程的光滑效应。我们得到的正则化估计推广到分数布朗运动Kusuoka和Stroock以前的结果，可以看作是一个定量版本的存在性光滑密度下Hörmander的类型的条件。
Wasserstein Convergence for Empirical Measures of Subordinated Fractional Brownian Motions on the Flat Torus
该研究估计平坦环面上从属分数布朗运动经验测度在所有p≥1的Wasserstein距离下的收敛速率，结果与现有相关研究一致，并拓展至时间离散化情形。
Local times of stochastic differential equations driven by fractional Brownian motions
本文研究了分数阶布朗运动驱动的随机微分方程局部时的存在性和（Hölder）正则性.特别地，我们证明了在一维和粗糙的情况下H<1/2，当地时间的Hölder指数（在t）是1-H，其中H是驱动分数布朗运动的Hurst参数。
An Itô-Wentzell formula for the fractional Brownian motion
本文证明了分数布朗运动的Itô-Wentzell公式，并以此推导出由该过程驱动的一类随机微分方程的存在性与唯一性结果。
A short note on a class of statistics for estimation of the Hurst index of fractional Brownian motion
基于离散轨道的二阶增量，提出了一类适合于分数布朗运动Hurst指数估计的统计量。
Small-time kernel expansion for solutions of stochastic differential equations driven by fractional Brownian motions
本文证明了在一定的假设条件下，对于具有Hurst参数$H>1/2$的$d$维分数布朗运动，由它驱动的随机微分方程的解的密度具有类似于布朗运动情形的短时展开.
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