Estimating the Copula of a class of Time-Changed Brownian Motions: A non-parametric Approach

A class of copulae associated with Brownian motion processes and their maxima

The main objective of this paper consists in creating a new class of copulae from various joint distributions occurring in connection with certain Brownian motion processes. We focus our attention on the distributions of univariate Brownian motions having a drift parameter and their maxima and on correlated bivariate Brownian motions by considering the maximum value of one of them. The copulae generated therefrom and their associated density functions are explicitly given as well as graphically represented.

二元时变L{é}vy模型的级数表示

Series representations for bivariate time-changed L{é}vy models

在本文中，我们分析了一个L{é}vy模型的基础上两个流行的概念-从属和L{é}vy copula。更精确地说，我们考虑了一个二维L{é}vy过程，使得每个分量是一个时变（从属）布朗运动，从属运动之间的依赖关系通过某个L{é}vy copula来描述.我们证明了我们的模型，可以有效地用于模拟目的的一系列表示，并提供了一些基于真实数据的实际例子

On the long range dependence of time-changed mixed fractional Brownian motion model

A time-changed mixed fractional Brownian motion is an iterated process constructed as the superposition of mixed fractional Brownian motion and other process. In this paper we consider mixed fractional Brownian motion of parameters a, b and H\in(0, 1) time-changed by two processes, gamma and tempered stable subordinators. We present their main properties paying main attention to the long range dependence. We deduce that the fractional Brownian motion time-changed by gamma and tempered stable subordinators has long range dependence property for all H\in(0, 1).

On the fractional mixed fractional Brownian motion Time Changed by Inverse alpha Stable Subordinator

A time-changed fractional mixed fractional Brownian motion by inverse alpha stable subordinator with index alpha in (0, 1) is an iterated process L constructed as the superposition of fractional mixed fractional Brownian motion N(a, b) and an independent inverse α-stable subordinator Talpha. In this paper we prove that the process LT alpha(a, b) is of long range dependence property under a smooth condition on the Hirsh index H1 and H2. We deduce that the fractional mixed fractional Brownian motion has long range dependence for every H1 < H2.

低频Lévy Copula估计

Low Frequency Lévy Copula Estimation

设$X$是一个$d$ -维Lévy过程，其中有Lévy三元组$(Σ,ν,α)$和$d\geq 2$ .在给定低频观测值$(X_t)_{t=1,\ldots,n}$的情况下，估计了$X$跳变的相关结构。Lévy测度$ν$描述了时间单位内的平均跳跃行为。因此，目的是通过估计$ν$的Lévy copula $\mathfrak{C}$来估计$ν$的依赖结构，参见。Kallsen和Tankov \cite{KalTan}。我们使用Neumann和Reiffel \cite{NeuRei}以及Nickl和Reiffel \cite{NicRei}中提出的一维设置中的低频技术，基于上述$n$观测值构造Lévy copula估计量$\widehat{\mathfrak{C}}_n$。在这样做，我们证明了$$\widehat{\mathfrak{C}}_n\to \mathfrak{C},\quad n\to\infty$$一致的紧集有界远离零的收敛速度$\sqrt{\log n}$。这种收敛在相当一般的假设下成立，其中也包括具有任意Blumenthal-Getoor指数$0\leqβ\leq 2$的$Σ\neq 0$和$ν$的Lévy三元组。注意，在低频观测方案中，统计上很难区分无限多个小跳跃和布朗运动部分。因此，相当慢的收敛速度$\sqrt{\log n}$并不奇怪。在复合泊松过程（CPP）的…