Catherine Donati-Martin, Marc Yor
We give a representation of the Gamma subordinator as a Krein functional of Brownian motion, using the known representations for stable subordinators and Esscher transforms. In particular, we have obtained Krein representations of the subordinators which govern the two parameter Poisson-Dirichlet family of distributions.
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In a previous paper, we have shown that the gamma subordinators may be represented as inverse local times of certain diffusions. In the present paper, we give such representations for other subordinators whose Lévy densities are of the form $ \frac{\mathcal{C}}{(\sinh(y))^γ}$, $0 < γ< 2$, and the more general family obtained from those by exponential tilting.
In this paper we deal with some open problems concerned with gamma subordinators. In particular, we provide a representation for the moments of the inverse gamma subordinator. Then, we focus on $λ$-potentials and we study the governing equations associated with gamma subordinators and inverse processes. Such representations are given in terms of higher transcendental functions, also known as Volterra functions.
Lévy processes with jumps governed by lower incomplete gamma subordinator and its variations
本文研究三类由独立Lévy从属过程时间变换的Lévy过程，推导其分布性质、讨论长程相关性，给出保险领域应用及从属过程模拟样本路径。
We exhibit, in the form of some identities in law, some connections between tilted stable subordinators, time-changed by independent Gamma processes and the occupation times of Bessel spiders, or their bridges. These identities in law are then explained thanks to excursion theory.
In the paper we consider time-changed Poisson processes where the time is expressed by compound Poisson-Gamma subordinators $G(N(t))$ and derive the expressions for their hitting times. We also study the time-changed Poisson processes where the role of time is played by the processes of the form $G(N(t)+at)$ and by the iteration of such processes.
A law of the iterated logarithm for the number of blocks in regenerative compositions generated by gamma-like subordinators
本文研究由类伽马从属过程生成的可再生随机组合，证明了当样本量趋于无穷时组合中块数满足重对数律，并附带证明了布朗运动与正则变化函数卷积的重对数律。
For $\widetilde{\cal R} = 1 - \exp(- {\cal R})$ a random closed set obtained by exponential transformation of the closed range ${\cal R}$ of a subordinator, a regenerative composition of generic positive integer $n$ is defined by recording the sizes of clusters of $n$ uniform random points as they are separated by the points of $\widetilde{\cal R}$. We focus on the number of parts $K_n$ of the composition when $\widetilde{\cal R}$ is derived from a gamma subordinator. We prove logarithmic asymptotics of the moments and central limit theorems for $K_n$ and other functionals of the composition s…
This paper describes how one can use the well-known Bayesian prior to posterior analysis of the Dirichlet process, and less known results for the gamma process, to address the formidable problem of assessing the distribution of linear functionals of Dirichlet processes. In particular, in conjunction with a gamma identity, we show easily that a generalized Cauchy-Stieltjes transform of a linear functional of a Dirichlet process is equivalent to the Laplace functional of a class of, what we define as, beta-gamma processes. This represents a generalization of the Markov-Krein identity for mean fu…
We study absolute-continuity properties of a class of stochastic processes, including the gamma and the Dirichlet processes. We prove that the laws of a general class of non-linear transformations of such processes are locally equivalent to the law of the original process and we compute explicitly the associated Radon-Nikodym densities. This work unifies and generalizes to random non-linear transformations several previous results on quasi-invariance of gamma and Dirichlet processes.
Subordinating a multivariate Lévy process, the subordinate, with a univariate subordinator gives rise to a pathwise construction of a new Lévy process, provided the subordinator and the subordinate are independent processes. The variance-gamma model in finance was generated accordingly from a Brownian motion and a gamma process. Alternatively, multivariate subordination can be used to create Lévy processes, but this requires the subordinate to have independent components. In this paper, we show that there exists another operation acting on pairs $(T,X)$ of Lévy processes which creates a Lévy p…
We start by defining a subordinator by means of the lower-incomplete gamma function. It can be considered as an approximation of the stable subordinator, easier to be handled thank to its finite activity. A tempered version is also considered in order to overcome the drawback of infinite moments. Then, we study Lévy processes time-changed by these subordinators, with particular attention to the Brownian case. An approximation of the fractional derivative (as well as of the fractional power of operators) arises from the analysis of governing equations. Finally, we show that time-changing the fr…
We define and study fractional versions of the well-known Gamma subordinator $Γ:=\{Γ(t),$ $t\geq 0\},$ which are obtained by time-changing $% Γ$ by means of an independent stable subordinator or its inverse. Their densities are proved to satisfy differential equations expressed in terms of fractional versions of the shift operator (with fractional parameter greater or less than one, in the two cases). As a consequence, the fractional generalization of some Gamma subordinated processes (i.e. the Variance Gamma, the Geometric Stable and the Negative Binomial) are introduced and the corresponding…
Asymptotics for densities of exponential functionals of subordinators
本文研究从属过程指数泛函的密度等渐近性，在Lévy测度正增长条件下，借助改进的Bernstein-gamma函数Stirling型渐近性，用鞍点法得到非经典Tauberian渐近结果，并证明一类密度在复平面锥内解析。
In this paper the running average of a subordinator with a tempered stable distribution is considered. We investigate a family of previously unexplored infinite-activity subordinators induced by the probability distribution of the running average process and determine their jump intensity measures. Special cases including gamma processes and inverse Gaussian processes are discussed. Then we derive easily implementable formulas for the distribution functions, cumulants, and moments, as well as provide explicit estimates for their asymptotic behaviors. Numerical experiments are conducted for ill…
Multivariate Generalized Counting Process via Gamma Subordination
本文提出了一种通过伽马子ordination的多元广义计数过程，研究了其数学性质及在冲击模型中的应用。
In this paper, we extend recent work on the functions that we call Bernstein-gamma to the class of bivariate Bernstein-gamma functions. In the more general bivariate setting, we determine Stirling-type asymptotic bounds which generalise, improve upon and streamline those found for the univariate Bernstein-gamma functions. Then, we demonstrate the importance and power of these results through an application to exponential functionals of Lévy processes. In more detail, for a subordinator (a non-decreasing Lévy process) $(X_s)_{s\geq 0}$, we study its \textit{exponential functional}, $\int_0^t e^…
On the Multivariate Generalized Counting Process and its Time-Changed Variants
本文提出多元广义计数过程及其经稳定、gamma等从属子时间变换的变体，推导其分布、Lévy测度与协方差等性质，并展示其在冲击模型中的应用。
Weak variance generalised gamma convolution processes are multivariate Brownian motions weakly subordinated by multivariate Thorin subordinators. Within this class, we extend a result from strong to weak subordination that a driftless Brownian motion gives rise to a self-decomposable process. Under moment conditions on the underlying Thorin measure, we show that this condition is also necessary. We apply our results to some prominent processes such as the weak variance alpha-gamma process, and illustrate the necessity of our moment conditions in some cases.
Let $\{L(t),t\geq 0\}$ be a Lévy process with representative random variable $L(1)$ defined by the infinitely divisible logarithmic series distribution. We study here the transition probability and Lévy measure of this process. We also define two subordinated processes. The first one, $Y(t)$, is a Negative-Binomial process $X(t)$ directed by Gamma process. The second process, $Z(t)$, is a Logarithmic Lévy process $L(t)$ directed by Poisson process. For them, we prove that the Bernstein functions of the processes $L(t)$ and $Y(t)$ contain the iterated logarithmic function. In addition, the Lévy…
We show that if $(X_s, s\geq 0)$ is a right-continuous process, $Y_t=\int_0^t\d s X_s$ its integral process and $τ= (τ_{\ell}, \ell \geq 0)$ a subordinator, then the time-changed process $(Y_{τ_{\ell}}, \ell\geq 0)$ allows to retrieve the information about $(X_{τ_{\ell}}, \ell\geq 0)$ when $τ$ is stable, but not when $τ$ is a gamma subordinator. This question has been motivated by a striking identity in law involving the Bessel clock taken at an independent inverse Gaussian variable.
The Poisson-Kingman distributions, $\mathrm{PK}(ρ)$, on the infinite simplex, can be constructed from a Poisson point process having intensity density $ρ$ or by taking the ranked jumps up till a specified time of a subordinator with Lévy density $ρ$, as proportions of the subordinator. As a natural extension, we replace the Poisson point process with a negative binomial point process having parameter $r>0$ and Lévy density $ρ$, thereby defining a new class $\mathrm{PK}^{(r)}(ρ)$ of distributions on the infinite simplex. The new class contains the two-parameter generalisation $\mathrm{PD}(α, θ)…
In this paper, we introduce a space fractional negative binomial (SFNB) process by subordinating the space fractional Poisson process to a gamma subordinator. Its one-dimensional distributions are derived in terms of generalized Wright functions and their governing equations are obtained. It is a Lévy process and the corresponding Lévy measure is given. Extensions to the case of distributed order SFNB process, where the fractional index follows a two-point distribution, is analyzed in detail. The connections of the SFNB process to a space fractional Polya-type process is also pointed out. More…
Generalized Counting Process with Random Drift and Different Brownian Clocks
本文引入了具有确定性和随机漂移的广义计数过程（GCP），推导了其概率律和分数阶微分方程，并研究了在多种布朗时钟及不完全伽马函数相关从属过程下的时间变更GCP及其分数阶积分。
We unify and extend a number of approaches related to constructing multivariate Variance-Gamma (V.G.) models for option pricing. An overarching model is derived by subordinating multivariate Brownian motion to a subordinator from the Thorin (1977) class of generalised Gamma convolution subordinators. A class of models due to Grigelionis (2007), which contains the well-known Madan-Seneta V.G. model, is of this type, but our multivariate generalization is considerably wider, allowing in particular for processes with infinite variation and a variety of dependencies between the underlying processe…
Consider a list of labeled objects that are organized in a heap. At each time, object $j$ is selected with probability $p_j$ and moved to the top of the heap. This procedure defines a Markov chain on the set of permutations which is referred to in the literature as Move-to-Front rule. The present contribution focuses on the stationary search cost, namely the position of the requested item in the heap when the Markov chain is in equilibrium. We consider the scenario where the number of objects is infinite and the probabilities $p_j$'s are defined as the normalization of the increments of a subo…
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.
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).
Classical beta ensembles and related eigenvalues processes at high temperature and the Markov--Krein transform
本文利用 Markov–Krein 变换刻画经典 beta 系综及其特征值过程在高温下的极限，识别出三类系综对应的极限测度及测度值过程。
This paper develops some general calculus for GGC and Dirichlet process means functionals. It then proceeds via an investigation of positive Linnik random variables, and more generally random variables derived from compositions of a stable subordinator with GGC subordinators, to establish various distributional equivalences between these models and phenomena connected to local times and occupation times of what are defined as randomly skewed Bessel processes and bridges. This yields a host of interesting identities and explicit density formula for these models. Randomly skewed Bessel processes…
Moments of exponential functionals of Lévy processes on a deterministic horizon -- identities and explicit expressions
本文研究确定性期限上 Lévy 过程指数泛函的复矩，建立两个卷积恒等式并给出对称过程与从属子的显式矩公式。
Many real time-series exhibit behavior adequate to long range dependent data. Additionally very often these time-series have constant time periods and also have characteristics similar to Gaussian processes although they are not Gaussian. Therefore there is need to consider new classes of systems to model these kind of empirical behavior. Motivated by this fact in this paper we analyze two processes which exhibit long range dependence property and have additional interesting characteristics which may be observed in real phenomena. Both of them are constructed as the superposition of fractional…
This paper considers the class of Lévy processes that can be written as a Brownian motion time changed by an independent Lévy subordinator. Examples in this class include the variance gamma model, the normal inverse Gaussian model, and other processes popular in financial modeling. The question addressed is the precise relation between the standard first passage time and an alternative notion, which we call first passage of the second kind, as suggested by Hurd (2007) and others. We are able to prove that standard first passage time is the almost sure limit of iterations of first passage of th…
In the information-based pricing framework of Brody, Hughston and Macrina, the market filtration $\{ \mathcal F_t\}_{t\geq 0}$ is generated by an information process $\{ ξ_t\}_{t\geq0}$ defined in such a way that at some fixed time $T$ an $\mathcal F_T$-measurable random variable $X_T$ is "revealed". A cash flow $H_T$ is taken to depend on the market factor $X_T$, and one considers the valuation of a financial asset that delivers $H_T$ at $T$. The value $S_t$ of the asset at any time $t\in[0,T)$ is the discounted conditional expectation of $H_T$ with respect to $\mathcal F_t$, where the expect…
Let $\{Γ_t, \, t\ge 0\}$ be the Gamma subordinator. Using a moment identification due to Bertoin-Yor (2002), we observe that for every $t > 0$ and $α\in (0,1)$ the random variable $Γ_t^{-α}$ is distributed as the exponential functional of some spectrally negative Lévy process. This entails that all size-biased samplings of Fréchet distributions are self-decomposable and that the extreme value distribution $F_ξ$ is infinitely divisible if and only if $ξ\not\in (0,1),$ solving problems raised by Steutel (1973) and Bondesson (1992). We also review different analytical and probabilistic interpreta…
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