Topic

# Subordinator

About: Subordinator is a(n) research topic. Over the lifetime, 771 publication(s) have been published within this topic receiving 15383 citation(s).

##### Papers published on a yearly basis

##### Papers

More filters

••

TL;DR: The two-parameter Poisson-Dirichlet distribution with a single parameter is known as the size-biased random permutation (SBNP) as discussed by the authors, which was introduced by Engen in the context of species diversity and rediscovered by Perman and the authors in the study of excursions of Bessel processes.

Abstract: The two-parameter Poisson-Dirichlet distribution, denoted $\mathsf{PD}(\alpha, \theta)$ is a probability distribution on the set of decreasing positive sequences with sum 1. The usual Poisson-Dirichlet distribution with a single parameter $\theta$, introduced by Kingman, is $\mathsf{PD}(0, \theta)$. Known properties of $\mathsf{PD}(0, \theta)$, including the Markov chain description due to Vershik, Shmidt and Ignatov, are generalized to the two-parameter case. The size-biased random permutation of $\mathsf{PD}(\alpha, \theta)$ is a simple residual allocation model proposed by Engen in the context of species diversity, and rediscovered by Perman and the authors in the study of excursions of Brownian motion and Bessel processes. For $0 < \alpha < 1, \mathsf{PD}(\alpha, 0)$ is the asymptotic distribution of ranked lengths of excursions of a Markov chain away from a state whose recurrence time distribution is in the domain of attraction of a stable law of index $\alpha$. Formulae in this case trace back to work of Darling, Lamperti and Wendel in the 1950s and 1960s. The distribution of ranked lengths of excursions of a one-dimensional Brownian motion is $\mathsf{PD}(1/2, 0)$, and the corresponding distribution for a Brownian bredge is $\mathsf{PD}(1/2, 1/2)$. The $\mathsf{PD}(\alpha, 0)$ and $\mathsf{PD}(\alpha, \alpha)$ distributions admit a similar interpretation in terms of the ranked lengths of excursions of a semistable Markov process whose zero set is the range of a stable subordinator of index $\alpha$.

1,170 citations

••

TL;DR: In this paper, a general framework for the rigorous construction of partition-valued and discrete measure-valued coalescent Markov processes allowing infinitely many massses was presented, where the restriction of the partition to each finite subset ofis a Markov chain with transition rates of a simple form determined by the moments of a finite measureon the unit interval.

Abstract: k−2 � 1 − xb−k � � dx� . Call this process a � -coalescent. Discrete measure-valued processes derived from the � -coalescent model a system of masses undergoing coalescent collisions. Kingman's coalescent, which has numerous applications in population genetics, is the δ0-coalescent for δ0 a unit mass at 0. The coalescent recently derived by Bolthausen and Sznit- man from Ruelle's probability cascades, in the context of the Sherrington- Kirkpatrick spin glass model in mathematical physics, is the U-coalescent for U uniform on � 0� 1� .F or� = U, and whenever an infinite number of masses are present, each collision in a � -coalescent involves an infinite number of masses almost surely, and the proportion of masses involved exists as a limit almost surely and is distributed proportionally to � . The two-parameter Poisson-Dirichlet family of random discrete distributions derived from a stable subordinator, and corresponding exchangeable ran- dom partitions ofgoverned by a generalization of the Ewens sampling formula, are applied to describe transition mechanisms for processes of coalescence and fragmentation, including the U-coalescent and its time reversal. 1. Introduction. Markovian coalescent models for the evolution of a sys- tem of masses by a random process of binary collisions were introduced by Marcus (29) and Lushnikov (28). See (3) for a recent survey of the scientific lit- erature of these models and their relation to Smoluchowski's mean-field theory of coagulation phenomena. Evans and Pitman (15) gave a general framework for the rigorous construction of partition-valued and discrete measure-valued coalescent Markov processes allowing infinitely many massses and treated the binary coalescent model where each pair of masses x and y is subject to a coa- lescent collision at rate κ� xyfor a suitable rate kernel κ. This paper studies a family of partition-valued Markov processes, with state space the compact set of all partitions of � �= � 1� 2 ���� � , such that the restriction of the partition to each finite subset ofis a Markov chain with transition rates of a simple form determined by the moments of a finite measureon the unit interval. The case � = δ 0 , a unit mass at 0, is Kingman's coalescent in which every

561 citations

••

TL;DR: In this article, the scaling limit of a continuous-time random walk is shown to be an operator Levy motion subordinated to the hitting time process of a classical stable subordinator.

Abstract: A continuous-time random walk is a simple random walk subordinated to a renewal process used in physics to model anomalous diffusion. In this paper we show that, when the time between renewals has infinite mean, the scaling limit is an operator Levy motion subordinated to the hitting time process of a classical stable subordinator. Density functions for the limit process solve a fractional Cauchy problem, the generalization of a fractional partial differential equation for Hamiltonian chaos. We also establish a functional limit theorem for random walks with jumps in the strict generalized domain of attraction of a full operator stable law, which is of some independent interest.

493 citations

••

476 citations

••

TL;DR: In this article, it was shown that the cumulative number of trades is a better stochastic clock than the volume for generating virtually perfect normality in returns, and this clock can be modeled nonparametrically, allowing both the time-change and price processes to take the form of jump diffusions.

Abstract: The goal of this paper is to show that normality of asset returns can be recovered through a stochastic time change. Clark (1973) addressed this issue by representing the price process as a subordinated process with volume as the lognormally distributed subordinator. We extend Clark's results and find the following: (i) stochastic time changes are mathematically much less constraining than subordinators; (ii) the cumulative number of trades is a better stochastic clock than the volume for generating virtually perfect normality in returns; (iii) this clock can be modeled nonparametrically, allowing both the time-change and price processes to take the form of jump diffusions. The relations among trading volume, stock prices, and price volatility, the subject of empirical and theoretical studies over many years, have lately received renewed attention with the increased availability of high frequency data. A vast amount of research has focused on issues such as news arrivals, volume, and price changes or volatility moves, usually outside any framework of general or even partial equilibrium. Is the normality of returns-a key issue, for example, in the mean-variance paradigm for portfolio choice, or the recent study of the problems of risk management (e.g., in Value at Risk)-verified at any time horizon? The evidence accumulated from a number of studies that document the presence of leptokurtosis and skewness in the distribution of returns of a wide variety of financial assets suggests that the answer is no. Studies as early as, for example, Fama (1965), showed that daily returns are more long tailed than the normal density, with the distribution of returns approaching normality as the holding period is extended to one month. In the same manner, volatility smiles and other observed

413 citations