Showing papers on "Subordinator published in 1999"

Coalescents With Multiple Collisions

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

Subordinators: Examples and Applications

Abstract: 0. Foreword 1. Elements on subordinators 1.1. Definitions and first properties 1.2. The Levy-Khintchine formula 1.3. The renewal measure 1.4. The range of a subordinator 2. Regenerative property 2.1. Regenerative sets 2.2. Connection with Markov processes 3. Asymptotic behaviour of last passage times 3.1. Asymptotic behaviour in distribution 3.1.1. The self-similar case 3.1.2. The Dynkin-Lamperti theorem 3.2. Asymptotic sample path behaviour 4. Rates of growth of local time 4.1. Law of the iterated logarithm 4.2. Modulus of continuity 5. Geometric properties of regenerative sets 5.1. Fractal dimensions 5.1.1. Box-counting dimension 5.1.2. Hausdorff and packing dimensions 5.2. Intersections with a regenerative set 5.2.1. Equilibrium measure and capacity 5.2.2. Dimension criteria 5.2.3. Intersection of independant regenerative sets 6. Burgers equation with Brownian initial velocity 6.1. Burgers equation and the Hopf-Cole Solution 6.2. Brownian initial velocity 6.3. Proof of the theorem 7. Random covering 7.1. Setting 7.2. The Laplace exponent of the uncovered set 7.3. Some properties of the uncovered set 8. Levy processes 8.1. Local time at a fixed point 8.2. Local time at the supremum 8.3. The spectrally negative case 8.4. Bochner’s subordination for Levy processes 9. Occupation times of a linear Brownian motion 9.1. Occupation times and subordinators 9.2. Levy measure and Laplace exponent 9.2.1. Levy measure via excursion theory 9.2.2. Laplace exponent via the Sturm-Liouville equation 9.2.3. Spectral representation of the Laplace exponent 9.3. The zero set of a one-dimensional diffusion References

A risky asset model with strong dependence through fractal activity time

Abstract: The geometric Brownian motion (Black–Scholes) model for the price of a risky asset stipulates that the log returns are i.i.d. Gaussian. However, typical log returns data shows a leptokurtic distribution (much higher peak and heavier tails than the Gaussian) as well as evidence of strong dependence. In this paper a subordinator model based on fractal activity time is proposed which simply explains these observed features in the data, and whose scaling properties check out well on various data sets.

Slow points and fast points of local times

Abstract: Let $L$ be a local time. It is well known that there exist a law of the iterated logarithm and a modulus of continuity for $L$. Motivated by the case of real Brownian motion, we study the existence of fast points and slow points of $L$. We prove the existence of such points by considering the right-continuous inverse of $L$, which is a subordinator.

Stochastic orders in preservation properties by Bernstein-type operators

Abstract: In this paper, we are concerned with preservation properties of first and second order by an operator L representable in terms of a stochastic process Z with non-decreasing right-continuous paths. We introduce the derived operator D of L and the derived process V of Z in order to characterize the preservation of absolute continuity and convexity. To obtain different characterizations of the preservation of convexity, we introduce two kinds of duality, the first referring to the process Z and the second to the derived process V. We illustrate the preceding results by considering some examples of interest both in probability and in approximation theory - namely, mixtures, centred subordinators, Bernstein polynomials and beta operators. In most of them, we find bidensities to describe the duality between the derived processes. A unified approach based on stochastic orders is given.