Showing papers on "Subordinator published in 2006"
TL;DR: In this paper, the stochastic foundations for ultra-low diffusion were developed based on random walks with a random waiting time between jumps whose probability tail falls off at a logarithmic rate.
132 citations
TL;DR: In this paper, a generalized risk model driven by a nondecreasing Levy process is presented. But unlike the classical case that models the individual claim size distribution and obtains from it the aggregate claims distribution, here the aggregate claim distribution is known in closed form, it is simply the one-dimensional distribution of a subordinator.
Abstract: Dufresne et al. (1991) introduced a general risk model defined as the limit of compound Poisson processes. Such a model is either a compound Poisson process itself or a process with an infinite number of small jumps. Later, in a series of now classical papers, the joint distribution of the time of ruin, the surplus before ruin, and the deficit at ruin was studied (Gerber and Shiu 1997, 1998a, 1998b; Gerber and Landry 1998). These works use the classical and the perturbed risk models and hint that the results can be extended to gamma and inverse Gaussian risk processes. In this paper we work out this extension to a generalized risk model driven by a nondecreasing Levy process. Unlike the classical case that models the individual claim size distribution and obtains from it the aggregate claims distribution, here the aggregate claims distribution is known in closed form. It is simply the one-dimensional distribution of a subordinator. Embedded in this wide family of risk models we find the gamma, in...
92 citations
TL;DR: In this paper, the authors prove that the Harnack inequality is valid for the nonnegative harmonic functions of X, obtained by subordinating Brownian motion with a subordinator with a positive drift.
Abstract: Let X be a Levy process in
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$$\mathbb{R}^{d} $$
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$$d \geqslant 3$$
, obtained by subordinating Brownian motion with a subordinator with a positive drift. Such a process has the same law as the sum of an independent Brownian motion and a Levy process with no continuous component. We study the asymptotic behavior of the Green function of X near zero. Under the assumption that the Laplace exponent of the subordinator is a complete Bernstein function we also describe the asymptotic behavior of the Green function at infinity. With an additional assumption on the Levy measure of the subordinator we prove that the Harnack inequality is valid for the nonnegative harmonic functions of X.
78 citations
TL;DR: In this article, the authors studied the number of parts K n of a random composition and other related functionals under the assumption that R = Φ(S • ), where (S t, t ≥ 0) is a subordinator and Φ:[0,∞] → [0, 1] is a diffeomorphism.
Abstract: A random composition of n appears when the points of a random closed set R ⊂ [0,1] are used to separate into blocks n points sampled from the uniform distribution. We study the number of parts K n of this composition and other related functionals under the assumption that R = Φ(S • ), where (S t , t ≥ 0) is a subordinator and Φ:[0,∞] → [0, 1] is a diffeomorphism. We derive the asymptotics of K n when the Levy measure of the subordinator is regularly varying at 0 with positive index. Specializing to the case of exponential function Φ(x) = 1 - e -x , we establish a connection between the asymptotics of K n and the exponential functional of the subordinator.
67 citations
TL;DR: In this article, it was shown that S n /(nr) converges in distribution to a limiting variable, S, characterized via an exponential integral of a certain subordinator, when the underlying genealogical tree of the sample is modeled by a coalescent process with mutation rate r > 0.
Abstract: We present recursions for the total number, S n , of mutations in a sample of n individuals, when the underlying genealogical tree of the sample is modelled by a coalescent process with mutation rate r > 0. The coalescent is allowed to have simultaneous multiple collisions of ancestral lineages, which corresponds to the existence of large families in the underlying population model. For the subclass of A-coalescent processes allowing for multiple collisions, such that the measure A(dx)/x is finite, we prove that S n /(nr) converges in distribution to a limiting variable, S, characterized via an exponential integral of a certain subordinator. When the measure A(dx)/x 2 is finite, the distribution of S coincides with the stationary distribution of an autoregressive process of order 1 and is uniquely determined via a stochastic fixed-point equation of the form S = AS + B, with specific independent random coefficients A and B. Examples are presented in which explicit representations for (the density of) S are available. We conjecture that S n /E(S n ) → 1 in probability if the measure A(dx)/x is infinite.
50 citations
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TL;DR: In this paper, the CGMY and Meixner processes are described as time changed Brownian motions and the required time changes may be generated by simulating the requisite one-sided stable subordinator and throwing away some of the jumps as described in Rosinski (2001).
Abstract: We describe the CGMY and Meixner processes as time changed Brownian motions. The CGMY uses a time change absolutely continuous with respect to the one-sided stable $(Y/2)$ subordinator while the Meixner time change is absolutely continuous with respect to the one sided stable $(1/2)$ subordinator$.$ The required time changes may be generated by simulating the requisite one-sided stable subordinator and throwing away some of the jumps as described in Rosinski (2001).
47 citations
TL;DR: In this paper, a continuous-time model with stationary increments for asset price {Pt} is proposed, which allows for skewness of returns and resembles closely that of Madan, Carr, and Chang (1998).
Abstract: A continuous-time model with stationary increments for asset price {Pt} is an extension of the symmetric subordinator model of Heyde (1999), and allows for skewness of returns. In the setting of independent variance-gamma-distributed returns the model resembles closely that of Madan, Carr, and Chang (1998). A simple choice of parameters renders {e_rr Pt} a familiar martingale. We then specify the activity time process, {Tt}, for which [Tt ? t] is asymptotically self-similar and [xt}, with zt = Tt ? 7}_i, is gamma distributed. This results in a skew variance-gamma distribution for each log price increment (return) Xt and a model for {Xt} which incorporates long-range dependence in squared returns. Our approach mirrors that for the (symmetric) Student process model of Heyde and Leonenko (2005), to which the present work is intended as a complement and a sequel. One intention is to compare, partly on the basis of fitting to data, versions of the general model wherein the returns have either (symmetric) t-distributions or variance-gamma distributions.
46 citations
TL;DR: In this article, the authors analyzed the behavior of a so-called cumulant M-estimator, in case this Levy density is characterized by a Euclidean (finite-dimensional) parameter.
Abstract: Key words and Phrases: cumulant, empirical characteristic function, Levy process, self-decomposable distribution, stationary process. Consider a stationary sequence of random variables with infinitely divisible marginal law, characterized by its Levy density. We analyze the behavior of a so-called cumulant M-estimator, in case this Levy density is characterized by a Euclidean (finite-dimensional) parameter. Under mild conditions, we prove consistency and asymptotic normality of the estimator. The estimator is considered in the situation where the data are increments of a subordinator as well as the situation where the data consist of a discretely sampled Ornstein Uhlenbeck process induced by the subordinator. We illustrate our results for the Gamma-process and the Inverse-Gaussian-OU-process. For these processes we also explain how the estimator can be computed numerically.
38 citations
TL;DR: In this paper, the authors present the random-variable formalism of anomalous diffusion processes and elucidate the role of the so-called inverse-time stochastic process, the main mathematical tool that allows modifying the dynamics of standard relaxation processes and give rise to the nonexponential decay of modes.
Abstract: The paper presents the random-variable formalism of the anomalous diffusion processes. The emphasis is on a rigorous presentation of asymptotic behaviour of random walk processes with infinite mean random time intervals between jumps. We elucidate the role of the so-called inverse-time stochastic process, the main mathematical tool that allows us to modify the dynamics of standard relaxation processes and give rise to the nonexponential decay of modes. In particular, we show that the Brownian motion in combination with an appropriate inverse-time process may lead not only to exponential but also to the nonexponential relaxation responses.
38 citations
TL;DR: In this article, 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 Open image in new window.
Abstract: For Open image in new window a random closed set obtained by exponential transformation of the closed range Open image in new window 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 Open image in new window . We focus on the number of parts K n of the composition when Open image in new window 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 such as the number of singletons, doubletons, etc. This study complements our previous work on asymptotics of these functionals when the tail of the Levy measure is regularly varying at 0+.
37 citations
TL;DR: Gnedin et al. as mentioned in this paper considered the case when the tail of the Levy measure of S is slowly varying and provided an asymptotic analysis of the fluctuations of K n, as n → ∞, for a wide spectrum of situations.
TL;DR: The scaling limit of Bouchaud's trap model in this article is the fractional-kinetics process, that is the time change of a $d$-dimensional Brownian motion by the inverse of an independent stable subordinator.
Abstract: We give the ``quenched'' scaling limit of Bouchaud's trap model in ${d\ge 2}$. This scaling limit is the fractional-kinetics process, that is the time change of a $d$-dimensional Brownian motion by the inverse of an independent $\alpha$-stable subordinator.
TL;DR: In this article, it was shown that if limn→∞ϕn(λ)=ϕ(λ) for every λ>0, then the spectra of the semigroups of Open image in new window and Xϕ,D are all discrete.
Abstract: Let X={Xt,t≥0} be a symmetric Markov process in a state space E and D an open set of E. Let S(n)={S(n)t, t ≥ 0} be a subordinator with Laplace exponent ϕn and S={St,t≥0} a subordinator with Laplace exponent ϕ. Suppose that X is independent of S and S(n). In this paper we consider the subordinate processes Open image in new window and Open image in new window and their subprocesses Open image in new window and Xϕ,D killed upon leaving D. Suppose that the spectra of the semigroups of Open image in new window and Xϕ,D are all discrete, with Open image in new window being the eigenvalues of the generator of Open image in new window and Open image in new window being the eigenvalues of the generator of Xϕ,D. We show that, if limn→∞ϕn(λ)=ϕ(λ) for every λ>0, then
TL;DR: In this article, the authors give a new example of duality between fragmentation and coagulation operators, and show that these Markov chains can be encoded naturally by certain random recursive trees, and use this representation to give an alternative and more concrete proof of the coaggulation-fragmentation duality.
Abstract: In this paper we give a new example of duality between fragmentation and coagulation operators. Consider the space of partitions of mass (i.e., decreasing sequences of nonnegative real numbers whose sum is 1) and the twoparameter family of Poisson–Dirichlet distributions PD(α, θ) that take values in this space. We introduce families of random fragmentation and coagulation operators Frag α and Coag α,θ , respectively, with the following property: if the input to Frag α has PD(α, θ) distribution, then the output has PD(α, θ + 1) distribution, while the reverse is true for Coag α,θ . This result may be proved using a subordinator representation and it provides a companion set of relations to those of Pitman between PD(α, θ) and PD(αβ, θ ). Repeated application of the Fragα operators gives rise to a family of fragmentation chains. We show that these Markov chains can be encoded naturally by certain random recursive trees, and use this representation to give an alternative and more concrete proof of the coagulation–fragmentation duality.
TL;DR: In this article, the Gamma subordinator is represented as a Krein functional of Brownian motion, using the known representations for stable subordinators and Esscher transforms, in particular the Krein representations of the subordinators which govern the two parameter Poisson Dirichlet family of distributions.
Abstract: 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 [23].
TL;DR: Aldous and Pitman as discussed by the authors studied asymptotic distributions as n → ∞, of various functional of a uniform random mapping of the set {1,..., n}, by constructing a mapping-walk and showing these random walks converge weakly to a reflecting Brownian bridge.
Abstract: Author(s): Aldous, D; Pitman, J | Abstract: Aldous and Pitman (1994) studied asymptotic distributions as n → ∞, of various functional of a uniform random mapping of the set {1,..., n}, by constructing a mapping-walk and showing these random walks converge weakly to a reflecting Brownian bridge. Two different ways to encode a mapping as a walk lead to two different decompositions of the Brownian bridge, each defined by cutting the path of the bridge at an increasing sequence of recursively defined random times in the zero set of the bridge. The random mapping asymptotics entail some remarkable identities involving the random occupation measures of the bridge fragments defined by these decompositions. We derive various extensions of these identities for Brownian and Bessel bridges, and characterize the distributions of various path fragments involved, using the Levy-lto theory of Poisson processes of excursions for a self-similar Markov process whose zero set is the range of a stable subordinator of index α ∈ (0,1).
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18 Sep 2006
TL;DR: In this paper, exact expressions for the probability distribution of linear functionals of the two-parameter Poisson-Dirichlet process PD(�,�) were derived from the application of an inversion formula for a generalized Cauchy-Stieltjes transform.
Abstract: The present paper provides exact expressions for the probability distribution of linear functionals of the two-parameter Poisson-Dirichlet process PD(�,�). Distributional results that follow from the application of an inversion formula for a (generalized) Cauchy- Stieltjes transform are achieved. Moreover, several interesting integral identities are obtained by exploiting a correspondence between the mean functional of a Poisson-Dirichlet process and the mean functional of a suitable Dirichlet process. Finally, some distributional char- acterizations in terms of mixture representations are illustrated. Our formulae are relevant to occupation time phenomena connected with Brownian motion and more general Bessel processes, as well as to models arising in Bayesian nonparametric statistics. 1. Introduction Let (Pi)i≥1, with P1 > P2 > . . . > 0 and P ∞=1 Pk = 1, denote a sequence of (random) ranked probabilities having the two-parameter (α, θ) Poisson-Dirichlet law, denoted as PD(α, θ) for 0 ≤ α < 1 and θ ≥ 0. A description, as well as a thorough investigation on its properties, is provided in (36). See also (28), (30) and (33). Equivalently, letting Vk, for any k ≥ 1, denote independent random variables such that Vk has beta(1−α, θ+kα) distribution, the PD(α, θ) law is defined as the ranked values of the stick-breaking sequence W1 = V1, Wk = Vk Q k−1 j=1 (1 −Vj) for k ≥ 2. Interestingly PD(α, θ) laws can also be obtained by manipulating random probabilities of the type Pi = Ji/ ˜ T, where ˜ T = P ∞=1 Ji and the sequence (Ji)i≥1 stands for the ranked jumps of a subordinator. If the Ji's are the ranked jumps of a gamma subordinator, then the total mass ˜ T has a gamma distribution with shape θ and scale 1 and (Pi)i≥1 follows a PD(0, θ) law. At the other extreme, letting the Ji's be the ranked jumps of a stable subordinator of index 0 < α < 1, (Pi)i≥1 follows a PD(α,0) distribution. For both α and θ positive, the PD(α, θ) model arises by first taking the ranked jumps governed by the stable subordinator conditioned on their total mass ˜ T and then mixing over a power tempered stable law proportional to t −θ fα(t), where fα(t)
01 Jan 2006-Review of Cognitive Linguistics. Published under the auspices of the Spanish Cognitive Linguistics Association
TL;DR: The authors investigates how simultaneity between two events, a main clause event and a subordinate clause event, is coded in English, focusing on as-clauses but also contrast them with while-claauses.
Abstract: This paper investigates how simultaneity between two events, a main clause event and a subordinate clause event, is coded in English. It focuses on as-clauses but also contrasts them with while-clauses. It argues that as-clauses evoke path events, i.e. events susceptible to change. It also points out that as-clauses define a family resemblance network in that different, though related, variants can be recognised. While-clauses are argued to generally evoke larger and more stable temporal configurations, e.g. properties. The different behaviour of as-clauses and while-clauses is related to the different lexical status of as vs. while. The former is analysed as a subordinator unspecified for temporality whereas the latter is regarded as a temporal subordinator by default. Finally, the use of progressive aspect is discussed. It is argued to function as a “slow motion” marker in as-clauses and/or to signal a contrast between the temporal expanses of the main and as-clauses. By contrast, it takes on a transience-highlighting function in while-clauses.
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TL;DR: In this article, a general calculus for GGC and Dirichlet process means functionals is developed via an investigation of positive Linnik random variables, and more generally random variables derived from compositions of a stable subordinator with GGC subordinators.
Abstract: 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 and bridges may be seen as a randomization of their p-skewed counterparts developed in Barlow, Pitman and Yor (1989) and Pitman and Yor (1997), and are shown to naturally arise via exponential tilting As a special result it is shown that the occupation time of a p-skewed random Bessel process or (generalized) bridge is equivalent in distribution to the occupation time of a non-trivial randomly skewed process
TL;DR: In this article, a simpler and shorter proof of Kesten's result for the probabilities with which a subordinator hits points is given, where the probability of hitting a point is proportional to the probability that the subordinator is hit.
Abstract: We give a simpler and shorter proof of Kesten's result for the probabilities with which a subordinator hits points.
TL;DR: In this article, the authors define a bivariate Levy process by subordination of a Brownian motion and investigate a generalization of the bivariate Variance Gamma process proposed in Luciano and Schoutens [8] as a price process.
Abstract: The purpose of this paper is to define a bivariate Levy process by subordination of a Brownian motion. In particular we investigate a generalization of the bivariate Variance Gamma process proposed in Luciano and Schoutens [8] as a price process. Our main contribution here is to introduce a bivariate subordinator with correlated Gamma margins. We characterize the process and study its dependence structure. At the end wealso propose an exponential Levy price model based on our process.
25 May 2006
TL;DR: In this paper, a simpler and shorter proof of Kesten's result for the probabilities with which a subordinator hits points is given, where the probability of hitting a point is proportional to the probability that the subordinator is hit.
Abstract: We give a simpler and shorter proof of Kesten's result for the probabilities with which a subordinator hits points.
01 Jan 2006
TL;DR: In this article, the authors introduced moduli of smoothness techniques to deal with Berry? Esseen bounds, and illustrate them by considering standardized subordinators with finite variance, and they showed that the optimal rate of convergence can be simply written in terms of the first modulus, depending on the characteristic random variable of the subordinator.
Abstract: We introduce moduli of smoothness techniques to deal with Berry? Esseen bounds, and illustrate them by considering standardized subordinators with finite variance. Instead of the classical Berry?Esseen smoothing inequality, we give an easy inequality involving the second modulus. Under finite third moment assumptions, such an inequality provides the main term of the approximation with small constants, even asymptotically sharp constants in the lattice case. Under infinite third moment assumptions, we show that the optimal rate of convergence can be simply written in terms of the first modulus of smoothness of an appropriate function, depending on the characteristic random variable of the subordinator. The preceding results are extended to standardized L?evy processes with finite variance.
TL;DR: In this paper, two main results are presented in relation to subor-dination, self-decomposability and semi-stability of the subordinated process arising from a semi-stable subordinator.
Abstract: Two main results are presented in relation to subor- dination, self-decomposability and semi-stability. One of the re- sult is that strict semi-stability of subordinand process by self- decomposable subordinator gives semi-selfdecomposability of the subordinated process. The second result is a su-cient condition for any subordinated process arising from a semi-stable subordi- nand and a semi-stable subordinator to be semi- selfdecomposable.
01 Jan 2006
TL;DR: In this article, the authors consider processes of the form X(t) = ˜ X(�(t)) where X is a self-similar process with stationary increments and is a deterministic subordinator with a periodic activity function a = � 0 > 0.
Abstract: We consider processes of the form X(t) = ˜ X(�(t)) where ˜ X is a self-similar process with stationary increments andis a deterministic subordinator with a periodic activity function a = � 0 > 0. Such processes have been proposed as models for high-frequency financial data, such as currency exchange rates, where there are known to be daily and weekly periodic fluctuations in the volatility, captured here by the periodic activity function. We review an existing estimator for the activity function then propose three new methods for estimating it and present some experimental studies of their performance. We finish with an application to some foreign exchange and FTSE100 futures data.
TL;DR: In this paper, the authors obtained law-of-the-iterated-logarithm results for the stable subordinator problem with respect to stable subordinators with index 0.
Abstract: Let $(X(t),\ t\ge 0)$ with $X(0)=0$ be a stable subordinator with index $0 0$. We obtain law‐of‐the‐iterated‐logarithm results for $(X(t_k)),(Y(t_k))$ and $Z(t_k)$, properly normalized.