Showing papers on "Subordinator published in 2004"

Limit theorems for continuous-time random walks with infinite mean waiting times

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.

Ruin probabilities and decompositions for general perturbed risk processes

Abstract: We study a general perturbed risk process with cumulative claims modelled by a subordinator with finite expectation, and the perturba- tion being a spectrally negative Levy process with zero expectation. We derive a Pollaczek-Hinchin type formula for the survival proba- bility of that risk process, and give an interpretation of the formula based on the decomposition of the dual risk process at modified ladder epochs.

A Parallel between Brownian Bridges and Gamma Bridges

Abstract: Some properties of the Gamma bridges (obtained by conditioning the Gamma subordinator to take a given value at a given time) are investigated; similarities with the Brownian bridges are emphasized.

Asymptotic laws for compositions derived from transformed subordinators

Abstract: A random composition of $n$ appears when the points of a random closed set $\widetilde{\mathcal{R}}\subset[0,1]$ 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 $\widetilde{\mathcal{R}}=\phi(S_{\bullet})$, where $(S_t,t\geq0)$ is a subordinator and $\phi:[0,\infty]\to[0,1]$ is a diffeomorphism. We derive the asymptotics of $K_n$ when the L\'{e}vy measure of the subordinator is regularly varying at 0 with positive index. Specializing to the case of exponential function $\phi(x)=1-e^{-x}$, we establish a connection between the asymptotics of $K_n$ and the exponential functional of the subordinator.

Sharp bounds on the density, Green function and jumping function of subordinate killed BM

Abstract: Subordination of a killed Brownian motion in a domain D ⊂ R d via an α/2-sta- ble subordinator gives rise to a process Zt whose infinitesimal generator is −(−� |D) α/2 , the fractional power of the negative Dirichlet Laplacian. In this paper we establish upper and lower estimates for the density, Green function and jumping function of Zt when D is either a bounded C 1,1 domain or an exterior C 1,1 domain. Our estimates are sharp in the sense that the upper and lower estimates differ only by a multiplicative constant.

Ruin probabilities for competing claim processes

Abstract: Let C1,C2,...,Cm be independent subordinators with finite expectations and denote their sum by C. Consider the classical risk process X(t) = x+ct C(t). The ruin probability is given by the well known Pollaczek-Hinchin formula. If ruin occurs, however, it will be caused by a jump of one of the subordinators whose sum constitutes C. Formulae for the probability that ruin is caused by Ci are derived. These formulae can be extended to perturbed risk processes of the

Asymptotic laws for regenerative compositions: gamma subordinators and the like

Abstract: 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 such as the number of singletons, doubletons, etc. This study complements our previous work on asymptotics of these functionals when the tail of the L\'evy measure is regularly varying at $0+$.

Regenerative partition structures

Abstract: We consider Kingman's partition structures which are regenerative with respect to a general operation of random deletion of some part. Prototypes of this class are the Ewens partition structures which Kingman characterised by regeneration after deletion of a part chosen by size-biased sampling. We associate each regenerative partition structure with a corresponding regenerative composition structure, which (as we showed in a previous paper) can be associated in turn with a regenerative random subset of the positive halfline, that is the closed range of a subordinator. A general regenerative partition structure is thus represented in terms of the Laplace exponent of an associated subordinator. We also analyse deletion properties characteristic of the two-parameter family of partition structures.

Scaling limit results for the sum of many inverse Lévy subordinators

Abstract: The first passage time process of a Levy subordinator with heavy-tailed Levy measure has long-range dependent paths The random fluctuations that appear under two natural schemes of summation and time scaling of such stochastic processes are shown to converge weakly The limit process is fractional Brownian motion in one case and a non-Gaussian and non-stable process in the other case The latter appears to be of independent interest as a random process that arises under the influence of coexisting Gaussian and stable domains of attraction and is known from other applications to provide a bridge between fractional Brownian motion and stable Levy motion

The Riesz–Bessel Fractional Diffusion Equation

Abstract: This paper examines the properties of a fractional diffusion equation defined by the composition of the inverses of the Riesz potential and the Bessel potential. The first part determines the conditions under which the Green function of this equation is the transition probability density function of a Levy motion. This Levy motion is obtained by the subordination of Brownian motion, and the Levy representation of the subordinator is determined. The second part studies the semigroup formed by the Green function of the fractional diffusion equation. Applications of these results to certain evolution equations is considered. Some results on the numerical solution of the fractional diffusion equation are also provided. [ABSTRACT FROM AUTHOR] Copyright of Applied Mathematics & Optimization is the property of Springer Science & Business Media B.V. and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts)

Sharp Bounds for Green and Jumping Functions of Subordinate Killed Brownian Motions

Abstract: In this paper we obtain sharp bounds for the Green function and jumping function of a subordinate killed Brownian motion in a bounded $C^{1,1}$ domain, where the subordinating process is a subordinator whose Laplace exponent has certain asymptotic behavior at infinity.

Two recursive decompositions of Brownian bridge

Abstract: Aldous and Pitman (1994) studied asymptotic distributions, as n tends to infinity, of various functionals of a uniform random mapping of a set of n elements, by constructing a mapping-walk and showing these mapping-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 theory of Poisson processes of excursions for a self-similar Markov process whose zero set is the range of a stable subordinator of index between 0 and 1

Indifference pricing and the minimal entropy martingale measure in a stochastic volatility model with jumps

Abstract: We use the dynamic programming approach to derive an equation for the utility indierence price of Markovian claims in a stochastic volatility model proposed by Barndor-Nielsen and Shephard (3). The pricing equation is a Black & Scholes equation with a nonlinear integral term involving the risk preferences of the investor. Passing to the zero risk aversion limit, we present a Feynman-Kac representation of the minimal entropy price. The density of the minimal entropy martingale measure is found via the Girsanov transform of the Brownian motion and a subordinator process controlling the jumps in the volatility model. The density is represented by the logarithm of the value function for an investor with exponential utility and no claim issued, and a Feynman-Kac representation of this function is provided. We calculate the function explicitly in a special case, and show some properties in the general case.

Ruin probabilities and decompositions for general perturbed risk processes

Abstract: We study a general perturbed risk process with cumulative claims modelled by a subordinator with finite expectation, with the perturbation being a spectrally negative Levy process with zero expectation. We derive a Pollaczek-Hinchin type formula for the survival probability of that risk process, and give an interpretation of the formula based on the decomposition of the dual risk process at modified ladder epochs.

Random coverings and self-similar Markov processes

Abstract: This thesis is composed of two parts. The first deals with the construction of a random set which has the property of regeneration. Precisely, we construct random intervals from the partial records of a Poisson point process; these are used to partially cover $\mathbb(R)^+.$ The purpose of this work is to study the random set $\Rs$ that is left uncovered. We give integral tests to decide whether the random set $\Rs$ has a positive Lebesgue measure, has isolated points or if it is bounded. We show that $\Rs$ is, indeed, a regenerative set and characterize its law via the potential measure of the subordinator associated to $\Rs$. We obtain formulas to estimate some fractal dimensions of $\Rs.$ The second part consists of some contributions to the theory of positive self--similar Markov processes. To obtain the results of this part, we use Lamperti's transformation which establishes a bijection between this class of processes and real--valued Levy processes. Firstly, we are interested in the behavior at infinity of increasing self--similar Markov processes. In this vein, under some hypotheses, we find a deterministic function $f$ such that the liminf, as $t$ goes to infinity, of the quotient $X_t/f(t)$ is finite and different from 0 with probability $1.$ We obtain an analogous result which determines the behavior near of 0 of the process $X$ started from 0. Secondly, we study the different ways to construct a positive self--similar Markov process $\widetilde(X)$ for which 0 is a regular and recurrent point. To this end, we give some conditions that enable us to ensure that a such process exists and to determine its resolvent. Next, we make a systematic study of the Ito excursion measure $\exc$ of the process $\widetilde(X)$. In particular, we give a description of $\exc$ similar to that of Imhof for Ito's excursion measure of Brownian motion; we determine the law under $\exc$ of the normalized excursion and the image under time reversal of $\exc$. Furthermore, we construct and describe a process which is in weak duality with the process $\widetilde(X).$ We obtain some estimations of tail probabilities of the law of an exponential functional of a Levy process.