scholarly journals Pareto Lévy Measures and Multivariate Regular Variation

2012 ◽  
Vol 44 (1) ◽  
pp. 117-138 ◽  
Author(s):  
Irmingard Eder ◽  
Claudia Klüppelberg
Keyword(s):  
Regular Variation ◽  
Lévy Process ◽  
Tail Dependence ◽  
Levy Process ◽  
Dependence Structure ◽  
Lévy Measure ◽  
Multivariate Distributions ◽  
One Dimensional ◽  
The One ◽  
Lévy Measures

We consider regular variation of a Lévy process X := (Xt)t≥0 in with Lévy measure Π, emphasizing the dependence between jumps of its components. By transforming the one-dimensional marginal Lévy measures to those of a standard 1-stable Lévy process, we decouple the marginal Lévy measures from the dependence structure. The dependence between the jumps is modeled by a so-called Pareto Lévy measure, which is a natural standardization in the context of regular variation. We characterize multivariate regularly variation of X by its one-dimensional marginal Lévy measures and the Pareto Lévy measure. Moreover, we define upper and lower tail dependence coefficients for the Lévy measure, which also apply to the multivariate distributions of the process. Finally, we present graphical tools to visualize the dependence structure in terms of the spectral density and the tail integral for homogeneous and nonhomogeneous Pareto Lévy measures.

Pareto Lévy Measures and Multivariate Regular Variation

2012 ◽  
Vol 44 (01) ◽  
pp. 117-138 ◽  
Author(s):  
Irmingard Eder ◽  
Claudia Klüppelberg
Keyword(s):  
Regular Variation ◽  
Lévy Process ◽  
Tail Dependence ◽  
Levy Process ◽  
Dependence Structure ◽  
Lévy Measure ◽  
Multivariate Distributions ◽  
One Dimensional ◽  
The One ◽  
Lévy Measures

We consider regular variation of a Lévy process X := ( X t) t≥0 in with Lévy measure Π, emphasizing the dependence between jumps of its components. By transforming the one-dimensional marginal Lévy measures to those of a standard 1-stable Lévy process, we decouple the marginal Lévy measures from the dependence structure. The dependence between the jumps is modeled by a so-called Pareto Lévy measure, which is a natural standardization in the context of regular variation. We characterize multivariate regularly variation of X by its one-dimensional marginal Lévy measures and the Pareto Lévy measure. Moreover, we define upper and lower tail dependence coefficients for the Lévy measure, which also apply to the multivariate distributions of the process. Finally, we present graphical tools to visualize the dependence structure in terms of the spectral density and the tail integral for homogeneous and nonhomogeneous Pareto Lévy measures.

scholarly journals One-dimensional distributions of subordinators with upper truncated Lévy measure, and applications

2009 ◽  
Vol 41 (2) ◽  
pp. 367-392 ◽  
Author(s):  
Shai Covo
Keyword(s):  
Distribution Function ◽  
Lévy Process ◽  
Marginal Distribution ◽  
Levy Process ◽  
Gamma Process ◽  
Lévy Measure ◽  
Original Process ◽  
One Dimensional ◽  
Analogous Formula

Given a pure-jump subordinator (i.e. nondecreasing Lévy process with no drift) with continuous Lévy measure ν, we derive a formula for the distribution function Fs (x; t) at time t of the associated subordinator whose Lévy measure is the restriction of ν to (0,s]. It will be expressed in terms of ν and the marginal distribution function F (⋅; t) of the original process. A generalization concerning an arbitrary truncation of ν will follow. Under certain conditions, an analogous formula will be obtained for the nth derivative, ∂nFs (x; t) ∂ xn. The requirement that ν is continuous is shown to have no intrinsic meaning. A number of interesting results involving the size ordered jumps of subordinators will be derived. An appropriate approximation for the small jumps of a gamma process will be considered, leading to a revisiting of the generalized Dickman distribution.

scholarly journals One-dimensional distributions of subordinators with upper truncated Lévy measure, and applications

2009 ◽  
Vol 41 (02) ◽  
pp. 367-392 ◽  
Author(s):  
Shai Covo
Keyword(s):  
Distribution Function ◽  
Lévy Process ◽  
Marginal Distribution ◽  
Levy Process ◽  
Gamma Process ◽  
Lévy Measure ◽  
Original Process ◽  
One Dimensional ◽  
Analogous Formula

Given a pure-jump subordinator (i.e. nondecreasing Lévy process with no drift) with continuous Lévy measure ν, we derive a formula for the distribution functionFs(x;t) at timetof the associated subordinator whose Lévy measure is the restriction of ν to (0,s]. It will be expressed in terms of ν and the marginal distribution functionF(⋅;t) of the original process. A generalization concerning an arbitrary truncation of ν will follow. Under certain conditions, an analogous formula will be obtained for thenth derivative, ∂nFs(x;t) ∂xn. The requirement that ν is continuous is shown to have no intrinsic meaning. A number of interesting results involving the size ordered jumps of subordinators will be derived. An appropriate approximation for the small jumps of a gamma process will be considered, leading to a revisiting of the generalized Dickman distribution.

scholarly journals Applications of factorization embeddings for Lévy processes

2006 ◽  
Vol 38 (03) ◽  
pp. 768-791 ◽  
Author(s):  
A. B. Dieker
Keyword(s):  
Lévy Process ◽  
Joint Distribution ◽  
Lévy Processes ◽  
Levy Processes ◽  
Levy Process ◽  
Tail Asymptotics ◽  
Lévy Measure ◽  
Risk Models ◽  
General Properties ◽  
Phase Type

We give three applications of the Pecherskii-Rogozin-Spitzer identity for Lévy processes. First, we find the joint distribution of the supremum and the epoch at which it is ‘attained’ if a Lévy process has phase-type upward jumps. We also find the characteristics of the ladder process. Second, we establish general properties of perturbed risk models, and obtain explicit fluctuation identities in the case that the Lévy process is spectrally positive. Third, we study the tail asymptotics for the supremum of a Lévy process under different assumptions on the tail of the Lévy measure.

scholarly journals Optimal Control with Absolutely Continuous Strategies for Spectrally Negative Lévy Processes

2012 ◽  
Vol 49 (01) ◽  
pp. 150-166 ◽  
Author(s):  
Andreas E. Kyprianou ◽  
Ronnie Loeffen ◽  
José-Luis Pérez
Keyword(s):  
Control Problem ◽  
Optimal Strategy ◽  
Lévy Process ◽  
Control Strategies ◽  
Levy Process ◽  
Lévy Measure ◽  
Absolutely Continuous ◽  
Completely Monotone ◽  
Hamilton Jacobi Bellman ◽  
Monotone Density

In the last few years there has been renewed interest in the classical control problem of de Finetti (1957) for the case where the underlying source of randomness is a spectrally negative Lévy process. In particular, a significant step forward was made by Loeffen (2008), who showed that a natural and very general condition on the underlying Lévy process which allows one to proceed with the analysis of the associated Hamilton-Jacobi-Bellman equation is that its Lévy measure is absolutely continuous, having completely monotone density. In this paper we consider de Finetti's control problem, but with the restriction that control strategies are absolutely continuous with respect to the Lebesgue measure. This problem has been considered by Asmussen and Taksar (1997), Jeanblanc-Picqué and Shiryaev (1995), and Boguslavskaya (2006) in the diffusive case, and Gerber and Shiu (2006) for the case of a Cramér-Lundberg process with exponentially distributed jumps. We show the robustness of the condition that the underlying Lévy measure has a completely monotone density and establish an explicit optimal strategy for this case that envelopes the aforementioned existing results. The explicit optimal strategy in question is the so-called refraction strategy.

scholarly journals Multivariate Markov Families of Copulas

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
Ludger Overbeck ◽  
Wolfgang M. Schmidt
Keyword(s):  
Markov Property ◽  
Dimensional Case ◽  
Dependence Structure ◽  
Multivariate Process ◽  
One Dimensional ◽  
Multivariate Case ◽  
Finite Dimensional ◽  
Different Dimensions ◽  
The One ◽  
Dimension One

AbstractFor the Markov property of a multivariate process, a necessary and suficient condition on the multidimensional copula of the finite-dimensional distributions is given. This establishes that the Markov property is solely a property of the copula, i.e., of the dependence structure. This extends results by Darsow et al. [11] from dimension one to the multivariate case. In addition to the one-dimensional case also the spatial copula between the different dimensions has to be taken into account. Examples are also given.

scholarly journals Tail Dependence for Heavy-Tailed Scale Mixtures of Multivariate Distributions

2009 ◽  
Vol 46 (4) ◽  
pp. 925-937 ◽  
Author(s):  
Haijun Li ◽  
Yannan Sun
Keyword(s):  
Regular Variation ◽  
Tail Dependence ◽  
Heavy Tail ◽  
Sufficient Condition ◽  
Elliptical Distributions ◽  
Multivariate Regular Variation ◽  
Multivariate Distributions ◽  
Scale Mixtures ◽  
Heavy Tailed ◽  
General Method

The tail dependence of multivariate distributions is frequently studied via the tool of copulas. In this paper we develop a general method, which is based on multivariate regular variation, to evaluate the tail dependence of heavy-tailed scale mixtures of multivariate distributions, whose copulas are not explicitly accessible. Tractable formulae for tail dependence parameters are derived, and a sufficient condition under which the parameters are monotone with respect to the heavy tail index is obtained. The multivariate elliptical distributions are discussed to illustrate the results.

scholarly journals A Lévy Process Reflected at a Poisson Age Process

2006 ◽  
Vol 43 (1) ◽  
pp. 221-230 ◽  
Author(s):  
Offer Kella ◽  
Onno Boxma ◽  
Michel Mandjes
Keyword(s):  
Stationary Distribution ◽  
Lévy Process ◽  
Initial Conditions ◽  
Random Variables ◽  
Levy Process ◽  
Independent Random Variables ◽  
Constant Multiple ◽  
The Stability ◽  
The One ◽  
Age Process

We consider a Lévy process with no negative jumps, reflected at a stochastic boundary that is a positive constant multiple of an age process associated with a Poisson process. We show that the stability condition for this process is identical to the one for the case of reflection at the origin. In particular, there exists a unique stationary distribution that is independent of initial conditions. We identify the Laplace-Stieltjes transform of the stationary distribution and observe that it satisfies a decomposition property. In fact, it is a sum of two independent random variables, one of which has the stationary distribution of the process reflected at the origin, and the other the stationary distribution of a certain clearing process. The latter is itself distributed as an infinite sum of independent random variables. Finally, we discuss the tail behavior of the stationary distribution and in particular observe that the second distribution in the decomposition always has a light tail.

scholarly journals Risk Theory with the Generalized Inverse Gaussian Lévy Process

2004 ◽  
Vol 34 (2) ◽  
pp. 361-377 ◽  
Author(s):  
Manuel Morales
Keyword(s):  
Lévy Process ◽  
Risk Model ◽  
Generalized Inverse ◽  
Compound Poisson Process ◽  
Levy Process ◽  
Inverse Gaussian ◽  
Lévy Measure ◽  
Compound Poisson ◽  
Inverse Gaussian Process ◽  
Generalized Inverse Gaussian

Dufresne et al. (1991) introduced a general risk model defined as the limit of compound Poisson processes. Such model is either a compound Poisson process itself or a strictly increasing Lévy process. Their construction is based on a non-negative non-increasing function Q that governs the jumps of the process. This function, it turns out, is the tail of the Lévy measure of the process. We discuss an illustration of their model using a generalized Inverse Gaussian (GIG) Lévy process. This increasing Lévy process has the gamma and the inverse Gaussian process as particular cases. Although mathematically more complex, the GIG Lévy process keeps some of the nice properties of the simpler gamma process.

For which functions are 𝑓(𝑋_{𝑡})-𝔼𝕗(𝕏_{𝕥}) and 𝕘(𝕏_{𝕥})/𝔼𝕘(𝕏_{𝕥}) martingales?

2021 ◽  
Vol 105 (0) ◽  
pp. 79-91
Author(s):  
F. Kühn ◽  
R. Schilling
Keyword(s):  
Lebesgue Measure ◽  
Lévy Process ◽  
Levy Process ◽  
Content Type ◽  
One Dimensional ◽  
Exponential Moments ◽  
Bounded Functions

Let X = ( X t ) t ≥ 0 X=(X_t)_{t\geq 0} be a one-dimensional Lévy process such that each X t X_t has a C b 1 C^1_b -density w. r. t. Lebesgue measure and certain polynomial or exponential moments. We characterize all polynomially bounded functions f : R → R f\colon \mathbb {R}\to \mathbb {R} , and exponentially bounded functions g : R → ( 0 , ∞ ) g\colon \mathbb {R}\to (0,\infty ) , such that f ( X t ) − E f ( X t ) f(X_t)-\mathbb {E} f(X_t) , resp. g ( X t ) / E g ( X t ) g(X_t)/\mathbb {E} g(X_t) , are martingales.

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