Remarks on the absolute maximum of a Lévy process

2002 ◽  
Vol 39 (02) ◽  
pp. 282-295
Author(s):  
Mykola Bratiychuk
Keyword(s):  
Asymptotic Behaviour ◽  
Lévy Process ◽  
Risk Process ◽  
Levy Process ◽  
Absolute Maximum ◽  
Lévy Measure ◽  
Process With Independent Increments ◽  
Independent Increments ◽  
The Absolute

Asymptotic behaviour of the distribution of the absolute maximum of a process with independent increments is studied depending on the properties of the Lévy measure of the process. Some applications to the risk process are also considered.

Remarks on the absolute maximum of a Lévy process

2002 ◽  
Vol 39 (2) ◽  
pp. 282-295
Author(s):  
Mykola Bratiychuk
Keyword(s):  
Asymptotic Behaviour ◽  
Lévy Process ◽  
Risk Process ◽  
Levy Process ◽  
Absolute Maximum ◽  
Lévy Measure ◽  
Process With Independent Increments ◽  
Independent Increments ◽  
The Absolute

Asymptotic behaviour of the distribution of the absolute maximum of a process with independent increments is studied depending on the properties of the Lévy measure of the process. Some applications to the risk process are also considered.

scholarly journals An Optimal Dividends Problem with a Terminal Value for Spectrally Negative Lévy Processes with a Completely Monotone Jump Density

2009 ◽  
Vol 46 (01) ◽  
pp. 85-98 ◽  
Author(s):  
R. L. Loeffen
Keyword(s):  
Lévy Process ◽  
Risk Process ◽  
Levy Process ◽  
Lévy Measure ◽  
Ruin Time ◽  
Optimal Dividends ◽  
Spectrally Negative Lévy Process ◽  
Completely Monotone ◽  
Monotone Density ◽  
Spectrally Negative Lévy Processes

We consider a modified version of the classical optimal dividends problem of de Finetti in which the objective function is altered by adding in an extra term which takes account of the ruin time of the risk process, the latter being modeled by a spectrally negative Lévy process. We show that, with the exception of a small class, a barrier strategy forms an optimal strategy under the condition that the Lévy measure has a completely monotone density. As a prerequisite for the proof, we show that, under the aforementioned condition on the Lévy measure, theq-scale function of the spectrally negative Lévy process has a derivative which is strictly log-convex.

scholarly journals An Optimal Dividends Problem with a Terminal Value for Spectrally Negative Lévy Processes with a Completely Monotone Jump Density

2009 ◽  
Vol 46 (1) ◽  
pp. 85-98 ◽  
Author(s):  
R. L. Loeffen
Keyword(s):  
Lévy Process ◽  
Risk Process ◽  
Levy Process ◽  
Lévy Measure ◽  
Ruin Time ◽  
Optimal Dividends ◽  
Spectrally Negative Lévy Process ◽  
Completely Monotone ◽  
Monotone Density ◽  
Spectrally Negative Lévy Processes

We consider a modified version of the classical optimal dividends problem of de Finetti in which the objective function is altered by adding in an extra term which takes account of the ruin time of the risk process, the latter being modeled by a spectrally negative Lévy process. We show that, with the exception of a small class, a barrier strategy forms an optimal strategy under the condition that the Lévy measure has a completely monotone density. As a prerequisite for the proof, we show that, under the aforementioned condition on the Lévy measure, the q-scale function of the spectrally negative Lévy process has a derivative which is strictly log-convex.

scholarly journals Last Exit Before an Exponential Time for Spectrally Negative Lévy Processes

2009 ◽  
Vol 46 (02) ◽  
pp. 542-558 ◽  
Author(s):  
E. J. Baurdoux
Keyword(s):  
Laplace Transform ◽  
Lévy Process ◽  
Risk Process ◽  
Levy Process ◽  
Risk Theory ◽  
Exponential Time ◽  
Main Motivation ◽  
Spectrally Negative Lévy Process ◽  
The Laplace Transform ◽  
Spectrally Negative Lévy Processes

Chiu and Yin (2005) found the Laplace transform of the last time a spectrally negative Lévy process, which drifts to ∞, is below some level. The main motivation for the study of this random time stems from risk theory: what is the last time the risk process, modeled by a spectrally negative Lévy process drifting to ∞, is 0? In this paper we extend the result of Chiu and Yin, and we derive the Laplace transform of the last time, before an independent, exponentially distributed time, that a spectrally negative Lévy process (without any further conditions) exceeds (upwards or downwards) or hits a certain level. As an application, we extend a result found in Doney (1991).

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 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.

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 The moments of the discounted loss and the discounted dividends for a spectrally negative Lévy risk process

2015 ◽  
Vol 52 (03) ◽  
pp. 665-687
Author(s):  
Esther Frostig
Keyword(s):  
Lévy Process ◽  
Infinite Horizon ◽  
Risk Process ◽  
Levy Process ◽  
Dual Model ◽  
Exit Times ◽  
Barrier Strategy ◽  
Spectrally Negative Lévy Process

Consider a spectrally negative risk process where, on ruin, the deficit is immediately paid, and the process restarts from 0. When the process reaches a threshold b, all the surplus above b is paid as dividend. Applying the theory of exit times for a spectrally negative Lévy process and its reflection at the maximum and at the minimum, we obtain recursive formulae for the following moments. (i) The moments of the discounted loss until the process reaches b. This is equivalent to the moments of the discounted dividends in the dual model under the barrier strategy. (ii) The moments of the discounted loss for models with and without a dividend barrier for the infinite horizon. (iii) The moments of the discounted dividends for the infinite horizon.

Sign in / Sign up

Export Citation Format

Share Document