Optimal barrier strategy for spectrally negative Lévy process discounted by a class of exponential Lévy processes

2018 ◽  
Vol 12 (2) ◽  
pp. 326-337
Author(s):  
Huanqun Jiang
Keyword(s):  
Lévy Process ◽  
Lévy Processes ◽  
Levy Processes ◽  
Levy Process ◽  
Barrier Strategy ◽  
Dividend Payment ◽  
Surplus Process ◽  
Spectrally Negative Lévy Process ◽  
Exponential Lévy Process ◽  
Spectrally Negative Lévy Processes

AbstractIn this paper, we extend the optimality of the barrier strategy for the dividend payment problem to the setting that the underlying surplus process is a spectrally negative Lévy process and the discounting factor is an exponential Lévy process. The proof of the main result uses the fluctuation identities of spectrally negative Lévy processes. This extends recent results of Eisenberg for the case where the accumulated interest rate and surplus process are independent Brownian motions with drift.

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 Oscillation of modes of some semi-stable Lévy processes

1993 ◽  
Vol 132 ◽  
pp. 141-153 ◽  
Author(s):  
Toshiro Watanabe
Keyword(s):  
Lévy Process ◽  
Lévy Processes ◽  
Levy Processes ◽  
Levy Process ◽  
Stable Processes ◽  
Unimodal Lévy Process ◽  
Oscillating Mode

In this paper it is shown that there is a unimodal Levy process with oscillating mode. After the author first constructed an example of such a self-decomposable process, Sato pointed out that it belongs to the class of semi-stable processes with β < 0. We prove that all non-symmetric semi-stable self-decomposable processes with β < 0 have oscillating modes.

scholarly journals Spectrally Negative Lévy Processes Perturbed by Functionals of their Running Supremum

2012 ◽  
Vol 49 (4) ◽  
pp. 1005-1014 ◽  
Author(s):  
Andreas E. Kyprianou ◽  
Curdin Ott
Keyword(s):  
General Class ◽  
Lévy Process ◽  
Levy Process ◽  
Spectrally Negative Lévy Process ◽  
Rate Functions ◽  
Current Value ◽  
Insurance Model ◽  
Tax Payments ◽  
Risk Insurance ◽  
Spectrally Negative Lévy Processes

In the setting of the classical Cramér–Lundberg risk insurance model, Albrecher and Hipp (2007) introduced the idea of tax payments. More precisely, if X = {Xt: t≥ 0} represents the Cramér–Lundberg process and, for all t≥ 0, St=sup_{s≤ t}Xs, then Albrecher and Hipp studied Xt - γ St,t≥ 0, where γ∈(0,1) is the rate at which tax is paid. This model has been generalised to the setting that X is a spectrally negative Lévy process by Albrecher, Renaud and Zhou (2008). Finally, Kyprianou and Zhou (2009) extended this model further by allowing the rate at which tax is paid with respect to the process S = {St: t≥ 0} to vary as a function of the current value of S. Specifically, they considered the so-called perturbed spectrally negative Lévy process, Ut:=Xt -∫(0,t]γ(S_u)dSu,t≥ 0, under the assumptions that γ:[0,∞)→ [0,1) and ∫0∞ (1-γ(s))d s =∞. In this article we show that a number of the identities in Kyprianou and Zhou (2009) are still valid for a much more general class of rate functions γ:[0,∞)→∝. Moreover, we show that, with appropriately chosen γ, the perturbed process can pass continuously (i.e. creep) into (-∞, 0) in two different ways.

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.

Variance Optimal Stopping for Geometric Lévy Processes

2015 ◽  
Vol 47 (01) ◽  
pp. 128-145 ◽  
Author(s):  
Kamille Sofie Tågholt Gad ◽  
Jesper Lund Pedersen
Keyword(s):  
Optimal Stopping ◽  
Lévy Process ◽  
Lévy Processes ◽  
Levy Processes ◽  
Levy Process ◽  
Stopping Times ◽  
Optimal Stopping Problem ◽  
Maximum Variance ◽  
Geometric Levy Process

The main result of this paper is the solution to the optimal stopping problem of maximizing the variance of a geometric Lévy process. We call this problem the variance problem. We show that, for some geometric Lévy processes, we achieve higher variances by allowing randomized stopping. Furthermore, for some geometric Lévy processes, the problem has a solution only if randomized stopping is allowed. When randomized stopping is allowed, we give a solution to the variance problem. We identify the Lévy processes for which the allowance of randomized stopping times increases the maximum variance. When it does, we also solve the variance problem without randomized stopping.

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.

Multiple points for symmetric Lévy processes

1978 ◽  
Vol 83 (1) ◽  
pp. 83-90 ◽  
Author(s):  
John Hawkes
Keyword(s):  
Markov Process ◽  
Euclidean Space ◽  
Lévy Process ◽  
Lévy Processes ◽  
Transition Function ◽  
Levy Processes ◽  
Levy Process ◽  
Dimensional Euclidean Space ◽  
Multiple Points ◽  
Image Position

Let Xt be a Lévy process in Rd, d-dimensional euclidean space. That is X is a Markov process whose transition function satisfies

scholarly journals Necessity of weak subordination for some strongly subordinated Lévy processes

2021 ◽  
Vol 58 (4) ◽  
pp. 868-879
Author(s):  
Boris Buchmann ◽  
Kevin W. Lu
Keyword(s):  
Lévy Process ◽  
Lévy Processes ◽  
Levy Processes ◽  
Levy Process

AbstractConsider the strong subordination of a multivariate Lévy process with a multivariate subordinator. If the subordinate is a stack of independent Lévy processes and the components of the subordinator are indistinguishable within each stack, then strong subordination produces a Lévy process; otherwise it may not. Weak subordination was introduced to extend strong subordination, always producing a Lévy process even when strong subordination does not. Here we prove that strong and weak subordination are equal in law under the aforementioned condition. In addition, we prove that if strong subordination is a Lévy process then it is necessarily equal in law to weak subordination in two cases: firstly when the subordinator is deterministic, and secondly when it is pure-jump with finite activity.

scholarly journals A Liouville theorem for Lévy generators

Positivity ◽  
2020 ◽  
Author(s):  
Franziska Kühn
Keyword(s):  
Weak Solution ◽  
Laplace Equation ◽  
Lévy Process ◽  
Lévy Processes ◽  
Infinitesimal Generator ◽  
Polynomial Growth ◽  
Liouville Theorem ◽  
Levy Processes ◽  
Levy Process ◽  
Regularity Estimates

AbstractUnder mild assumptions, we establish a Liouville theorem for the “Laplace” equation $$Au=0$$ A u = 0 associated with the infinitesimal generator A of a Lévy process: If u is a weak solution to $$Au=0$$ A u = 0 which is at most of (suitable) polynomial growth, then u is a polynomial. As a by-product, we obtain new regularity estimates for semigroups associated with Lévy processes.

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