scholarly journals On the optimal dividend problem for a spectrally negative Lévy process

2007 ◽  
Vol 17 (1) ◽  
pp. 156-180 ◽  
Author(s):  
Florin Avram ◽  
Zbigniew Palmowski ◽  
Martijn R. Pistorius
Keyword(s):  
Lévy Process ◽  
Levy Process ◽  
Spectrally Negative Lévy Process ◽  
Optimal Dividend

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 ON THE OPTIMAL DIVIDEND PROBLEM FOR A SPECTRALLY POSITIVE LÉVY PROCESS

2014 ◽  
Vol 44 (3) ◽  
pp. 635-651 ◽  
Author(s):  
Chuancun Yin ◽  
Yuzhen Wen ◽  
Yongxia Zhao
Keyword(s):  
Lévy Process ◽  
Risk Model ◽  
Levy Process ◽  
Dual Model ◽  
Threshold Strategy ◽  
Classical Risk Model ◽  
Surplus Process ◽  
Optimal Dividend ◽  
Special Cases ◽  
The Classical Risk Model

AbstractIn this paper we study the optimal dividend problem for a company whose surplus process evolves as a spectrally positive Lévy process before dividends are deducted. This model includes the dual model of the classical risk model and the dual model with diffusion as special cases. We assume that dividends are paid to the shareholders according to an admissible strategy whose dividend rate is bounded by a constant. The objective is to find a dividend policy so as to maximize the expected discounted value of dividends which are paid to the shareholders until the company is ruined. We show that the optimal dividend strategy is formed by a threshold strategy.

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 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.

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

2012 ◽  
Vol 49 (04) ◽  
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 = {X t : t≥ 0} represents the Cramér–Lundberg process and, for all t≥ 0, S t =sup_{s≤ t}X s , then Albrecher and Hipp studied X t - γ S t ,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 = {S t : t≥ 0} to vary as a function of the current value of S. Specifically, they considered the so-called perturbed spectrally negative Lévy process, U t :=X t -∫(0,t]γ(S_u)dS u ,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 The moments of the discounted loss and the discounted dividends for a spectrally negative Lévy risk process

2015 ◽  
Vol 52 (3) ◽  
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.

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