Distributional Study of De Finetti's Dividend Problem for a General Lévy Insurance Risk Process

2007 ◽  
Vol 44 (02) ◽  
pp. 428-443 ◽  
Author(s):  
A. E. Kyprianou ◽  
Z. Palmowski
Keyword(s):  
Net Present Value ◽  
Risk Process ◽  
Excursion Theory ◽  
Insurance Risk ◽  
Present Value ◽  
Deficit At Ruin ◽  
Spectrally Negative Lévy Process ◽  
The Deficit At Ruin ◽  
The Laplace Transform ◽  
Classical Control

We provide a distributional study of the solution to the classical control problem due to De Finetti (1957), Gerber (1969), Azcue and Muler (2005), and Avram et al. (2007), which concerns the optimal payment of dividends from an insurance risk process prior to ruin. Specifically, we build on recent work in the actuarial literature concerning calculations of the nth moment of the net present value of dividends paid out in the optimal strategy as well as the moments of the deficit at ruin and the Laplace transform of the red period. The calculations we present go much further than the existing literature, in that our calculations are valid for a general spectrally negative Lévy process as opposed to the classical Cramér–Lundberg process with exponentially distributed jumps. Moreover, the technique we use appeals principally to excursion theory rather than integro-differential equations and, for the case of the nth moment of the net present value of dividends, makes a new link with the distribution of integrated exponential subordinators.

scholarly journals Distributional Study of De Finetti's Dividend Problem for a General Lévy Insurance Risk Process

2007 ◽  
Vol 44 (02) ◽  
pp. 428-443 ◽  
Author(s):  
A. E. Kyprianou ◽  
Z. Palmowski
Keyword(s):  
Net Present Value ◽  
Risk Process ◽  
Excursion Theory ◽  
Insurance Risk ◽  
Present Value ◽  
Deficit At Ruin ◽  
Spectrally Negative Lévy Process ◽  
The Deficit At Ruin ◽  
The Laplace Transform ◽  
Classical Control

We provide a distributional study of the solution to the classical control problem due to De Finetti (1957), Gerber (1969), Azcue and Muler (2005), and Avram et al. (2007), which concerns the optimal payment of dividends from an insurance risk process prior to ruin. Specifically, we build on recent work in the actuarial literature concerning calculations of the nth moment of the net present value of dividends paid out in the optimal strategy as well as the moments of the deficit at ruin and the Laplace transform of the red period. The calculations we present go much further than the existing literature, in that our calculations are valid for a general spectrally negative Lévy process as opposed to the classical Cramér–Lundberg process with exponentially distributed jumps. Moreover, the technique we use appeals principally to excursion theory rather than integro-differential equations and, for the case of the nth moment of the net present value of dividends, makes a new link with the distribution of integrated exponential subordinators.

scholarly journals Distributional Study of De Finetti's Dividend Problem for a General Lévy Insurance Risk Process

2007 ◽  
Vol 44 (2) ◽  
pp. 428-443 ◽  
Author(s):  
A. E. Kyprianou ◽  
Z. Palmowski
Keyword(s):  
Net Present Value ◽  
Risk Process ◽  
Excursion Theory ◽  
Insurance Risk ◽  
Present Value ◽  
Deficit At Ruin ◽  
Spectrally Negative Lévy Process ◽  
The Deficit At Ruin ◽  
The Laplace Transform ◽  
Classical Control

We provide a distributional study of the solution to the classical control problem due to De Finetti (1957), Gerber (1969), Azcue and Muler (2005), and Avram et al. (2007), which concerns the optimal payment of dividends from an insurance risk process prior to ruin. Specifically, we build on recent work in the actuarial literature concerning calculations of the nth moment of the net present value of dividends paid out in the optimal strategy as well as the moments of the deficit at ruin and the Laplace transform of the red period. The calculations we present go much further than the existing literature, in that our calculations are valid for a general spectrally negative Lévy process as opposed to the classical Cramér–Lundberg process with exponentially distributed jumps. Moreover, the technique we use appeals principally to excursion theory rather than integro-differential equations and, for the case of the nth moment of the net present value of dividends, makes a new link with the distribution of integrated exponential subordinators.

scholarly journals A reinsurance risk model with a threshold coverage policy: the Gerber–Shiu penalty function

2017 ◽  
Vol 54 (1) ◽  
pp. 267-285 ◽  
Author(s):  
Onno J. Boxma ◽  
Esther Frostig ◽  
David Perry
Keyword(s):  
Risk Model ◽  
Risk Process ◽  
Insurance Risk ◽  
Deficit At Ruin ◽  
Scale Functions ◽  
The Deficit At Ruin ◽  
The Laplace Transform ◽  
Surplus Before Ruin ◽  
Set Up ◽  
Spectrally Negative Lévy Processes

AbstractWe consider a Cramér–Lundberg insurance risk process with the added feature of reinsurance. If an arriving claim finds the reserve below a certain threshold γ, or if it would bring the reserve below that level, then a reinsurer pays part of the claim. Using fluctuation theory and the theory of scale functions of spectrally negative Lévy processes, we derive expressions for the Laplace transform of the time to ruin and of the joint distribution of the deficit at ruin and the surplus before ruin. We specify these results in much more detail for the threshold set-up in the case of proportional reinsurance.

scholarly journals General tax Structures and the Lévy Insurance Risk Model

2009 ◽  
Vol 46 (04) ◽  
pp. 1146-1156 ◽  
Author(s):  
Andreas E. Kyprianou ◽  
Xiaowen Zhou
Keyword(s):  
Net Present Value ◽  
Risk Model ◽  
General Structure ◽  
Excursion Theory ◽  
Insurance Risk ◽  
Present Value ◽  
Scale Functions ◽  
Exit Problem ◽  
Tax Payments ◽  
Insurance Risk Model

In the spirit of Albrecher and Hipp (2007), and Albrecher, Renaud, and Zhou (2008) we consider a Lévy insurance risk model with tax payments of a more general structure than in the aforementioned papers, which was also considered in Albrecher, Borst, Boxma, and Resing (2009). In terms of scale functions, we establish three fundamental identities of interest which have stimulated a large volume of actuarial research in recent years. That is to say, the two-sided exit problem, the net present value of tax paid until ruin, as well as a generalized version of the Gerber–Shiu function. The method we appeal to differs from Albrecher and Hipp (2007), and Albrecher, Renaud, and Zhou (2008) in that we appeal predominantly to excursion theory.

scholarly journals General tax Structures and the Lévy Insurance Risk Model

2009 ◽  
Vol 46 (4) ◽  
pp. 1146-1156 ◽  
Author(s):  
Andreas E. Kyprianou ◽  
Xiaowen Zhou
Keyword(s):  
Net Present Value ◽  
Risk Model ◽  
General Structure ◽  
Excursion Theory ◽  
Insurance Risk ◽  
Present Value ◽  
Scale Functions ◽  
Exit Problem ◽  
Tax Payments ◽  
Insurance Risk Model

In the spirit of Albrecher and Hipp (2007), and Albrecher, Renaud, and Zhou (2008) we consider a Lévy insurance risk model with tax payments of a more general structure than in the aforementioned papers, which was also considered in Albrecher, Borst, Boxma, and Resing (2009). In terms of scale functions, we establish three fundamental identities of interest which have stimulated a large volume of actuarial research in recent years. That is to say, the two-sided exit problem, the net present value of tax paid until ruin, as well as a generalized version of the Gerber–Shiu function. The method we appeal to differs from Albrecher and Hipp (2007), and Albrecher, Renaud, and Zhou (2008) in that we appeal predominantly to excursion theory.

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 Gerber–Shiu distribution at Parisian ruin for Lévy insurance risk processes

2016 ◽  
Vol 53 (2) ◽  
pp. 572-584 ◽  
Author(s):  
Erik J. Baurdoux ◽  
Juan Carlos Pardo ◽  
José Luis Pérez ◽  
Jean-François Renaud
Keyword(s):  
Risk Process ◽  
Levy Processes ◽  
Insurance Company ◽  
Excursion Theory ◽  
Insurance Risk ◽  
Scale Functions ◽  
Surplus Process ◽  
Parisian Ruin ◽  
Risk Processes ◽  
Spectrally Negative Lévy Processes

Abstract Inspired by the works of Landriault et al. (2011), (2014), we study the Gerber–Shiu distribution at Parisian ruin with exponential implementation delays for a spectrally negative Lévy insurance risk process. To be more specific, we study the so-called Gerber–Shiu distribution for a ruin model where at each time the surplus process goes negative, an independent exponential clock is started. If the clock rings before the surplus becomes positive again then the insurance company is ruined. Our methodology uses excursion theory for spectrally negative Lévy processes and relies on the theory of so-called scale functions. In particular, we extend the recent results of Landriault et al. (2011), (2014).

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