scholarly journals Convexity of Ruin Probability and Optimal Dividend Strategies for a General Lévy Process

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Chuancun Yin ◽  
Kam Chuen Yuen ◽  
Ying Shen
Keyword(s):  
Lévy Process ◽  
Levy Process ◽  
Probability Of Ruin ◽  
Optimal Dividends ◽  
Time Of Ruin ◽  
Optimal Dividend ◽  
Convexity Properties ◽  
The Time Of Ruin ◽  
Cash Reserves ◽  
A Company

We consider the optimal dividends problem for a company whose cash reserves follow a general Lévy process with certain positive jumps and arbitrary negative jumps. The objective is to find a policy which maximizes the expected discounted dividends until the time of ruin. Under appropriate conditions, we use some recent results in the theory of potential analysis of subordinators to obtain the convexity properties of probability of ruin. We present conditions under which the optimal dividend strategy, among all admissible ones, takes the form of a barrier strategy.

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 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.

Stochastic differential equations for ruin probabilities

1995 ◽  
Vol 32 (01) ◽  
pp. 74-89 ◽  
Author(s):  
Christian Max Møller
Keyword(s):  
Differential Equations ◽  
Point Processes ◽  
Diffusion Processes ◽  
Marked Point ◽  
Marked Point Processes ◽  
Ruin Probabilities ◽  
Probability Of Ruin ◽  
Time Of Ruin ◽  
Markov Structure ◽  
The Time Of Ruin

The present paper proposes a general approach for finding differential equations to evaluate probabilities of ruin in finite and infinite time. Attention is given to real-valued non-diffusion processes where a Markov structure is obtainable. Ruin is allowed to occur upon a jump or between the jumps. The key point is to define a process of conditional ruin probabilities and identify this process stopped at the time of ruin as a martingale. Using the theory of marked point processes together with the change-of-variable formula or the martingale representation theorem for point processes, we obtain differential equations for evaluating the probability of ruin. Numerical illustrations are given by solving a partial differential equation numerically to obtain the probability of ruin over a finite time horizon.

scholarly journals Optimal Dividends in the Dual Model with Diffusion

2008 ◽  
Vol 38 (02) ◽  
pp. 653-667 ◽  
Author(s):  
Benjamin Avanzi ◽  
Hans U. Gerber
Keyword(s):  
Poisson Process ◽  
Wiener Process ◽  
Compound Poisson Process ◽  
Dual Model ◽  
Optimal Dividends ◽  
Sample Paths ◽  
Optimal Dividend ◽  
The Individual ◽  
A Company ◽  
Poisson Parameter

In the dual model, the surplus of a company is a Lévy process with sample paths that are skip-free downwards. In this paper, the aggregate gains process is the sum of a shifted compound Poisson process and an independent Wiener process. By means of Laplace transforms, it is shown how the expectation of the discounted dividends until ruin can be calculated, if a barrier strategy is applied, and how the optimal dividend barrier can be determined. Conditions for optimality are discussed and several numerical illustrations are given. Furthermore, a family of models is analysed where the individual gain amount distribution is rescaled and compensated by a change of the Poisson parameter.

scholarly journals Optimal Dividends in the Dual Model with Diffusion

2008 ◽  
Vol 38 (2) ◽  
pp. 653-667 ◽  
Author(s):  
Benjamin Avanzi ◽  
Hans U. Gerber
Keyword(s):  
Poisson Process ◽  
Wiener Process ◽  
Compound Poisson Process ◽  
Dual Model ◽  
Optimal Dividends ◽  
Sample Paths ◽  
Optimal Dividend ◽  
The Individual ◽  
A Company ◽  
Poisson Parameter

In the dual model, the surplus of a company is a Lévy process with sample paths that are skip-free downwards. In this paper, the aggregate gains process is the sum of a shifted compound Poisson process and an independent Wiener process. By means of Laplace transforms, it is shown how the expectation of the discounted dividends until ruin can be calculated, if a barrier strategy is applied, and how the optimal dividend barrier can be determined. Conditions for optimality are discussed and several numerical illustrations are given. Furthermore, a family of models is analysed where the individual gain amount distribution is rescaled and compensated by a change of the Poisson parameter.

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 Risque de décès et risque de ruine: Réflexions sur la mesure du risque de ruine

1992 ◽  
Vol 22 (1) ◽  
pp. 107-119 ◽  
Author(s):  
Par Marc-Henri Amsler
Keyword(s):  
Closed Form ◽  
Probability Of Ruin ◽  
Closed Form Solutions ◽  
Time Of Ruin ◽  
The Time Of Ruin ◽  
Severity Of Ruin

SummaryThe paper proposes a three parameter measure by which the risk of a portfolio can be assessed. The parameters are: the probability of ruin, the severity of ruin (i.e. the amount of the deficit when ruin occurs) and the time of ruin. This type of analysis does not lend itself to closed form solutions, but it can be easily carried out on a PC. The author presents some theoretical and practical examples.

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.

Stochastic differential equations for ruin probabilities

1995 ◽  
Vol 32 (1) ◽  
pp. 74-89 ◽  
Author(s):  
Christian Max Møller
Keyword(s):  
Differential Equations ◽  
Point Processes ◽  
Diffusion Processes ◽  
Marked Point ◽  
Marked Point Processes ◽  
Ruin Probabilities ◽  
Probability Of Ruin ◽  
Time Of Ruin ◽  
Markov Structure ◽  
The Time Of Ruin

The present paper proposes a general approach for finding differential equations to evaluate probabilities of ruin in finite and infinite time. Attention is given to real-valued non-diffusion processes where a Markov structure is obtainable. Ruin is allowed to occur upon a jump or between the jumps. The key point is to define a process of conditional ruin probabilities and identify this process stopped at the time of ruin as a martingale. Using the theory of marked point processes together with the change-of-variable formula or the martingale representation theorem for point processes, we obtain differential equations for evaluating the probability of ruin.Numerical illustrations are given by solving a partial differential equation numerically to obtain the probability of ruin over a finite time horizon.

scholarly journals Optimality of refraction strategies for a constrained dividend problem

2019 ◽  
Vol 51 (03) ◽  
pp. 633-666
Author(s):  
Mauricio Junca ◽  
Harold A. Moreno-Franco ◽  
José Luis Pérez ◽  
Kazutoshi Yamazaki
Keyword(s):  
Lebesgue Measure ◽  
Control Strategies ◽  
Risk Models ◽  
Complementary Slackness ◽  
Absolutely Continuous ◽  
Numerical Examples ◽  
Time Of Ruin ◽  
Optimal Dividend ◽  
The Time Of Ruin ◽  
Complementary Slackness Conditions

AbstractWe consider de Finetti’s problem for spectrally one-sided Lévy risk models with control strategies that are absolutely continuous with respect to the Lebesgue measure. Furthermore, we consider the version with a constraint on the time of ruin. To characterize the solution to the aforementioned models, we first solve the optimal dividend problem with a terminal value at ruin and show the optimality of threshold strategies. Next, we introduce the dual Lagrangian problem and show that the complementary slackness conditions are satisfied, characterizing the optimal Lagrange multiplier. Finally, we illustrate our findings with a series of numerical examples.

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