Minimising expected discounted capital injections by reinsurance in a classical risk model

2011 ◽  
Vol 2011 (3) ◽  
pp. 155-176 ◽  
Author(s):  
Julia Eisenberg ◽  
Hanspeter Schmidli
Keyword(s):  
Risk Model ◽  
Classical Risk Model ◽  
Capital Injections

scholarly journals Optimal Control of Capital Injections by Reinsurance with a Constant Rate of Interest

2011 ◽  
Vol 48 (3) ◽  
pp. 733-748 ◽  
Author(s):  
Julia Eisenberg ◽  
Hanspeter Schmidli
Keyword(s):  
Diffusion Approximation ◽  
Value Function ◽  
Risk Model ◽  
Classical Risk Model ◽  
Surplus Process ◽  
The Individual ◽  
Reinsurance Treaty ◽  
Riskless Asset ◽  
Capital Injections ◽  
The Value Function

We consider a classical risk model and its diffusion approximation, where the individual claims are reinsured by a reinsurance treaty with deductible b ∈ [0, b̃]. Here b = b̃ means ‘no reinsurance’ and b= 0 means ‘full reinsurance’. In addition, the insurer is allowed to invest in a riskless asset with some constant interest rate m > 0. The cedent can choose an adapted reinsurance strategy {bt}t≥0, i.e. the parameter can be changed continuously. If the surplus process becomes negative, the cedent has to inject additional capital. Our aim is to minimise the expected discounted capital injections over all admissible reinsurance strategies. We find an explicit expression for the value function and the optimal strategy using the Hamilton-Jacobi-Bellman approach in the case of a diffusion approximation. In the case of the classical risk model, we show the existence of a ‘weak’ solution and calculate the value function numerically.

scholarly journals Optimal Control of Capital Injections by Reinsurance with a Constant Rate of Interest

2011 ◽  
Vol 48 (03) ◽  
pp. 733-748
Author(s):  
Julia Eisenberg ◽  
Hanspeter Schmidli
Keyword(s):  
Diffusion Approximation ◽  
Value Function ◽  
Risk Model ◽  
Classical Risk Model ◽  
Surplus Process ◽  
The Individual ◽  
Reinsurance Treaty ◽  
Riskless Asset ◽  
Capital Injections ◽  
The Value Function

We consider a classical risk model and its diffusion approximation, where the individual claims are reinsured by a reinsurance treaty with deductible b ∈ [0, b̃]. Here b = b̃ means ‘no reinsurance’ and b= 0 means ‘full reinsurance’. In addition, the insurer is allowed to invest in a riskless asset with some constant interest rate m > 0. The cedent can choose an adapted reinsurance strategy {b t } t≥0, i.e. the parameter can be changed continuously. If the surplus process becomes negative, the cedent has to inject additional capital. Our aim is to minimise the expected discounted capital injections over all admissible reinsurance strategies. We find an explicit expression for the value function and the optimal strategy using the Hamilton-Jacobi-Bellman approach in the case of a diffusion approximation. In the case of the classical risk model, we show the existence of a ‘weak’ solution and calculate the value function numerically.

scholarly journals A VaR-Type Risk Measure Derived from Cumulative Parisian Ruin for the Classical Risk Model

Risks ◽  
2018 ◽  
Vol 6 (3) ◽  
pp. 85 ◽  
Author(s):  
Mohamed Lkabous ◽  
Jean-François Renaud
Keyword(s):  
Risk Model ◽  
Risk Measures ◽  
Risk Measure ◽  
Short Paper ◽  
Classical Risk Model ◽  
Parisian Ruin ◽  
The Classical Risk Model

In this short paper, we study a VaR-type risk measure introduced by Guérin and Renaud and which is based on cumulative Parisian ruin. We derive some properties of this risk measure and we compare it to the risk measures of Trufin et al. and Loisel and Trufin.

An adaptive premium policy with a Bayesian motivation in the classical risk model

2012 ◽  
Vol 51 (2) ◽  
pp. 370-378 ◽  
Author(s):  
David Landriault ◽  
Christiane Lemieux ◽  
Gordon E. Willmot
Keyword(s):  
Risk Model ◽  
Classical Risk Model ◽  
The Classical Risk Model

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.

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