scholarly journals Dependent Risk Models with Bivariate Phase-Type Distributions

2009 ◽  
Vol 46 (01) ◽  
pp. 113-131 ◽  
Author(s):  
Andrei L. Badescu ◽  
Eric C. K. Cheung ◽  
David Landriault
Keyword(s):  
Fluid Flow ◽  
Penalty Function ◽  
Risk Model ◽  
Fluid Flows ◽  
Dependence Structure ◽  
Deficit At Ruin ◽  
Phase Type ◽  
The Deficit At Ruin ◽  
Discounted Penalty Function ◽  
Risk Processes

In this paper we consider an extension of the Sparre Andersen insurance risk model by relaxing one of its independence assumptions. The newly proposed dependence structure is introduced through the premise that the joint distribution of the interclaim time and the subsequent claim size is bivariate phase-type (see, e.g. Assaf et al. (1984) and Kulkarni (1989)). Relying on the existing connection between risk processes and fluid flows (see, e.g. Badescu et al. (2005), Badescu, Drekic and Landriault (2007), Ramaswami (2006), and Ahn, Badescu and Ramaswami (2007)), we construct an analytically tractable fluid flow that leads to the analysis of various ruin-related quantities in the aforementioned risk model. Using matrix-analytic methods, we obtain an explicit expression for the Gerber–Shiu discounted penalty function (see Gerber and Shiu (1998)) when the penalty function depends on the deficit at ruin only. Finally, we investigate how some ruin-related quantities involving the surplus immediately prior to ruin can also be analyzed via our fluid flow methodology.

scholarly journals Dependent Risk Models with Bivariate Phase-Type Distributions

2009 ◽  
Vol 46 (1) ◽  
pp. 113-131 ◽  
Author(s):  
Andrei L. Badescu ◽  
Eric C. K. Cheung ◽  
David Landriault
Keyword(s):  
Fluid Flow ◽  
Penalty Function ◽  
Risk Model ◽  
Fluid Flows ◽  
Dependence Structure ◽  
Deficit At Ruin ◽  
Phase Type ◽  
The Deficit At Ruin ◽  
Discounted Penalty Function ◽  
Risk Processes

In this paper we consider an extension of the Sparre Andersen insurance risk model by relaxing one of its independence assumptions. The newly proposed dependence structure is introduced through the premise that the joint distribution of the interclaim time and the subsequent claim size is bivariate phase-type (see, e.g. Assaf et al. (1984) and Kulkarni (1989)). Relying on the existing connection between risk processes and fluid flows (see, e.g. Badescu et al. (2005), Badescu, Drekic and Landriault (2007), Ramaswami (2006), and Ahn, Badescu and Ramaswami (2007)), we construct an analytically tractable fluid flow that leads to the analysis of various ruin-related quantities in the aforementioned risk model. Using matrix-analytic methods, we obtain an explicit expression for the Gerber–Shiu discounted penalty function (see Gerber and Shiu (1998)) when the penalty function depends on the deficit at ruin only. Finally, we investigate how some ruin-related quantities involving the surplus immediately prior to ruin can also be analyzed via our fluid flow methodology.

scholarly journals Approximations for the Gerber-Shiu expected discounted penalty function in the compound poisson risk model

2007 ◽  
Vol 39 (2) ◽  
pp. 385-406 ◽  
Author(s):  
Susan M Pitts ◽  
Konstadinos Politis
Keyword(s):  
Penalty Function ◽  
Risk Model ◽  
Classical Risk Model ◽  
Deficit At Ruin ◽  
Expected Discounted Penalty Function ◽  
Time Of Ruin ◽  
Special Cases ◽  
The Deficit At Ruin ◽  
The Time Of Ruin ◽  
Expected Discounted Penalty

In the classical risk model with initial capital u, let τ(u) be the time of ruin, X+(u) be the risk reserve just before ruin, and Y+(u) be the deficit at ruin. Gerber and Shiu (1998) defined the function mδ(u) =E[e−δ τ(u)w(X+(u), Y+(u)) 1 (τ(u) < ∞)], where δ ≥ 0 can be interpreted as a force of interest and w(r,s) as a penalty function, meaning that mδ(u) is the expected discounted penalty payable at ruin. This function is known to satisfy a defective renewal equation, but easy explicit formulae for mδ(u) are only available for certain special cases for the claim size distribution. Approximations thus arise by approximating the desired mδ(u) by that associated with one of these special cases. In this paper a functional approach is taken, giving rise to first-order correction terms for the above approximations.

On the Expected Discounted Penalty Function for a Risk Model with Thinning Process

2010 ◽  
Vol 29-32 ◽  
pp. 1156-1161
Author(s):  
Wen Guang Yu
Keyword(s):  
Penalty Function ◽  
Risk Model ◽  
Standard Wiener Process ◽  
Deficit At Ruin ◽  
Expected Discounted Penalty Function ◽  
Time Of Ruin ◽  
Special Cases ◽  
Discounted Penalty Function ◽  
Thinning Process ◽  
Expected Discounted Penalty

This paper studies the expected discounted penalty function for a risk model in which the arrival of insurance policies is a Poisson process and the process of claim occurring is -thinning process. Using backward differential argument, we derive the integro-differential equation satisfied by the expected discounted penalty function when the stochastic discount interest process is perturbed by standard Wiener process and Poisson-Geometric process. Applications of the integral equation are given to the Laplace transform of the time of ruin, the deficit at ruin, the surplus immediately before ruin occurs. In some special cases with exponential distributions, closed form expressions for these quantities are obtained.

scholarly journals RATIO MONOTONICITY FOR TAIL PROBABILITIES IN THE RENEWAL RISK MODEL

2011 ◽  
Vol 25 (2) ◽  
pp. 171-185
Author(s):  
Georgios Psarrakos ◽  
Michael Tsatsomeros
Keyword(s):  
Risk Model ◽  
Asymptotic Result ◽  
Height Distribution ◽  
Tail Probabilities ◽  
Deficit At Ruin ◽  
Ladder Height ◽  
Phase Type ◽  
The Deficit At Ruin ◽  
Renewal Risk ◽  
Monotonicity Results

A renewal model in risk theory is considered, where $\overline{H}(u,y)$ is the tail of the distribution of the deficit at ruin with initial surplus u and $\overline{F}(y)$ is the tail of the ladder height distribution. Conditions are derived under which the ratio $\overline{H}(u,y)/\overline{F}(u+y)$ is nondecreasing in u for any y≥0. In particular, it is proven that if the ladder height distribution is stable and DFR or phase type, then the above ratio is nondecreasing in u. As a byproduct of this monotonicity, an upper bound and an asymptotic result for $\overline{H}(u,y)$ are derived. Examples are given to illustrate the monotonicity results.

scholarly journals Approximations for the Gerber-Shiu expected discounted penalty function in the compound poisson risk model

2007 ◽  
Vol 39 (02) ◽  
pp. 385-406
Author(s):  
Susan M Pitts ◽  
Konstadinos Politis
Keyword(s):  
Penalty Function ◽  
Risk Model ◽  
Classical Risk Model ◽  
Deficit At Ruin ◽  
Expected Discounted Penalty Function ◽  
Time Of Ruin ◽  
Special Cases ◽  
The Deficit At Ruin ◽  
The Time Of Ruin ◽  
Expected Discounted Penalty

In the classical risk model with initial capital u, let τ(u) be the time of ruin, X +(u) be the risk reserve just before ruin, and Y +(u) be the deficit at ruin. Gerber and Shiu (1998) defined the function m δ(u) =E[e−δ τ(u) w(X +(u), Y +(u)) 1 (τ(u) &lt; ∞)], where δ ≥ 0 can be interpreted as a force of interest and w(r,s) as a penalty function, meaning that m δ(u) is the expected discounted penalty payable at ruin. This function is known to satisfy a defective renewal equation, but easy explicit formulae for m δ(u) are only available for certain special cases for the claim size distribution. Approximations thus arise by approximating the desired m δ(u) by that associated with one of these special cases. In this paper a functional approach is taken, giving rise to first-order correction terms for the above approximations.

scholarly journals The Gerber-Shiu Expected Penalty Function for the Risk Model with Dependence and a Constant Dividend Barrier

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Donghai Liu ◽  
Zaiming Liu ◽  
Dan Peng
Keyword(s):  
Linear Combination ◽  
Penalty Function ◽  
Integrodifferential Equation ◽  
Risk Model ◽  
Dependence Structure ◽  
Claim Amount ◽  
Compound Poisson ◽  
Compound Poisson Risk Model ◽  
Discounted Penalty Function ◽  
Constant Dividend Barrier

We consider a compound Poisson risk model with dependence and a constant dividend barrier. A dependence structure between the claim amount and the interclaim time is introduced through a Farlie-Gumbel-Morgenstern copula. An integrodifferential equation for the Gerber-Shiu discounted penalty function is derived. We also solve the integrodifferential equation and show that the solution is a linear combination of the Gerber-Shiu function with no barrier and the solution of an associated homogeneous integrodifferential equation.

Expected discounted penalty function for a phase-type risk model with stochastic return on investment and random observation periods

Kybernetes ◽  
2018 ◽  
Vol 47 (7) ◽  
pp. 1420-1434
Author(s):  
Wenyan Zhuo ◽  
Honglin Yang ◽  
Xu Chen
Keyword(s):  
Penalty Function ◽  
Return On Investment ◽  
Risk Model ◽  
Insurance Company ◽  
Content Type ◽  
Expected Discounted Penalty Function ◽  
Sinc Method ◽  
Phase Type ◽  
Discounted Penalty Function ◽  
Expected Discounted Penalty

Purpose The purpose of this paper is to build a phase-type risk model with stochastic return on investment and random observation periods to characterize the ruin quantities under which the insurance company may take effective investment strategies to avoid bankruptcy. Design/methodology/approach By the Markov property and Ito’s formula, this paper derives the integro-differential equations in which the interclaim times follow a phase-type distribution. Using the sinc method, this paper obtains the approximate solutions of the expected discounted penalty function. The numerical examples are given to verify the robustness of the proposed sinc method. Findings This paper discloses the relationship between the investment strategy and initial surplus level. The insurance company with a high initial surplus level prefers high risk portfolios to earn more profit. Contrarily, the insurance company would invest low risk portfolios to avoid bankruptcy. In addition, this paper shows that a short observation period would bring higher ruin probability. Originality/value The risk model is distinct in that a phase-type risk model is constructed with stochastic return on investment and random observation periods. These considerations in the risk model are in sharp contrast to the setting in which the stochastic return on investment is observed continuously. In practice, the insurance company only can periodically observe the surplus level to check the balance of the book. This setting, therefore, is difficult to adopt. This paper develops a sinc method to solve the approximate solutions of the expected discounted penalty function.

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