scholarly journals Some Explicit Solutions for the Joint Density of the Time of Ruin and the Deficit at Ruin

2008 ◽  
Vol 38 (01) ◽  
pp. 259-276 ◽  
Author(s):  
David C.M. Dickson
Keyword(s):  
Risk Model ◽  
Joint Density ◽  
Explicit Solutions ◽  
Classical Risk Model ◽  
Exponential Distributions ◽  
Deficit At Ruin ◽  
Time Of Ruin ◽  
The Deficit At Ruin ◽  
The Time Of Ruin ◽  
The Classical Risk Model

Using probabilistic arguments we obtain an integral expression for the joint density of the time of ruin and the deficit at ruin. For the classical risk model, we obtain the bivariate Laplace transform of this joint density and invert it in the cases of individual claims distributed as Erlang(2) and as a mixture of two exponential distributions. As a consequence, we obtain explicit solutions for the density of the time of ruin.

scholarly journals Some Explicit Solutions for the Joint Density of the Time of Ruin and the Deficit at Ruin

2008 ◽  
Vol 38 (1) ◽  
pp. 259-276 ◽  
Author(s):  
David C.M. Dickson
Keyword(s):  
Risk Model ◽  
Joint Density ◽  
Explicit Solutions ◽  
Classical Risk Model ◽  
Exponential Distributions ◽  
Deficit At Ruin ◽  
Time Of Ruin ◽  
The Deficit At Ruin ◽  
The Time Of Ruin ◽  
The Classical Risk Model

Using probabilistic arguments we obtain an integral expression for the joint density of the time of ruin and the deficit at ruin. For the classical risk model, we obtain the bivariate Laplace transform of this joint density and invert it in the cases of individual claims distributed as Erlang(2) and as a mixture of two exponential distributions. As a consequence, we obtain explicit solutions for the density of the time of ruin.

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.

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.

On higher-order properties of compound geometric distributions

2002 ◽  
Vol 39 (02) ◽  
pp. 324-340 ◽  
Author(s):  
Gordon E. Willmot
Keyword(s):  
Equilibrium Distribution ◽  
Risk Model ◽  
Higher Order ◽  
Residual Lifetime ◽  
Classical Risk Model ◽  
Time Of Ruin ◽  
The Laplace Transform ◽  
The Time Of Ruin ◽  
Stop Loss ◽  
The Classical Risk Model

An explicit convolution representation for the equilibrium residual lifetime distribution of compound zero-modified geometric distributions is derived. Second-order reliability properties are seen to be essentially preserved under geometric compounding, and complement results of Brown (1990) and Cai and Kalashnikov (2000). The convolution representation is then extended to thenth-order equilibrium distribution. This higher-order convolution representation is used to evaluate the stop-loss premium and higher stop-loss moments of the compound zero-modified geometric distribution, as well as the Laplace transform of thekth moment of the time of ruin in the classical risk model.

scholarly journals The Joint Density of the Surplus Before and After Ruin in the Sparre Andersen Model

2007 ◽  
Vol 44 (3) ◽  
pp. 695-712 ◽  
Author(s):  
Susan M. Pitts ◽  
Konstadinos Politis
Keyword(s):  
Classical Model ◽  
Joint Density ◽  
Deficit At Ruin ◽  
Sparre Andersen Model ◽  
Andersen Model ◽  
Time Of Ruin ◽  
Collective Risk ◽  
Before And After ◽  
The Deficit At Ruin ◽  
The Time Of Ruin

Gerber and Shiu (1997) have studied the joint density of the time of ruin, the surplus immediately before ruin, and the deficit at ruin in the classical model of collective risk theory. More recently, their results have been generalised for risk models where the interarrival density for claims is nonexponential, but belongs to the Erlang family. Here we obtain generalisations of the Gerber-Shiu (1997) results that are valid in a general Sparre Andersen model, i.e. for any interclaim density. In particular, we obtain a generalisation of the key formula in that paper. Our results are made more concrete for the case where the distribution between claim arrivals is phase-type or the integrated tail distribution associated with the claim size distribution belongs to the class of subexponential distributions. Furthermore, we obtain conditions for finiteness of the joint moments of the surplus before ruin and the deficit at ruin in the Sparre Andersen model.

scholarly journals Algorithmic Analysis of the Sparre Andersen Model in Discrete Time

2007 ◽  
Vol 37 (02) ◽  
pp. 293-317 ◽  
Author(s):  
Attahiru Sule Alfa ◽  
Steve Drekic
Keyword(s):  
Discrete Time ◽  
Risk Model ◽  
Probability Distributions ◽  
Computational Procedure ◽  
Deficit At Ruin ◽  
Time Of Ruin ◽  
Special Cases ◽  
The Deficit At Ruin ◽  
Algorithmic Analysis ◽  
The Time Of Ruin

In this paper, we show that the delayed Sparre Andersen insurance risk model in discrete time can be analyzed as a doubly infinite Markov chain. We then describe how matrix analytic methods can be used to establish a computational procedure for calculating the probability distributions associated with fundamental ruin-related quantities of interest, such as the time of ruin, the surplus immediately prior to ruin, and the deficit at ruin. Special cases of the model, namely the ordinary and stationary Sparre Andersen models, are considered in several numerical examples.

Sign in / Sign up

Export Citation Format

Share Document