On higher-order properties of compound geometric distributions

2002 ◽  
Vol 39 (2) ◽  
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 the nth-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 the kth moment of the time of ruin in the classical risk model.

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 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.

The density of the time of ruin in the classical risk model with a constant dividend barrier

2013 ◽  
Vol 8 (1) ◽  
pp. 63-78 ◽  
Author(s):  
Shuanming Li ◽  
Yi Lu
Keyword(s):  
Density Function ◽  
Risk Model ◽  
Special Functions ◽  
Laplace Transforms ◽  
Classical Risk Model ◽  
Dividend Payments ◽  
Time Of Ruin ◽  
Constant Dividend Barrier ◽  
The Time Of Ruin ◽  
The Classical Risk Model

AbstractIn this paper, we investigate the density function of the time of ruin in the classical risk model with a constant dividend barrier. When claims are exponentially distributed, we derive explicit expressions for the density function of the time of ruin and its decompositions: the density of the time of ruin without dividend payments and the density of the time of ruin with dividend payments. These densities are obtained based on their Laplace transforms, and expressed in terms of some special functions which are computationally tractable. The Laplace transforms are being inverted using a magnificent tool, the Lagrange inverse formula, developed in Dickson and Willmot (2005). Several numerical examples are given to illustrate our results.

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.

SOME ADVANCES ON THE ERLANG(n) DUAL RISK MODEL

2014 ◽  
Vol 45 (1) ◽  
pp. 127-150 ◽  
Author(s):  
Eugenio V. Rodríguez-Martínez ◽  
Rui M. R. Cardoso ◽  
Alfredo D. Egídio dos Reis
Keyword(s):  
Risk Model ◽  
Waiting Times ◽  
Dual Model ◽  
Time Of Ruin ◽  
Constant Rate ◽  
Dual Risk Model ◽  
The Laplace Transform ◽  
The Time Of Ruin ◽  
Renewal Risk ◽  
The Individual

AbstractThe dual risk model assumes that the surplus of a company decreases at a constant rate over time and grows by means of upward jumps, which occur at random times and sizes. It is said to have applications to companies with economical activities involved in research and development. This model is dual to the well-known Cramér-Lundberg risk model with applications to insurance. Most existing results on the study of the dual model assume that the random waiting times between consecutive gains follow an exponential distribution, as in the classical Cramér-Lundberg risk model. We generalize to other compound renewal risk models where such waiting times are Erlang(n) distributed. Using the roots of the fundamental and the generalized Lundberg's equations, we get expressions for the ruin probability and the Laplace transform of the time of ruin for an arbitrary single gain distribution. Furthermore, we compute expected discounted dividends, as well as higher moments, when the individual common gains follow a Phase-Type, PH(m), distribution. We also perform illustrations working some examples for some particular gain distributions and obtain numerical results.

On applications of residual lifetimes of compound geometric convolutions

2004 ◽  
Vol 41 (03) ◽  
pp. 802-815
Author(s):  
Gordon E. Willmot ◽  
Jun Cai
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Risk Model ◽  
Residual Lifetime ◽  
Waiting Time Distribution ◽  
Classical Risk Model ◽  
Virtual Waiting Time ◽  
Compound Geometric Distribution ◽  
Renewal Risk ◽  
The Classical Risk Model

We demonstrate that the residual lifetime distribution of a compound geometric distribution convoluted with another distribution, termed a compound geometric convolution, is again a compound geometric convolution. Conditions under which the compound geometric convolution is new worse than used (NWU) or new better than used (NBU) are then derived. The results are applied to ruin probabilities in the stationary renewal risk model where the convolution components are of particular interest, as well as to the equilibrium virtual waiting time distribution in the G/G/1 queue, an approximation to the equilibrium M/G/c waiting time distribution, ruin in the classical risk model perturbed by diffusion, and second-order reliability classifications.

scholarly journals Numerical Ruin Probability in the Dual Risk Model with Risk-Free Investments

Risks ◽  
2018 ◽  
Vol 6 (4) ◽  
pp. 110 ◽  
Author(s):  
Sooie-Hoe Loke ◽  
Enrique Thomann
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
Ruin Probability ◽  
Risk Model ◽  
Constant Force ◽  
Time Of Ruin ◽  
Dual Risk Model ◽  
The Laplace Transform ◽  
Force Of Interest ◽  
The Time Of Ruin

In this paper, a dual risk model under constant force of interest is considered. The ruin probability in this model is shown to satisfy an integro-differential equation, which can then be written as an integral equation. Using the collocation method, the ruin probability can be well approximated for any gain distributions. Examples involving exponential, uniform, Pareto and discrete gains are considered. Finally, the same numerical method is applied to the Laplace transform of the time of ruin.

scholarly journals On the Expected Discounted Penalty Function for the Classical Risk Model with Potentially Delayed Claims and Random Incomes

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Huiming Zhu ◽  
Ya Huang ◽  
Xiangqun Yang ◽  
Jieming Zhou
Keyword(s):  
Penalty Function ◽  
Risk Model ◽  
Claim Amount ◽  
Classical Risk Model ◽  
Main Claim ◽  
Expected Discounted Penalty Function ◽  
The Laplace Transform ◽  
Discounted Penalty Function ◽  
The Classical Risk Model ◽  
Expected Discounted Penalty

We focus on the expected discounted penalty function of a compound Poisson risk model with random incomes and potentially delayed claims. It is assumed that each main claim will produce a byclaim with a certain probability and the occurrence of the byclaim may be delayed depending on associated main claim amount. In addition, the premium number process is assumed as a Poisson process. We derive the integral equation satisfied by the expected discounted penalty function. Given that the premium size is exponentially distributed, the explicit expression for the Laplace transform of the expected discounted penalty function is derived. Finally, for the exponential claim sizes, we present the explicit formula for the expected discounted penalty function.

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 applications of residual lifetimes of compound geometric convolutions

2004 ◽  
Vol 41 (3) ◽  
pp. 802-815 ◽  
Author(s):  
Gordon E. Willmot ◽  
Jun Cai
Keyword(s):  
Waiting Time ◽  
Time Distribution ◽  
Risk Model ◽  
Residual Lifetime ◽  
Waiting Time Distribution ◽  
Classical Risk Model ◽  
Virtual Waiting Time ◽  
Compound Geometric Distribution ◽  
Renewal Risk ◽  
The Classical Risk Model

We demonstrate that the residual lifetime distribution of a compound geometric distribution convoluted with another distribution, termed a compound geometric convolution, is again a compound geometric convolution. Conditions under which the compound geometric convolution is new worse than used (NWU) or new better than used (NBU) are then derived. The results are applied to ruin probabilities in the stationary renewal risk model where the convolution components are of particular interest, as well as to the equilibrium virtual waiting time distribution in the G/G/1 queue, an approximation to the equilibrium M/G/c waiting time distribution, ruin in the classical risk model perturbed by diffusion, and second-order reliability classifications.

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