scholarly journals Asymptotics for the Solutions to Defective Renewal Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Kaiyong Wang ◽  
Yang Chen ◽  
Zhongquan Tan
Keyword(s):  
Renewal Equations ◽  
Heavy Tailed ◽  
Distribution Class ◽  
Defective Renewal Equations

This paper investigates the defective renewal equations under the nonconvolution equivalent distribution class. The asymptotics of the solution to the defective renewal equations have been given for the heavy-tailed and light-tailed cases, respectively.

scholarly journals On a general class of renewal risk process: analysis of the Gerber-Shiu function

2005 ◽  
Vol 37 (03) ◽  
pp. 836-856 ◽  
Author(s):  
Shuanming Li ◽  
José Garrido
Keyword(s):  
Laplace Transform ◽  
Process Analysis ◽  
Risk Process ◽  
Expected Discounted Penalty Function ◽  
Renewal Equations ◽  
The Laplace Transform ◽  
Discounted Penalty Function ◽  
Renewal Risk ◽  
Defective Renewal Equations ◽  
Expected Discounted Penalty

We consider a compound renewal (Sparre Andersen) risk process with interclaim times that have a K n distribution (i.e. the Laplace transform of their density function is a ratio of two polynomials of degree at most n ∈ N). The Laplace transform of the expected discounted penalty function at ruin is derived. This leads to a generalization of the defective renewal equations given by Willmot (1999) and Gerber and Shiu (2005). Finally, explicit results are given for rationally distributed claim severities.

scholarly journals Gerber–Shiu Function in a Class of Delayed and Perturbed Risk Model with Dependence

Risks ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 30 ◽  
Author(s):  
Franck Adékambi ◽  
Essodina Takouda
Keyword(s):  
Time Delay ◽  
Diffusion Process ◽  
Ruin Probability ◽  
Equilibrium Distribution ◽  
Risk Model ◽  
Numerical Examples ◽  
Renewal Equations ◽  
Generalized Mixed Equilibrium ◽  
Defective Renewal Equations ◽  
Occurrence Times

This paper considers the risk model perturbed by a diffusion process with a time delay in the arrival of the first two claims and takes into account dependence between claim amounts and the claim inter-occurrence times. Assuming that the time arrival of the first claim follows a generalized mixed equilibrium distribution, we derive the integro-differential Equations of the Gerber–Shiu function and its defective renewal equations. For the situation where claim amounts follow exponential distribution, we provide an explicit expression of the Gerber–Shiu function. Numerical examples are provided to illustrate the ruin probability.

Lundberg inequalities for renewal equations

2001 ◽  
Vol 33 (03) ◽  
pp. 674-689 ◽  
Author(s):  
Gordon E. Willmot ◽  
Jun Cai ◽  
X. Sheldon Lin
Keyword(s):  
Branching Processes ◽  
Geiger Counter ◽  
Upper And Lower Bounds ◽  
Exponential Inequalities ◽  
Renewal Equations ◽  
Additional Information ◽  
Special Cases ◽  
Age Dependent ◽  
Defective Renewal Equations ◽  
Generalized Type

Sharp upper and lower bounds are derived for the solution of renewal equations. These include as special cases exponential inequalities, some of which have been derived for specific renewal equations. Together with the well-known Cramér-Lundberg asymptotic estimate, these bounds give additional information about the behaviour of the solution. Nonexponential bounds, which are of use in connection with defective renewal equations, are also obtained. The results are then applied in examples involving the severity of insurance ruin, age-dependent branching processes, and a generalized type II Geiger counter.

On asymptotic equivalence among the solutions of some defective renewal equations

2013 ◽  
Vol 53 (4) ◽  
pp. 391-405 ◽  
Author(s):  
Qingwu Gao ◽  
Yu Liu ◽  
Georgios Psarrakos ◽  
Yuebao Wang
Keyword(s):  
Asymptotic Equivalence ◽  
Renewal Equations ◽  
Defective Renewal Equations

scholarly journals On a general class of renewal risk process: analysis of the Gerber-Shiu function

2005 ◽  
Vol 37 (3) ◽  
pp. 836-856 ◽  
Author(s):  
Shuanming Li ◽  
José Garrido
Keyword(s):  
Laplace Transform ◽  
Process Analysis ◽  
Risk Process ◽  
Expected Discounted Penalty Function ◽  
Renewal Equations ◽  
The Laplace Transform ◽  
Discounted Penalty Function ◽  
Renewal Risk ◽  
Defective Renewal Equations ◽  
Expected Discounted Penalty

We consider a compound renewal (Sparre Andersen) risk process with interclaim times that have a Kn distribution (i.e. the Laplace transform of their density function is a ratio of two polynomials of degree at most n ∈ N). The Laplace transform of the expected discounted penalty function at ruin is derived. This leads to a generalization of the defective renewal equations given by Willmot (1999) and Gerber and Shiu (2005). Finally, explicit results are given for rationally distributed claim severities.

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