scholarly journals Martingale Approach to Derive Lundberg-Type Inequalities

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1742
Author(s):  
Tautvydas Kuras ◽  
Jonas Sprindys ◽  
Jonas Šiaulys
Keyword(s):  
Upper Bound ◽  
Ruin Probability ◽  
Risk Model ◽  
Tail Probability ◽  
Martingale Approach ◽  
Renewal Risk Model ◽  
Renewal Risk ◽  
Wald's Identity

In this paper, we find the upper bound for the tail probability Psupn⩾0∑I=1nξI>x with random summands ξ1,ξ2,… having light-tailed distributions. We find conditions under which the tail probability of supremum of sums can be estimated by quantity ϱ1exp{−ϱ2x} with some positive constants ϱ1 and ϱ2. For the proof we use the martingale approach together with the fundamental Wald’s identity. As the application we derive a few Lundberg-type inequalities for the ultimate ruin probability of the inhomogeneous renewal risk model.

scholarly journals Ruin excursions, the G/G/∞ queue, and tax payments in renewal risk models

2011 ◽  
Vol 48 (A) ◽  
pp. 3-14
Author(s):  
Hansjörg Albrecher ◽  
Sem C. Borst ◽  
Onno J. Boxma ◽  
Jacques Resing
Keyword(s):  
Ruin Probability ◽  
Risk Model ◽  
Risk Models ◽  
Renewal Risk Model ◽  
Insurance Portfolio ◽  
Tax Payments ◽  
Renewal Risk

In this paper we investigate the number and maximum severity of the ruin excursion of the insurance portfolio reserve process in the Cramér–Lundberg model with and without tax payments. We also provide a relation of the Cramér–Lundberg risk model with the G/G/∞ queue and use it to derive some explicit ruin probability formulae. Finally, the renewal risk model with tax is considered, and an asymptotic identity is derived that in some sense extends the tax identity of the Cramér– Lundberg risk model.

scholarly journals Uniform Estimate of the Finite-Time Ruin Probability for All Times in a Generalized Compound Renewal Risk Model

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Qingwu Gao ◽  
Na Jin ◽  
Juan Zheng
Keyword(s):  
Finite Time ◽  
Ruin Probability ◽  
Risk Model ◽  
Uniform Estimate ◽  
Random Variables ◽  
Dependence Structure ◽  
Finite Time Ruin Probability ◽  
Renewal Risk Model ◽  
Renewal Risk ◽  
Compound Renewal Risk Model

We discuss the uniformly asymptotic estimate of the finite-time ruin probability for all times in a generalized compound renewal risk model, where the interarrival times of successive accidents and all the claim sizes caused by an accident are two sequences of random variables following a wide dependence structure. This wide dependence structure allows random variables to be either negatively dependent or positively dependent.

scholarly journals Ruin excursions, the G/G/∞ queue, and tax payments in renewal risk models

2011 ◽  
Vol 48 (A) ◽  
pp. 3-14 ◽  
Author(s):  
Hansjörg Albrecher ◽  
Sem C. Borst ◽  
Onno J. Boxma ◽  
Jacques Resing
Keyword(s):  
Ruin Probability ◽  
Risk Model ◽  
Risk Models ◽  
Renewal Risk Model ◽  
Insurance Portfolio ◽  
Tax Payments ◽  
Renewal Risk

In this paper we investigate the number and maximum severity of the ruin excursion of the insurance portfolio reserve process in the Cramér–Lundberg model with and without tax payments. We also provide a relation of the Cramér–Lundberg risk model with the G/G/∞ queue and use it to derive some explicit ruin probability formulae. Finally, the renewal risk model with tax is considered, and an asymptotic identity is derived that in some sense extends the tax identity of the Cramér– Lundberg risk model.

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