Edgeworth Type Expansion of Ruin Probability Under Levy Risk Processes in the Small Loading Asymptotics

2011 ◽  
Author(s):  
Yasutaka Shimizu
Keyword(s):  
Ruin Probability ◽  
Edgeworth Type Expansion ◽  
Risk Processes

scholarly journals A class of risk processes with delayed claims: ruin probability estimates under heavy tail conditions

2006 ◽  
Vol 43 (4) ◽  
pp. 916-926 ◽  
Author(s):  
Ayalvadi Ganesh ◽  
Giovanni Luca Torrisi
Keyword(s):  
Size Distribution ◽  
Ruin Probability ◽  
Heavy Tail ◽  
Claim Size ◽  
Probability Estimates ◽  
Claim Size Distribution ◽  
Risk Processes

We consider a class of risk processes with delayed claims, and we provide ruin probability estimates under heavy tail conditions on the claim size distribution.

scholarly journals Ruin Probabilities for Two Classes of Risk Processes

2005 ◽  
Vol 35 (1) ◽  
pp. 61-77 ◽  
Author(s):  
Shuanming Li ◽  
José Garrido
Keyword(s):  
Laplace Transform ◽  
Ruin Probability ◽  
Risk Model ◽  
Compound Poisson Process ◽  
Ruin Probabilities ◽  
Counting Processes ◽  
Arrival Times ◽  
The Laplace Transform ◽  
Sparre Andersen Risk Model ◽  
Risk Processes

We consider a risk model with two independent classes of insurance risks. We assume that the two independent claim counting processes are, respectively, Poisson and Sparre Andersen processes with generalized Erlang(2) claim inter-arrival times. The Laplace transform of the non-ruin probability is derived from a system of integro-differential equations. Explicit results can be obtained when the initial reserve is zero and the claim severity distributions of both classes belong to the Kn family of distributions. A relation between the ruin probability and the distribution of the supremum before ruin is identified. Finally, the Laplace transform of the non-ruin probability of a perturbed Sparre Andersen risk model with generalized Erlang(2) claim inter-arrival times is derived when the compound Poisson process converges weakly to a Wiener process.

scholarly journals How Much We Gain by Surplus-Dependent Premiums—Asymptotic Analysis of Ruin Probability

Risks ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 157
Author(s):  
Jing Wang ◽  
Zbigniew Palmowski ◽  
Corina Constantinescu
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Ruin Probability ◽  
Ruin Probabilities ◽  
Insurance Company ◽  
Case Scenario ◽  
Mathematical Functions ◽  
Linear Ordinary Differential Equations ◽  
Risk Processes ◽  
Premium Risk

In this paper, we generate boundary value problems for ruin probabilities of surplus-dependent premium risk processes, under a renewal case scenario, Erlang (2) claim arrivals, and a hypoexponential claims scenario, Erlang (2) claim sizes. Applying the approximation theory of solutions of linear ordinary differential equations, we derive the asymptotics of the ruin probabilities when the initial reserve tends to infinity. When considering premiums that are linearly dependent on reserves, representing, for instance, returns on risk-free investments of the insurance capital, we firstly derive explicit solutions of the ordinary differential equations under considerations, in terms of special mathematical functions and integrals, from which we can further determine their asymptotics. This allows us to recover the ruin probabilities obtained for general premiums dependent on reserves. We compare them with the asymptotics of the equivalent ruin probabilities when the premium rate is fixed over time, to measure the gain generated by this additional mechanism of binding the premium rates with the amount of reserve owned by the insurance company.

scholarly journals Reinsurance and Solvency Capital: Mitigating Insurance Companies’ Ruin Probability

2022 ◽  
Vol 26 (1) ◽  
Author(s):  
Jorge Wilson Euphasio Junior ◽  
João Vinícius França Carvalho
Keyword(s):  
Ruin Probability ◽  
Compound Poisson Process ◽  
Correct Choice ◽  
Insurance Companies ◽  
Business Results ◽  
The Monte Carlo Method ◽  
Exponential Decline ◽  
Solvency Capital ◽  
Property Losses ◽  
Risk Processes

ABSTRACT Context: insurance companies are important to society, since they guarantee financial protection to individuals from property losses, in addition to fostering the capital market through the allocation of guarantee assets. Thus, it is essential to evaluate the instruments that guarantee their long-term financial solvency. Among them are the adoption of reinsurance treaties, the sizing of the solvency capital, and the actuarial modeling of risk processes, which allow the measurement of the ruin probability. Objective: estimate the ruin probability in risk processes with the adoption of reinsurance contracts (quota share and excess of loss), compared to scenarios without such treaties. Methods: the Cramér-Lundberg process was simulated using the Monte Carlo method, adjusting several probabilistic distributions to the severity of the compound Poisson process, which is calibrated with a set of 3,917,863 real microdata, from 30 insurance lines of business. Results: it was found that, although each branch presents particularities in the claim severity, the correct choice of reinsurance (proportional or not) implies the reduction of the ruin probability for a fixed solvency capital. Conclusion: the appropriate choice of the reinsurance contract, especially when there is evidence of high kurtosis in the claim values, intensifies the exponential decline in the relationship between the solvency capital and the ruin probability.

scholarly journals Exponential convergence rate of ruin probabilities for level-dependent Lévy-driven risk processes

2019 ◽  
Vol 56 (4) ◽  
pp. 1244-1268 ◽  
Author(s):  
Pierre-Olivier Goffard ◽  
Andrey Sarantsev
Keyword(s):  
Convergence Rate ◽  
Ruin Probability ◽  
Lyapunov Functions ◽  
Risk Model ◽  
Exponential Convergence ◽  
Jump Diffusion ◽  
Ruin Probabilities ◽  
Exponential Rate ◽  
Risk Processes

AbstractWe find explicit estimates for the exponential rate of long-term convergence for the ruin probability in a level-dependent Lévy-driven risk model, as time goes to infinity. Siegmund duality allows us to reduce the problem to long-term convergence of a reflected jump-diffusion to its stationary distribution, which is handled via Lyapunov functions.

scholarly journals Threshold strategies for risk processes and their relation to queueing theory

2011 ◽  
Vol 48 (A) ◽  
pp. 29-38 ◽  
Author(s):  
Onno J. Boxma ◽  
Andreas Löpker ◽  
David Perry
Keyword(s):  
Queueing Theory ◽  
Ruin Probability ◽  
Risk Model ◽  
Insurance Company ◽  
Threshold Strategy ◽  
Ruin Time ◽  
Risk Processes

We consider a risk model with threshold strategy, where the insurance company pays off a certain percentage of the income as dividend whenever the current surplus is larger than a given threshold. We investigate the ruin time, ruin probability, and the total dividend, using methods and results from queueing theory.

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