Asymptotics of Ruin Probabilities for Risk Processes under Optimal Reinsurance and Investment Policies: The Large Claim Case

2004 ◽  
Vol 46 (1/2) ◽  
pp. 149-157 ◽  
Author(s):  
Hanspeter Schmidli
Keyword(s):  
Ruin Probabilities ◽  
Large Claim ◽  
Optimal Reinsurance ◽  
Investment Policies ◽  
Risk Processes

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 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 Ruin probabilities and decompositions for general perturbed risk processes

2004 ◽  
Vol 14 (3) ◽  
pp. 1378-1397 ◽  
Author(s):  
Miljenko Huzak ◽  
Mihael Perman ◽  
Hrvoje Šikić ◽  
Zoran Vondraček
Keyword(s):  
Ruin Probabilities ◽  
Risk Processes

scholarly journals Fourier/Laplace Transforms and Ruin Probabilities

2002 ◽  
Vol 32 (1) ◽  
pp. 91-105 ◽  
Author(s):  
Fátima D.P. Lima ◽  
Jorge M.A. Garcia ◽  
Alfredo D. Egídio Dos Reis
Keyword(s):  
Classical Model ◽  
Risk Process ◽  
Laplace Transforms ◽  
Ruin Probabilities ◽  
Ruin Theory ◽  
Easy Method ◽  
Arrival Times ◽  
The Real ◽  
Renewal Risk ◽  
Risk Processes

AbstractIn this paper we use Fourier/Laplace transforms to evaluate numerically relevant probabilities in ruin theory as an application to insurance. The transform of a function is split in two: the real and the imaginary parts. We use an inversion formula based on the real part only, to get the original function.By using an appropriate algorithm to compute integrals and making use of the properties of these transforms we are able to compute numerically important quantities either in classical or non-classical ruin theory. As far as the classical model is concerned the problems considered have been widely studied. In what concerns the non-classical model, in particular models based on more general renewal risk processes, there is still a long way to go. In either case the approach presented is an easy method giving good approximations for reasonable values of the initial surplus.To show this we compute numerically ruin probabilities in the classical model and in a renewal risk process in which claim inter-arrival times have an Erlang(2) distribution and compare to exact figures where available. We also consider the computation of the probability and severity of ruin in the classical model.

Sign in / Sign up

Export Citation Format

Share Document