scholarly journals Ruin Probabilities for Two Classes of Risk Processes

2005 ◽  
Vol 35 (01) ◽  
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 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 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.

Ruin Probabilities of Continuous-Time Risk Model with Dependent Claim Sizes and Interarrival Times

2020 ◽  
Vol 28 (Supp01) ◽  
pp. 69-80
Author(s):  
Nguyen Huy Hoang ◽  
Bao Quoc Ta
Keyword(s):  
Continuous Time ◽  
Ruin Probability ◽  
Risk Model ◽  
Random Variables ◽  
Ruin Probabilities ◽  
Dependent Random Variables ◽  
Arrival Times ◽  
Exponential Bound ◽  
Interarrival Times

In this paper we investigate an insurance continuous-time risk model when the claim sizes and inter-arrival times are m-dependent random variables. We provide an upper exponential bound for the ruin probability.

scholarly journals On the Ruin Probability Under a Class of Risk Processes

2002 ◽  
Vol 32 (1) ◽  
pp. 81-90 ◽  
Author(s):  
Wang Rongming ◽  
Liu Haifeng
Keyword(s):  
Laplace Transform ◽  
Renewal Process ◽  
Finite Time ◽  
Ruin Probability ◽  
Risk Model ◽  
Claim Amount ◽  
Classical Risk Model ◽  
Finite Time Ruin Probability ◽  
Risk Processes ◽  
The Classical Risk Model

AbstractIn this paper a class of risk processes in which claims occur as a renewal process is studied. A clear expression for Laplace transform of the finite time ruin probability is well given when the claim amount distribution is a mixed exponential. As its consequence, a well-known result about ultimate ruin probability in the classical risk model is obtained.

scholarly journals Ruin Probabilities in a Finite-Horizon Risk Model with Investment and Reinsurance

2012 ◽  
Vol 49 (04) ◽  
pp. 954-966
Author(s):  
R. Romera ◽  
W. Runggaldier
Keyword(s):  
Financial Market ◽  
Continuous Time ◽  
Ruin Probability ◽  
Risk Model ◽  
Finite Horizon ◽  
Ruin Probabilities ◽  
Optimal Solutions ◽  
Semi Markov Process ◽  
Insurance Model ◽  
Recursive Relations

A finite-horizon insurance model is studied where the risk/reserve process can be controlled by reinsurance and investment in the financial market. Our setting is innovative in the sense that we describe in a unified way the timing of the events, that is, the arrivals of claims and the changes of the prices in the financial market, by means of a continuous-time semi-Markov process which appears to be more realistic than, say, classical diffusion-based models. Obtaining explicit optimal solutions for the minimizing ruin probability is a difficult task. Therefore we derive a specific methodology, based on recursive relations for the ruin probability, to obtain a reinsurance and investment policy that minimizes an exponential bound (Lundberg-type bound) on the ruin probability.

scholarly journals Erlangian Approximations for Finite-Horizon Ruin Probabilities

2002 ◽  
Vol 32 (2) ◽  
pp. 267-281 ◽  
Author(s):  
Soren Asmussen ◽  
Florin Avram ◽  
Miguel Usabel
Keyword(s):  
Ruin Probability ◽  
Risk Model ◽  
Fluid Model ◽  
Ruin Probabilities ◽  
Approximation Procedure ◽  
Exponential Form ◽  
Probability Of Ruin ◽  
Phase Type ◽  
The Matrix ◽  
The Mean

AbstractFor the Cramér-Lundberg risk model with phase-type claims, it is shown that the probability of ruin before an independent phase-type time H coincides with the ruin probability in a certain Markovian fluid model and therefore has an matrix-exponential form. When H is exponential, this yields in particular a probabilistic interpretation of a recent result of Avram & Usabel. When H is Erlang, the matrix algebra takes a simple recursive form, and fixing the mean of H at T and letting the number of stages go to infinity yields a quick approximation procedure for the probability of ruin before time T. Numerical examples are given, including a combination with extrapolation.

scholarly journals Ruin Probabilities in a Finite-Horizon Risk Model with Investment and Reinsurance

2012 ◽  
Vol 49 (4) ◽  
pp. 954-966 ◽  
Author(s):  
R. Romera ◽  
W. Runggaldier
Keyword(s):  
Financial Market ◽  
Continuous Time ◽  
Ruin Probability ◽  
Risk Model ◽  
Finite Horizon ◽  
Ruin Probabilities ◽  
Optimal Solutions ◽  
Semi Markov Process ◽  
Insurance Model ◽  
Recursive Relations

A finite-horizon insurance model is studied where the risk/reserve process can be controlled by reinsurance and investment in the financial market. Our setting is innovative in the sense that we describe in a unified way the timing of the events, that is, the arrivals of claims and the changes of the prices in the financial market, by means of a continuous-time semi-Markov process which appears to be more realistic than, say, classical diffusion-based models. Obtaining explicit optimal solutions for the minimizing ruin probability is a difficult task. Therefore we derive a specific methodology, based on recursive relations for the ruin probability, to obtain a reinsurance and investment policy that minimizes an exponential bound (Lundberg-type bound) on the ruin probability.

scholarly journals Pricing the Zero-Coupon Bond and its Fair Premium Under a Structural Credit Risk Model with Jumps

2011 ◽  
Vol 48 (02) ◽  
pp. 404-419 ◽  
Author(s):  
Yinghui Dong ◽  
Guojing Wang ◽  
Rong Wu
Keyword(s):  
Laplace Transform ◽  
Credit Risk ◽  
Closed Form ◽  
Risk Model ◽  
Market Value ◽  
Credit Spread ◽  
The Laplace Transform ◽  
Credit Risk Model ◽  
Coupon Bond ◽  
Fair Premium

In this paper we consider a structural form credit risk model with jumps. We investigate the credit spread, the price, and the fair premium of the zero-coupon bond for the proposed model. The price and the fair premium of the bond are associated with the Laplace transform of default time and the firm's expected present market value at default. We give sufficient conditions under which the Laplace transform and the expected present market value of a firm at default are twice continuously differentiable. We derive closed-form expressions for them when the jumps have a hyperexponential distribution. Using the closed-form expressions, we obtain numerical solutions for the default probability, the credit spread, and the fair premium of the bond.

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.

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