Analysis of the ruin probability using Laplace transforms and Karamata-Tauberian theorem (original) (raw)
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Analysis of the ruin probability using Laplace transforms and Karamata Tauberian theorems
The classical result of Cramer-Lundberg states that if the rate of premium, c, exceeds the average of the claims paid per unit time, ‚", then the probability of ruin of an insurance company decays exponen- tially fast as the initial capital u ! 1. In this note, the asymptotic behavior of the probability of ruin is derived by means of inflnitesimal generators and Laplace transforms. Using these same tools, it is shown that the probability of ruin has an algebraic decay rate if the insurance company invests its capital in a risky asset with a price which follows a geometric Brownian motion. The latter result is shown to be valid not only for exponentially distributed claim amounts, as in Frolova et al. (2002), but, more generally, for any claim amount distribution that has a moment generating function deflned in a neighborhood of the origin.
How Much We Gain by Surplus-Dependent Premiums—Asymptotic Analysis of Ruin Probability
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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...
RUIN PROBABILITIES FOR RISK MODELS WITH CONSTANT INTEREST
In this paper, we consider risk models in continuous time of insurance companies. We show that the ruin probabilities still have exponential form whilst assuming that the claim sizes are dependent random variables. In addition, more general models of ruin probabilities where effects of interest are constant are considered. The proof follows the Martingale’s method
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In this paper a quantitative analysis of the ruin probability in finite time of discrete risk process with proportional reinsurance and investment of finance surplus is focused on. It is assumed that the total loss on a unit interval has a light-tailed distribution -- exponential distribution and a heavy-tailed distribution -- Pareto distribution. The ruin probability for finite-horizon 5 and 10 was determined from recurrence equations. Moreover for exponential distribution the upper bound of ruin probability by Lundberg adjustment coefficient is given. For Pareto distribution the adjustment coefficient does not exist, hence an asymptotic approximation of the ruin probability if an initial capital tends to infinity is given. Obtained numerical results are given as tables and they are illustrated as graphs.
Ruin probabilities in classical risk models with gamma claims
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In this paper we provide three equivalent expressions for ruin probabilities in a Cramér-Lundberg model with gamma distributed claims. The results are solutions of integro-differential equations, derived by means of (inverse) Laplace transforms. All the three formulas have infinite series forms, two involving Mittag-Leffler functions and the third one involving moments of the claims distribution. This last result applies to any other claim size distributions that exhibits finite moments.
Ruin probability for an insurer investing in several risky assets
The ruin probability of an insurer is studied for the classical Cramér-Lundberg model with finite exponential moments. The nonclassical property of the model considered in the paper is the possibility to invest in two different risky assets (which may be dependent) whose price processes are either described by geometric Brownian motions or are semimartingales with absolutely continuous characteristics with respect to Lebesgue measure. We study the ruin probability for the case where a free credit is not available in the money market and where the insurer can invest in a finite number of risky assets whose price processes are described by jointly independent Brownian motions.
Asymptotic tail probabilities of risk processes in insurance and finance
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In this thesis we are interested in the impact of economic and financial factors, such as interest rate, tax payment, reinsurance, and investment return, on insurance business. The underlying risk models of insurance business that we consider range from the classical compound Poisson risk model to the newly emerging and more general Lévy risk model. In these risk models, we assume that the claim-size distribution belongs to some distribution classes according to its asymptotic tail behavior. We consider both light-tailed and heavy-tailed cases. Our study is through asymptotic tail probabilities. Firstly, we study the asymptotic tail probability of discounted aggregate claims in the renewal risk model by introducing a constant force of interest. In this situation we focus on claims with TABLE OF CONTENTS LIST OF TABLES .