Two Types Of Bankruptcy For The Sparre Andersen Risk Model

This dissertation is devoted to the development of ruin theory in two kinds of Sparre Andersen risk models. One is a Sparre Andersen risk model in which the claim inter-arrival distribution is a mixture of an exponential distribution and an Erlang(n) distribution. Another is a Sparre Andersen risk model in which the claim inter-arrival distribution is a mixture of two Erlang distributions.Sparre Andersen (1957) introduced Saprre Andersen model which satisfies the following basic requirements:1. The inter-arrival times [T1, T2, …} is a sequence of independent random variables with common distribution K(i) satisfies A'(0) = 0.2. The claim amounts {Z1. Z2, …} is also a sequence of independent random variables with common distribution.For Sparre Andersen risk model, the discussion about it has been become more and more perfect. Dickson (1998) considered the Erlang(2) risk model, the expressions for the ruin probability and severity of ruin and for the probability of absorption by an upper barrier are derived. In addition, for general Erlang(n) risk model, an integro-diifcrontial equation for the probability of ultimate ruin are presented: Dickson arid Hipp (2001) consider the Erlang(2) risk model, and introduce the expectation of the discounted penalty H'(u) which determines the joint and the marginal distribution of the time to ruin (T), the surplus prior to ruin (U(T-)} and the deficit at ruin (|U(T)|). In that paper, they showed that W(u) satisfies an integro-differential equation, from which the Laplace transform of W(u) is obtained. And it is also showed that W(u) satisfies a defective renewal equation. In our first chapter, we study a more general Sparre Andersen risk model, for which the distribution of inter-arrival time is a mixture of an exponential distribution and an Erlang(n) distribution. We prove that the expected discounted penalty satisfies a higher-order integro-differential equation, thatTheorem 1.2.1 The function W(u) satisfies the integro-differential equationWe also show an asymptotic expressions as the initial surplus u tends to infinity. Theorem 1.2.2 If the function W*(.) is analytic on the complex plane except for the roots of Eq.(1.2.10). Then, .When n = 2, the Laplace of the expected discounted penalty W1 can be expressed Theorem 1.3.1. W\(u) has a Laplace transform,In Chapter 2. we consider the ruin probability of the risk model which be introduced in Chapter 1. If S = 0,w(x1, w2) = 1, the expectation of the discounted penalty W(u) reduces to the ruin probability (u). So the ruin probability also satisfies a higher-order integro-differential equation, from which we prove that (u) satisfies an defective renewal equation, and then the convolution formula for ruin probability is derived. Heavy-tailed risk has been interest of many recent papers in insurance and fiance, especially the Subexponential distribution and the class of S(v] (y >1). In this chapter, the asymptotic exponential and non-exponential behaviors of the ruin probability are examined. Especially, we investigate a local asymptotic behavior of the probability of ruin which individual claims size have a distribution that belongs to S(v) with v > 0. The main results:Theorem 2.3.2 Let satisfies the defective renewal equation,whereTheorem 2.3.2 The ruin probability (u) has the following expression(2.3.3)where 7(11) is defined in Theorem 2.2.2 and dx. Theorem 2.4.1 Let-R denote the negative root of equation L(a) = 0. ThenTheorem 2.4.2 Suppose thenwhere Q be the same as Lemma 2.4.1.Theorem 2.4.3 Let -v < 0 be the left abscissa of convergence of p*(a) and satisfies, then for any z > 0,(2.4.15) where R(u. u + z] = (u) - (u + z). Q be the same as Lemma 2.4.2.In the third chapter, we mainly extend the model in Chapter 1 and show the expectation of the discounted penalty satisfies a higher-order integro-differential equation. The result can be obtained, Theorem 3.2.1 When m > 2, the function W(u) satisfies the integro-dif...