The Bankruptcy Of Several Types Of Risk Models

This dissert at ion is devoted to t he development of ruin theory in two kinds of risk model. One is the common shock model for which the two claim number processes are correlated, that is. the times between claims of both classes relate to Poisson and Krlang (n) processes. Another is a Phase-type risk model in which the times between claims {T1. = 1. 2. ...} with a density function k(t) satisfying a linear differential equal ion.For the common shock risk model, many authors studies various aspects of the common shock model in recent years. Ambagaspitiya (1998) considered a general method of constructing a vector p (p 2) dependent claim numbers from a vector of independent random variables, and derived formulas to compute the correlated aggregate claim distribution for corresponding common shock model with p dependent classes of business. Cossotte and Marceau (2000) used a discrete-time approach to study how the common shodcaffects the finite-time ruin probabilities and the adjustment coefficient. Yuen.K. C., and Wang (2001) studied common shock risk model in which the two claim number processes are both Poisson processes by transforming into classical model. Yuen, Guo and Wu (2002) studied a more general common shock model for which one claim number process is Poisson process while the other is Erlang(2) process. This model is transformed into another risk model for which two claim number processes are independent. In our first chapter, we consider a more general common shock risk model in which one claim number process is Poisson process and the other is Erlang (n) process. In the first section of this chapter, we show that the expectation of the discounted penalty W(u) satisfies an integro-differential equation from which we derive the Laplace transform of W(u). The expectation of the distribution of the time to ruin (T), the surplus prior to ruin (S(T-)) and the deficit at ruin (S(T)). and we also show their distributions. An asymptotic result for W(u) is presented. In the second section, we show that the probability of ruin satisfies an defective renewal equation and an asymptotic expression as the initial u tends to infinity is obtained.In the last section of Chapter 1, we consider a special case that n = 2, and get somedifferent result as Yuen, Guo and Wu (2002). The main results:Theorem 1.2.2 The function (u)-satisfies the integro-differential equationwhere,Theorem 1.3.2 If *( )s analytic on the complex plane except for the roots ofIn second chapter, we consider a risk process in which inter-arrival times have a phase-type(2) distribution, a distribution with a density k(t) satisfying the following second order linear differential equation:The conditions are satisfied for all convolutions of two exponential distributions (with not necessarily equal means). This distribution is a special phase-type distributions. This risk model is a more general than it which is introduced by Dickson and Hipp (2000). They consider some ruin related problems. They consider the compound geometric representation of the infinite time survival probability, as well as the (defective) distributions of the surplus immediately prior to ruin and of the deficit at ruin. But we consider the asymptotic behavior of ruin probability of this model as the initial surplus u tends to infinity, we also show the probability of ruin satisfies a defective renewal equation. The asymptotic exponential and non-exponential behaviors of the ruin probability are examined. The main results:Thoerem 2.2.1 (v) satisfies a defective renewal equation.where.Thoerem 2.4.1 Let -R denote the negative root of equation (2.2.3) Thenwhere, L( ) denots the denominator of (2.2.2), h(u) be defined in (2.2.1). Theorem 2.4.2 Let P1 S, thenTheorem 2.4.3 Let -v < 0 be the left abscissa of convergence of p*( ) and satisfies evz (z) dz < 1. If P e 5(v), then for any z > 0, thenwhere, (z) be the same as Theorem 2.3.2.In the third chapter, we extend the work of Dickson and Hipp(2000), and consi...