Research On Ruin Probabilities In Several Classes Risk Model Based On Poisson-Geometric Process

Risk thoery,which is maily applied in finance,insurance,securities investment and the risk management, is an important part of insurance actuarial.This thesis maily study the multi-type risk model based the compound Poisson-Geometric process of literature[28] and problems are solved as follows:1.Do a study on U (t )uin 1 Kj i1(t )Yij in 1 Nj i1(t)XijW(t) = = = = the multi-type risk model, where, u≥0 is the initial reserve, { N i(t ); t≥0}are all the compound Poisson- Geometric process with parameterλi ,ρi, the counting process{ K i(t ); t≥0}is the Poisson process with parameterαi, i = 1, L ,n,{W (t ), t≥0}is Wiener process.We get the Lundberg exponent and the expression of ultimate ruin probability ,and get the Gerber-Shiu discounted penalty function with the degenerated double-type risk model.2.Do a study on the risk models with interest force.One of the risk models is dUδ(t )= Uδ(t)δdt+cdM(t)?dS(t), where, {M (t),t≥0)} is the Poisson process with parameterλ1 ,∑S (t )= iN =1(t)Xi, { N (t,t≥0)} is the compound Poisson-Geometric process with parameterλ2 ,ρ,and interest force is the constantδ,δ>0.We get the Integro-differential equations of non-ruin probability, to correct the error of literature[39];And anoher model is∫Uδ(t )= ueδt +cst(δ)?0t eδ(t?x)dS(t),∑∑= =11 ( )(1 )+=21( )(2)S (t )iN tXi Nj tXj,{ N i(t ); t≥0} are all the compound Poisson- Geometric process with parameterλi ,ρi, The integral equation satisfying the non-ruin probability with initial reserve u and the expression of non-ruin probability with 0 initial reserve are obtained.3. We generalize the risk model of literature[28] with Markov-modulated,the risk model is ,where, { I t}t≥0is a stationary homogeneous and irreducible Markov jump process,where,the premiun rate is modulated by the Markov process.That is ,when I t is on the state i ,the premiun rate is c It, i = 1, 2,L,n.And we get a recurive inequality about the probability with the stationary initial distribution.4. In this chapter we do a study on the double-type risk model,∑∑= +?=11 ( )(1 )?=21( )(2)U (t )uctiN tXi Nj tXjwhere, the counting process { N 1 (t);t≥0} is the compound Poisson-Geometric process with parameterλ,ρ, { N 2 (t);t≥0}is a renewal process with i.i.d. inter-claim times {V i}i≥1,we assume that Vi are generalized Erlang(n) distributed with different parametersλ1 ,λ2,L ,λn.Then Vi can be expressed as Vi = Vi1 +Vi2+L+Vin,where V ij is exponentially wih parameterλj.We decomposed the Gerber-Shiu discounted penalty function into two parts.And then we get the Integro-differential equations of Gerber-Shiu discounted penalty function,and the Lundberg equation is obtained by using the martingale method, the expression of Gerber-Shiu discounted penalty function with 0 initial reserve are obtained.