| Risk theory has become a very active branch of the probability theory and math-ematical statistics; and it is also one of the most active research areas. Risk theory describe insurance business process and the related numerical characteristics mainly by statistical method and mathematical model. It research the numerical characteristics of the insurance company. It also can be analyzing the ruin probability of the insurance company,and provide early warning system for insurance company in order to improve the management ability and their own competitiveness. In this thesis, we combines the two hot research risk theory and the integer-valued time series model. We mainly discuss three kinds of risk model which based on the integer-valued time series model.1. Risk model based on the INAR(1) Claim processNow we introduce the background of the risk model. Premiums are collected at a constant rate of c>0per period, the premium paid during the first T periods is cT, where T is a positive integer. Define the surplus process at time T as UT=u+cT-St, where u is the initial surplus at time0and ST is the aggregate claims paid up until time T. The premiums are to provide wage loss benefits (WLB) to disabled workers while they are unable to work. Assume the ith claimant of the jth period would receive WLB of Yi(j)per period while disabled. That is if a worker became disabled he would receiv Yi(j) WLB per period until he is able to return to work. Let Xj denote the number of claimants collecting WLB in period j. The aggregate loss process is described as where{Yi(j),i,j∈N+,j≤T} is a sequence of positive independent identically dis-tributed random variables with distribution function F(y). We rewrite ST as follows in order to be consistent with classical risk model, where Nt=X1+…+Xt, and {Yi, i∈N+} is a sequence of independent and identically distributed nonnegative random variables with distribution function F(y), which is independent of NT,T≥0. The number of claimants in each period depends on two components: the number of claimants continuing to collect from the previous period and the number of new claimants arriving during the period. Let εj denote the number of new disabled workers in period j, and εj has a Poisson distribution with mean λ. Suppose that each disabled worker collecting WLB at time j-1has probability α of continuing to collect benefit at time j, where α∈[0,1]. That is Xj=αοXj-1+εj, j∈N-where X0is the initial number of claimants collecting WLB at time0, and X0has a Poisson distribution with mean λ/1-α. In view of this, we can denote the risk model follows,Hence, given a sequence of i.i.d.Poisson r.v.’s {εj,j∈N+} with mean A and a Poisson r.v.’s X0with mean λ/1-α,X1,X2…, is a stationary sequence of Poisson r.v.’s with mean λ/1-α. However, it is not difficult to find that the loss increments {Si—Si-1,i∈N+} are dependent, so we called which was a dependent risk model.Considering the same practical problems, we give a approximation risk model with a independent stationary increment claim process. It has well theoretical background and easy to deal in the mathematics.In the dependent model, suppose that each disabled worker collecting WLB during the period t—1has probability α of continuing to collect benefit during the period t. Let Xj*denote the number of claimants for the jth disabled worker. So it is naturally to suppose that r.v.’s {Xj*, j∈N+} is a Geometric sequence with probability function P(Xj*=x)=(1-α)αx-1,,x=1,2....The mean, variance and moment generating function of the r.v. Xj*are1/1-α,α/(1-α)2and MXj*(s)=(1-α)es/1-αes respectively. Let NT*,=ε1+ε2+…+εT, then Nt denote the aggregate number of disabled workers in [0, T]. Because r.v.’s {εi,i=1,2,…,T} are assumed to be Poisson distribution with mean A in dependent model, so NT*is a Poisson random variable with parameter Tλ. From this, the model introduced before can be approximated by the following process. Define the surplus process at time T aswhere {Yi(j),i≥1,j≥1} is a sequence of independent and identically distributed nonnegative r.v. with distribution function F(y).Obviously, the loss increments of the approximation model are stationary and independent. The difference between the approximation model and the dependent model is how losses are counted. For the dependent model, the incremental loss is equal to the current number of disabled workers paying which is the amount of losses actually paid during the period. For while the approximation model, the incremental loss is dependent on the number of new disabled workers during the interval and the duration of each of these disabled worker.Actually, according to the definitions of St and ST*, we can get the following conclusion by the proposition of conditional expectation.Theorem1According to the definitions of St and ST*, we have()E(ST*)=E(ST)=TE(Y)/1-α,(?) Var(ST*)-Var(ST)=2αλ(1-αT)(E(Y))2/(1-α)3>0.From Theorem l(ii), we have Var{ST*)> Var(ST). Intuitively, the probability of ruin of the approximating model is bigger than that of the dependent model. In fact, we will confirm this result by simulation studies. Moreover, Var(ST*)-Var(ST)â†'2αλ(E(Y))2/(1-α)3as Tâ†'∞. So we could make the gap between the two models as small as possible by selecting appropriate parameters. Let T*=min{T: T≥0and UT*<0} denote the time of ruin, T*=∞ifUT*>0for all T. Let Ψ*(u)=Pr (T*<∞|U0*=u) denote the probability of ruin. The relative safety loading θ is denned by θ=cT/E(ST*)-1. Generally, assume that θ∈(0,1). Because the loss increments of the approximation model are stationary and independent, we can obtain corresponding theorems for the approximation model easily. Theorem2For the approximation model, assuming that αMY(r)<1, the ex-pression for c(r) is given by Particular, we could get an explicit expression for the adjustment coefficient R*.corollary1If Yi(j) is an exponential random variable with parameter β, i≥1,1≤j≤T, then R*=(1-α)β-λ/c.The connection between the adjustment coefficient and the probability of ruin is given in the following result.Theorem3Let Ψ*(u) denote the probability of ruin for the approximation model, then Ψ*(u) satisfie where T*denotes the time for ruin of the approximation model.Because UT**≤0,E [e-R*UT**|T*<∞]≥1. From Theorem3, we get a simplified upper bound for the probability of ruin.corollary2The probability of ruin Ψ*(u) satisfies Ψ*(u)≤e-R*This is the famous Lundberg inequality.At last, we compare the ruin probability of the dependent model with the approx-imation model by numerical simulations. Four representative claim-size distribution in insurance are considered, including Exponential distribution, Pareto distribution, Weibull distribution and Log Norm distribution. We know that ΨT(u)≤ΨT*(u) in most cases. That is to say that the approximate model is more conservative than the dependent model. Moreover, the difference between Ψy(u) and ΨT*(u) becomes smaller as the A become smaller. Generally, when λ≤0.1the gap of the approximation model and the dependent model can be ignore.2.Risk model with NGINAR(l) claim processWe considered the total numbers of claims satisfies a new geometric first-order integer-valued autoregressive (NGINAR(l)) model. In view of this, we build up the new risk model as where Yi is individual claim,{Yi, i≥1} is assumed to be a sequence of independent and identically distribution (i.i.d.)r.v.’s.; Xi is the number of claim during the i period, X1is a geometric distribution with μ/(1-μ)(μ>0). That is P(X1=x)=μx/(1-μ)x-1, x=0,1,2, E(X1)=μ. Var (X1)=μ(1+μ);{Wj,i≥1} and {εj, j≥2}are all assumed to be a sequence of independent and identically geometric distribution with α/1+α; Moreover,{εj, j≥2},{Wi, i≥1},{Xj-l, j≥2,j>l>1} are independently for each other. We called it the risk model with NGINAR(l) claim process.Let T0=inf {T|Ut <0, T≥0} denote the time of ruin, T0=∞if T≥0for all UT>0. LetΨ(u)=Pr (T0<∞|U0=u) denote the probability of ruin.Define the convex function, cT(r)=1/Tln(E[er(ST-cT)]). Under different approaches, Nyrhinen(1998)and Muller and Pflug(2001)have shown that the adjustment coefficient R is the solution to c(r)=lim/Tâ†'∞Ct(t)=0. The moment generating function and the probability generating function of r.v. X are given by Mx(r),ΦX(r),|r|≤1.The expression for ΦNT(r)=E[rX1+X2+…+XT]is provided in the following theo-rem.Theorem4Let ΦNt (r) denotes the probability generating function of r.v.NT,,then Φnt (r)=a1(r)a2(r)…aT-1(r)bT(r),T=1,2, where a1(r)=Φε(r), a2(r)=Φ￡(ra1(r)),…, an(r)=Φε(ran-1)); b1(r)-ΦX(r),b2(r)=ΦX(ra1(r)))…,bn(r)=ΦX(ran-1(r)); Φε(r)=1/1-α(1-r),ΦX(r)=1/1+μ(1-r).Simply, we use a,j, bj denote a,j(r), bj(r),j=1,2,.....respectively.Theorem5Functions {an,n∈N+} satisfies:(i) When0≤r≤1,{an,n∈N+}is a monotonically decreasing of bounded func-tion sequence, moreover, nâ†'∞lim an=a.(ii) When-1≤r<0,{an,n∈N+} is a nonmonotonic bounded function se-quence, moreover, nâ†'∞lim an=a, where a=1+α-(?).Theorem6Functions {bn,n E N+}satisfies:(i) When0≤r≤l,{bn, n∈N+}is a monotonically increasing of bounded function sequence, moreover, nâ†'∞lim bn=b.(ii) When-1≤r<0,{bn,n∈N+}is a nonmonotonic bounded function sequence, moreover, nâ†'∞lim bn=b,Theorem7c(r)=In a (MY (r))-cr. where a(MY(r))=1+α-(?).From theorem7, we differentiate the expression of c(r) derived for α, then This is to say, if α<α’, then r> r’An explicit expression for the adjustment coefficient R is provided when the r.v.Y have a explicit distribution. Usually, we assume Yis an exponential distribution with β, then MY(r)=β/β-r. Substitute it into theorem7, let c(r)=0, we getthen αβ3e2cr+r(l+α)ecr-β(1+α)ecr-r+β=0.3.Premium income randomized risk model with INAR(l) claim processIn fact, the premium of each period assumed to a constant is an ideal assumptions to mathematics dealing with problems convenience. In practice, it is difficult to achieve. This model considering premium income randomized situation. Not only each phase of the claim number has a autoregressive dependent relationship, but also the number of policy in every period has dependent relationship.This risk model comes from the unemployment insurance, and the one which is closely related to the reality. Define the surplus process at time T as UT=u+Vt—St, where T is a positive integer, u is the initial surplus at time0, VT is the premium paid during the first T periods and St is the aggregate claims paid up until time T. Set Vt=∑i=1NT Yi, where Nt=∑j=1T Xj is the number of policy during the first T periods, Yi is the individual premium. The number of policy in each period depends on two components: the number of customer continuing to buy policy from the previous period and the number of new policy arriving during the period. Let εj denote the number of new policy in period j, and εj has a Poisson distribution with mean A. Suppose the each customer at time j-1has probability α of continuing to buy policy at time j, that is Xj=αοXj-1+εj, j≥1.The premiums are to provide wage loss benefits (WLB) to disabled customers while they are unable to work. Assume the ith claimant would receive WLB of Yi’ per period while disabled. Set St=(?)Yi, whereNT’’=∑j=1T Xj’ is the number of disabled customers during the first T periods. The number of claimants in each period depends on two components: the number of claimants continuing to collect from the previous period and the number of new claimants arriving during the period. Let εj’ denote the number of new claims in period j, and εj’ has a Poisson distribution with mean A’. Suppose the each individual collecting WLB at time j-1has probability α’ of continuing to collect benefit at time j, that is Xj’=αοXj-1’+εj’, j≥1. In view of this, we can build up the new risk model as follows, Assumptions:(i){Yi,i∈N+},{Yi’,i∈N+},{εj,j∈N+},{εj’,i∈N+} are all assumed to be a sequence of independent and identically distribution (i.i.d.)r.v.’s.(ii) X0has a Poisson distribution with mean λ/1-α.(iii) X0’ has a Poisson distribution with mean λ/1-α.(iv) The premium in come per period satisfies the usual solvency condition: E (Vt)> E(YT).We get the following proposition easily.Proposition1(i)If X0~P(λ0), then E (NT)=T-(α+α2+…+αT)/1-αλ+α(1-αT)/1-αλMore specifically, if λ0=λ/1-α, then Xt~P(λ/1-α),(?)t∈{0,1,2,…}, and E (NT)=T E(X1)=Tλ/1-α.(ii)If X0’~P(λo0), then E(NT’)=T-(α’+α’2+…+α’T)1-α’λ’+α’(1-α’T)/1-α’λ More specifically, if λ0’=λ’/1-α, then Xt’~P(λ’/1-α),(?)t∈{0,1,2,…}, and E (NT’)=T E(X1’)=Tλ’/1-α’Let T0=min{T: T≥0and Ut <0} denote the time of ruin of the model with the understanding that T0=∞if UT≥0for all T>0. Let Ψ(u)=Pr (T0<∞|U0=u) denote the probability of ruin. The premium income per period is satisfies the usual solvency condition E (VT)> E(ST), that is Tλ/1-αEY1>Tλ’/1-α’EY1. The relative safety loading θ is defined by θ=E(VT)/E(ST)-1=(1-α’)λEY1/(1-α)λ’EY’-1. Generally, assume that θ∈(0,1). Define the convex function T(r)=1/Tln(E(er(ST-VT))). Using different approaches, Nyrhinen(1998)and Muller(2001) have shown that the Lundberg adjustment coefficient R is the solution to c(r)=Tâ†'∞lim cT(r)=0.The moment generating function and the probability generating function of r.v. Y are given by MY(r) and ΦY(r) respectively in this paper. The expression for c(r) is provided in the following theorem.Theorem8Assuming that α’MY1ξ(r)<1, the expression for c(r) is given by c(r)=λ(1-α)MY1(-r)/1-αMY1(-r)=λ’(1-α’)MY1’(r)/1-α’MY1’(r)-λ-λ’.An explicit expression for the adjustment coefficient R is provided in the next corollary when the r.v. Yi and Yi’ follow an exponential distribution.Corollary3If Yi is an exponential random variable with parameter δ, Yi’ is an exponential random variable with parameter δ’,i≥1,1≤j≤T, then R=λδ’(1-α)-λ’δ(1-α)/λ+λ’... |
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