Ruin Probability Under Claim Numbers With Compound PNB Process' Risk Model

Ruin Probability Under Claim Numbers with Compound PNB Process' Risk ModelFrom the study of probability theory, we know that Poisson distribution have a certain degree of universality in the description of random phenomena of both the natural sciences and social management activities[21]. In the study of claim numbers and claim amount's distribution, we usually use Poisson distribution. However, for a collection of policies with different quality, it's not suitable to use Poisson distribution to describe its characteristics . In this paper, firstly, we use the idea of the literature [19] and [20], to introduce a new Compound PNB distribution. Secondly, we give a new counting process, Compound PNB process. Finally, we discuss ruin probability under claim numbers with Compound PNB Process' risk model.Definition 1 If the generating function of random variablesξis G(t)=exp (?),where A, p is parameter , then we callξfollowscompound Poisson negative binomial distribution, recorded as PNB(λ,p).Theorem 1 Ifξfollows compound Poisson negative binomial distribution, then(1) The probability distribution ofξis(2) The expectation and variance ofξare(3) The dispersion parameter ofξisTheorem 2 If the generating function of random variables S is Q{t) , then the moment generating function of S is Definition 2 Supposeλ> 0, 0≤p < 1, the we call {N(t); t≥0} is a compound Poisson negative binomial process with parameter (λt,ρ), if {N(t)} satisfy the following conditions:1) N(0)=0;2) {N(t); t≥0} has independent stationary incremental variable;3) Fort > 0, then N(t)-PNB{(λt,ρ); and E[N(t)]=(?)Lemma 1 [18] Suppose process S(t)=(?)X_i, and {N(t); t≥0} is a countingprocess with independent stationary incremental variable; X_i (i=1, 2,···) is independent identical distribution random variables, and is independent of {N(t); t≥0}, then compound process {S(t); t≥0} has independent stationary incremental variable too.Theorem 3 Supposeλ>0, 0≤ρ<1,then there exist a random process {N(t); t≥1} with parameters (λt,ρ) such that:1) N(0)=0;2) {N(t); t≥0} has independent stationary incremental variable;3) for t > 0, then N(t)-PNB(λt,ρ), and E[N(t)]=(?)Theorem 4 Suppose claims process S(t)=(?)X_i,where {N(t);t≥0} is acompound Poisson negative binomial process; claim amount X_i(i=1, 2,···) is independent identical distribution random variables , and is independent of {N(t); t≥0}. If the density function of X_i is fx(x), distribution function is Fx(x), and moment generating function is M_x(Ï…), and X_i≥0, then1) claims process {S(t); t≥0} has independent stationary incremental variable;2) The moment generating function of S(t) is3) The expectation and variance of S(t) are μ₁=EX₁,μ₂=EX₁².We recorded {S(t); t≥0} as compound compound PNB process.Theorem 5 Suppose M_x(Ï…) is the moment generating function of individual claim amount X in compound PNB process which is defined on (0,Ï…₀), whereÏ…₀≤+∞.if c>(?), then adjustment equationλ[((?)_rM_x(Ï…)-1]=cÏ…has unique positive root and V∈(0,Ï…₀).Theorem 6 For u≥0,then (?)(u)=(?)...