Study On Ruin Problems Of Some Bidimensional Risk Models With Dependence

In modern insurance industry, insurance portfolios have become the main stream of their operations, thus, the study on bidimensional and multidimensional risk models is one of persistent hot spots in modern actuarial field. Due to their depicting the portfolio risk process characterization in more practical ways, various assumptions and computational complexities, bidimensional risk models theories have gained a lot of attention. On the other hand, the traditional assumption that the claim counts obey Levy processes does not conform the realities of the insurance practice. Thus, the studies on dependent risk models are also the spot issues in current years. Cossette et al. (2009,2010) introduced the integer-valued time series models into unidimen-sional risk models, got the generalized adjustment coefficient expressions and gained the approximations to ruin probabilities. Esmaeili and Kluppelberg (2010) introduced the Levy copula to describe the dependence structure of a bivariate compound Poisson process, a parametric model for the Levy copula and estimate the parameters of the full dependent model based on a maximum-likelihood approach. Motivated by those authors, we introduced three categories of dependent bivariate counting processes with their own dependent structures into bidimensional risk models, and discuss their ruin problems.1. Bidimensional risk models based on bivariate integer-valued time series Considering the following bidimensional discrete-time risk model: where u1 and u2 are the initial reserves of the two risk responsibilities, respective-ly; π1 and π2 are the two premium rate of the two risk responsibilities, respectively; (S1(n),S2(n)) are the aggregate bivariate claim amounts of the two risk responsibili-ties of n periods; {{N1i, N2i), i ∈ N} are the bivariate claim counting process; the claim sizes, {Xi,j,i,j ∈N+} and {Yi,k,i,k∈N+} are two independent sequences of i.i.d. nonnegative r.v.’s with d.f.’s F(x) and G(y), m.g.f.’s MX(t) and MY(s),respective-ly. Furthermore, we assume that the bivariate claim sizes are independent from the bivariate claim counts.Assume {(N1i,N2i),i∈N+} obey·BINMA(1) process·BINAR(1) process respectively, where α1,α2∈[0,1], {(ε1i,ε2i),i∈N} are a sequence of i.i.d. bivariate Poisson r.v.’s with parameter vector (λ1,λ2,λ), whose density is And the binomial thinning operator "o" is defined as follows: where {δi+1,i,j(1),i,j∈N+} and {δi+1,i,j(2),i,j∈N+} are two independent sequences of i.i.d. Bernoulli r.v.’s with means α1 and α2 , respectively.As for the two risk models, we get three expressions for adjustment coefficient functions:Porposition 1 For BINMA(1) risk models, c(t,s), c1(t) and c2(s) are c(t,s) = λ1(α1MX2(t) +α1MX(t)-1)+λ2{α2MY2(s)+α2MY(s)- 1) +λ1(α1MX2(t) α1MX(t))+(α2MY2(s)+α2MY(s)) 1]-π1t-π2s, c1 (t)＝(λ1+λ)(α1MX2(t)+α1MX(t)-1)-π1t, c2(s)=(λ2+λ)(λ2MY2(s)+α2MY(s)-1)-π2s, respectively, and where αi=1-αi,i=1,2.Proposition 2 For BINAR(1) risk models,if α1,α2∈[0,1) and α1Mx(t)<1, α2MY(s)<1,then the corresponding expressions for c(t,s), c1(t) and c2(s) are where c(t,s)=0, c1(t)= 0, and c2(s)=0 are the adjustment coefficient functions of Ψmax(u1,u2),Ψ1(u1) and Ψ2(u2),respectively. And the positive roots of c1(t) and c2(s) are the adjustment coefficients of the marginal ruin probabilities, respectively; the roots of c(t, s) are the continuous curve in the first quadrant.As for the two risk models mentioned above, we assume the bivariate claim sizes are mutually independent.However, in most practical cases, the two risk responsibilities claim coincidentally in an accident, the two claim sizes are always depend on each other, those cases are called the common shocks, many authors, e.g. Yuen et al. (2006), Esmaeili and Kluppelberg (2010) and etc., introduced copula distributions to meet the situations. So, we extend the BINMA(1) and BINAR(1) risk models to the more generalized forms:we single out the common shocks and then the bivariate integer-valued time series become the three-variate integer-valued time series models, then, (S1(n),S2(n)) can be decomposed into: where·{Nki⊥,i∈N+} are the claim counts of the kth risk responsibility claiming separately, k=1,2.? {NiH, i∈N+} are the common shock counts.? {N1i/1,i∈N+}, {N2i/1,i∈N+} and {Ni",i∈N+} are mutually independent.? {Xij,i,j∈N+} and {Yik,i, k∈ N+} are two mutually independent sequences of i.i.d. r.v.’s, their corresponding d.f.’s are F(x) (x > 0) and G(y) (y > 0), with means μ1 and μ2, and with variances ζ1 and ζ2, respectively.? {(Xij,Yij),i,j G N+} are a sequence of i.i.d. bivariate r.v.’s with copula distribu-tion C(x,y) = C(F(x), G(y)) with Carr(X,Y) = ρ∈ [-1,1], and with join m.g.f. MC{t,s).? The three-variate claim counting process and their corresponding claim sizes are mutually independent.Here, we assume that {(N1i⊥N2i⊥,Ni"),i∈N+} obey the following inter-valued time series models:? Three-variate INMA(l) process where {εki,i∈N} are two sequences of i.i.d. Poisson r.v.’s with parameters λk/(1+αk), k =1,2, respectively; {εi,i∈N} are a sequence of i.i.d. Poisson r.v.’s with parameter λ/(l + α). Furthermore, {ε1i,i∈N}, {ε2i,i∈N} and {εi,i∈N} are mutually inde-pendent. If and only if α1,α2,α∈[0,1], three-variate INMA(l) process is stationary.? Three-variate INAR(l) process where {ε’ki,i∈N} are two sequences of i.i.d. Poisson r.v.’s with parameters (1-γk)λke, k = 1,2, respectively; {ε‘i, i∈N} are a sequence of i.i.d. Poisson r.v.’s with parameter (1-γ)λ;furthermore, N10⊥, N20⊥ and N0" are the initial r.v.’s and they are the copies of ε’11, ε’21 and ε’1,respectively. Samely,{ε1i,i∈N}, {ε2i,i∈N} and {εi,i∈N} are mutually independent. If and only if γ1,γ2,γ∈ [0,1), the three-variate INAR(l) process is stationary.Proposition 3 If α1,α2,α∈[0,1], the corresponding expressions for adjustment coefficient functions of three-variate INMA (1) process are presented as follows:Proposition 4 If γ1Mx(t)<1,γ2My(s)< 1 and γMc(t,s)< 1, the cor-responding expressions for adjustment coefficient functions of three-variate INAR(1) model are presented as follows:Theorem 1 For BINMA(1), BINAR(1), three-variate INMA(1) and three-variate INAR(1) risk models,Theorem 2 For BINMA(1), BINAR(1), three-variate INMA(1) and three-variate INAR(1) risk models, where t* = {t; c(t,t) = 0,i > 0} is the adjustment coefficient.Theorem 3 For BINMA(l), BINAR(l), three-variate INMA(l) and three-variate INAR(l) risk models, where t’ = {t > 0;c1(t) = 0}, s’ ={s > 0; c2{s) = 0} are the corresponding adjustment coefficients of their marginal ruin probabilities, andIf F(x) and G(y) belong to the heavy-tailed family εRV(-β2,-β1), 1< β2< β1, let F(x) =1- F(x) and G(y) = 1- G(y), then we have following results:Theorem 4 For the BINMA(l) and BINAR(l) risk models, there exists any given γ1,γ2 >0, Ψmax(u1,u2,n)～E[N1(n)]E[N2(n)]F(u1)G(u2), Ψmax(u2,u2,n)～E[N1(n)](u1)]E(N1(n)]G(u2), Ψ(u1,n)～E[N1(n)]F(u1), Ψ(u2,n)～E[N2(n)]G(u2), Ψmin(u1,u2,n)≤Ψ1(u1),n+Ψ1(u2,n)-Ψmax(u1,u2,u3) hold uniformly for u1 > γ1E[N1(n)], u2 >γ2E[N2(n)].2. Bivariate Markov Binomial risk models based on copulaWe extend the uinvariate Markov Binomial risk models proposed by Cossette et al. (2003 and 2004) to bivariate ones. Considering the bivariate Markov Binomial risk models: where u1, u2, π1 and π2 are defined the same as before, {(Y1k,Y2k),k∈N+} are the bivariate claim process, where {(X1k,X2k), k∈N+} are a sequence of i.i.d. bivariate claim sizes with joint distribution C(x, y), and with margins F1(x1) and F2(x2), respectively; {(I1k,I2k), k∈ N} are the bivariate Markov Bernoulli process. Furthermore, {Imk,k∈N}, m =1,2 obey the marginal univariate Markov Bernoulli process, respectively, and {Imk, k∈N}, m= 1,2, have the probability transition matrices: respectively, and they are irreducible, where pij(m) = P{Im,k+i= j|Imk = i}, i,j∈ {0,1} and the sum is equal to 1 for every row, m =1,2;P{Im0 =1} = qm = 1- P{Im0 = 0} =1- qm (qm ∈(0,1)) are the probabilities of their own initial states, respectively;αm = 1-αm∈ [0.1) are the correlation coefficients for m = 1,2; and if αm = 0, {Imk, k∈N} are mutually independent two sequences of i.i.d. Bernoulli r.v.’s, m = l,2.We make copula onto {I1k, k∈N} and {^k, k eN}, and get the joint transition probability matrix of {(I1k,I2k),k∈N}, denoted Π, then where P(i,j)-(i’,j’),i,j,i’,j’∈{0,1} the probability of (I1k,I2k) transit from the state (i,j) to the state (i’,j’).And we assume that II is irreducible.Assume that the m.g.f.’s of (X1k, X2k), X1k and X2k are MC(t, s), M1(t) and M2(s), respectively, then the joint m.g.f. matrix of the bivariate compound Markov Bernoulli process {(Y1k-π1,Y2k-π2), k∈N+} are where for every i, j∈{0,1}, if i’≠j’∈ {0,1}, M(i，j)-(i’,j’) =p(i,j)M1(t)i’M2(s)j’ ×e-π1t-π2s,M(i,j)-(1,1) = p(i,j)-(1,1)Mc(t,s)e-π1t-π2s.At the same time, the marginal m.g.f. matrices of {(Y1k -π1),k∈N+} and {(Y2k-π2), k∈N+} are presented as follows: where for every i,j∈{0,1}, M(ij)(1) =qij(1)M1(t)je-π1t; and where for every i,j = 0,1, M(ij)(2) = qij(2) M2{s)je-π2s.Proposition 5 The three types of adjustment coefficient expressions of the bi-variate Markov Binomial risk model are presented as follows: c(t,s) =log{A(t,s)}, where Λ(t, s) = argsup{Λ|det (M(t, s) -ΛE) = 0}, i.e. Λ = Λ(i, s) is the maximum eigenvalue of M(t,s); c1(t)=log{Λ1(t)}, where Λ1(t) = argsup{Λ|det (M1(t)-ΛE) = 0}, i.e. Λ1(t) is the maximum eigen-value of M1(t); c2(s) = log{Λ2(s)}, where Λ2(s) = argsup{Λ | det (M2(s) -ΛE) = 0}, i.e.Λ2(s) is the maximum eigen-value of M2{s).Theorem 5 If there exist three kinds of adjustment coefficients for the bivariate Markov Binomial risk model, then Ψmin(u1,u2)≤Ψ1(u1)+Ψ2(u2)-Ψmax(u1,u2), where Δ= {t,s);c(t,s) = 0,t > 0,s > 0}, t* = {t;c{t,t) = 0,t > 0},t’ = {t;c1(t) = 0, t > 0}, and s’ = {s; c2(s) = 0, s > 0}.3. Bivariate Sparre Anderson risk models based on quasi renewal processConsidering the following bidimensional continuous-time risk model: where u1,u2,π1 and π2 are defined the same as before;{(X1j,X2j),k∈N+}are a sequence of i.i.d. bivariate nonnegative r.v.’s with the joint d.f. C(x, y), and with the margins F1(xx1) and F2(x2),respectively, which present the jth bivariate claim sizes; {N(t),t≥0} is the aggregate claim counting process.Let{Tj,j∈N+} be the inter-claim times, then N(t)=max{n:∑j=1 Tj≤t}. Based on the Beta-Gamma series models proposed by Lewis et al. (1983a,1989b), we extend the counting process to a generalized one:let{Tj,j∈N} be the Beta-Gamma time series with Gamma(0, σ) margins, this category of the counting process {N(t),t∈R+} is called the quasi renewal process. Let Y(Y>0)be a Gamma r.v. with parameter vector (θ,σ),there exists a Beta r.v. A acting on Y, such that A(Y)～Gamma(α0,σ),α∈[0,1], where given Y=y>0, A has the density i.e. (A|Y= y) obeys Beta(αθ,αθ;y),α=1-α, and (A(Y),Y) have joint density Based on Beta operator,we have the following Beta-Gamma time series models:·The stationary Beta-Gamma MA(1)model. Tj=Bj(εj-1)+εj,j∈N+, where{ε,j∈N} are a sequence of i.i.d. r.v.’s with Gamma(θ/1+β,σ) distribution,β∈ [0,1],{Bj,j∈N+} are a sequence of i.i.d. r.v.’s obeying Beta(βθ/1+β,θ/1+β) distribution, and Bj+1(εj) obeys Gamma(βθ/1+β,σ).If and only if β∈[0,1], the Beta-Gamma MA(1) is stationary.Proposition 6 If(T1,...,Tn) obeys the Beta-Gamma MA(1) process, the joint characterization function is ·The stationary Beta-Gamma AR(1) processSimilarly to the Beta-Gamma MA(1) process, there exists another Beta-Gamma time series model with the following dependent structure: Tj=Aj(Tj-1)+ej, j∈N, where T0 obeys Gamma(θ,σ), {ej, j∈N+} are a sequence of i.i.d. Gamma r.v.’s with parameter vector (αθ,σ),(α =1 - α, α∈[0,1]), given {ej,∈N+}, {Aj,j∈N+} are a sequence of i.i.d. Beta r.v.’s with the parameter vector (αθ,αθ;Tj-1), Aj(Tj-1) are Gamma(αθ,σ) r.v.’s, and {Tj, j ∈N+} have Gamma(θ,σ) margins. If and only if α∈[0,1), {Tj,j ∈N+} is stationary.Proposition 7 Beta-Gamma AR(1) process is Markovian of order 1, if α→1, {Tj,j =1,2,...} are dependent perfect, and if α= 0, they are mutually independent.Proposition 8 For the stationary Beta-Gamma AR(1) process, (Ti,..., Tn) has the following joint characterization function: whereProposition 9 Assume there exist t,s>0, such that Mx(t) = ∫0∞ etx dF(x) <∞, and hold, then the three adjustment coefficient functions of the Beta-Gamma MA(1) and the Beta-Gamma AR(1) risk models are presented as follows:Theorem 6 For the Beta-Gamma MA(1) and the Beta-Gamma MA(1) risk models, where Δ = {(t,s);c(t,s) = 0,t > 0,s > 0}.Theorem 7 For the Beta-Gamma MA(1) and the Beta-Gamma MA(1) risk models, where t* = {t; c(t,t) = 0, t > 0}.Theorem 8 For the Beta-Gamma MA(1) and the Beta-Gamma MA(1) risk models, Ψmin(u1,u2) ≤Ψ1(u1) +Ψ#2(u2)-Ψmax(u1,u2), the t’ ={t;c1(t)= 0,t > 0} and s’ = {s;c2(s) = 0, s > 0}.Theorem 9 Assume that F(x), G(y) belong to the heavy-tailed distribution family εRV(-β2,-β1), 1 < β2≤β1,and u1,u2>γt, then Ψ1(u1,t)～Λ(t)F(u1),u1→∞ Ψ2(u2,t)～Λ(t)F(u2),u2→∞ hold for some γ > 0.