Multivariable Life Model Based On Comonotonic Dependence Structure

In life insurance practice, when more than one life are involved they areoften assumed to be independent. Under this hypothesis, it is can obtain thatthe multivariable life function, pure premiums of life insurance policy and lifeannuity. However, this hypothesis is apart from the actual circumstance.In fact, it is deemed to be positively dependent among lives that are affectedby some same factors. In the sense of correlation order, prove that the comonotonic bivariable random vector has the strongest positively dependent structureamong binomial random vectors which have the same marginals. It also studythe probability distribution of the two-life model based on the correspondingdependent structure, and obtain the stochastic upper and lower bounds of theremaining life of two status in the sense of stochastic order. However, the scopelocalization of the two-life models have certain limitations.It is can't completelysatisfy the demand of the actuarial work. Exactly according to this defect, inthis paper we aim to extend the two-life model to the multivariable life model,We study the multivariable life function of multivariable life model based on thecomonotonic dependent structure, and obtain the actuarial present value of themodel. On the top of this basic, we obtain the stochastic upper and lower boundsof the remaining life of two status in the sense of stochastic order.Therefore, this paper got primarily four conclusions:1.It is prove that the comonotonic random vector has the strongest positively dependent structure among multifiber randon vecto which have the samemarginals. 2. It is obtain that the multivariable life function and mortality function oftwo status, which are suppose the remaining life of multivariable life model arecomonotonic. The Unite survival state(x₁x₂...x_n) tqx₁x₂...x_n=max{tqx₁,tqx₂,...tqx_n}μT(x₁x₂...x_n)(t)=μx₁(t)I(tqx₁x₂...x_n=tqx₁)++...+μx_n(t),I(tqx₁x₂...x_n=tqx_n)The finally survival state(?) t(?)=min{tqx₁, tqx₂,..., tqx_n}μ_T(?)(t)=μ_x1(t)I(t(?)=tqx₁)+...+μx_n(t)I(t(?)=tqx_n)We also prove that the remaining life of single life and multivariable life arecomonotonic.3. in the foundation of the second conclusion, we get the pure premiums ofthe multivariable life model which is given in the end of the death year, and lifeannuity actuarial present value of the dispersed circumstance.4. We obtain the stochastic upper and lower bounds of the remaining life oftwo status in the sense of stochastic order. T^#(x₁...x_n)≤st T^⊥(x₁...x_n)≤st T(x₁...x_n)≤st T^*(x₁...x_n) T^#(?)≤st T^⊥(?)≤st T(?)≤st T^*(?)...