Application Of The Kalman Algorithm In A Generalized Bühlmann-straub Model

The calculation of the credibility premiums is one of core issues in the actuarial mathematics. Currently the calculations of the credibility premiums generally use Bayesian estimation methods. Piet de Jong and Ben Zehnwirth[8] use the Kalman algorithm to derive the Buhlmann-Straub model in their article to obtain another derivation of the Buhlmann-Straub model——the Kalman algorithm. The Buhlmann-Straub model assumes that the risk level of each insured in every risk subset is independent and identically distributed. But in fact, this assumption is difficult to achieve. When the individuals in the groups are not entirely independent, the variance of the group mean will be greater than the individual variance divided by the number of individuals. Therefore, this paper presents a class of credibility model under the more general circumstances, in this model, all individuals in the groups are not fully independent and homogeneous. This model is called the generalized Buhlmann-Straub model. We use the Kalman algorithm to derive this model.First we introduce the classical Buhlmann model in the credibility theory. This model was proposed by Buhlmann[3] in1967. Buhlmann model assumes that under the given parameterΘ=θ, X1,X2,...，Xn are independent and identically distributed random variables. They have the same mean and variance. Mathematical notation is as follows: μ(θ)=E(Xi|Θ=θ);v(θ)=Var[Xi|Θ=θ]. By Bayesian methods, we can obtain the credibility premiums Xn+1:Then we introduce the Buhlmann-Straub model. This model was proposed by Buhlmann and Straub[4] in1970. This model assumes that for a given individual risk Θ=θ, the average loss for each risk unit Xj,X2,...,Xn is independent. They have the same conditional mean, but conditional variance are not the same. They are inversely proportional to the number of the risk units mi contained in the individual risk, ie E(Xi|Θ=θ)=μ(θ); Var (Xj｜Θ=θ)=v(θ)/m.By Bayesian methods, we can obtain the credibility premiums Xn+1: xn+i=zX+(1-z)μ,(0.2)Then we introduce the Kalman filter. Kalman filter consists of two parts. One part is the observation equation which describes the observation vector, The other part is the state equation which describes the state vector. Kalman filtering using equations expressed is as follows: x(n)=M(n)β(n)+u(n),(0.3) β(n)=H(n)β(n-1)+v(n).(0.4)In formula (0.3), kx1vector x(n) represents the observation vector at time n in a dynamic system; kxm matrix M(n) is a known matrix, called the observation matrix in the system; in formula (0.4), mx1vectorβ(n) is a vector that can not be observed, which means the system state vector under the time n； m×m matrix H(n) is a known matrix,called the state transition matrix,Shows the state transfer of the dynamic system between the time n一1and the time n.u(n) and v(n) are both white noise with0mean.Also the initial value of the dynamic systemwβ(0)and v(n)、u(n)(n≥0)are independent,and the white noise v(n)and u(n)are independent too.Finally,through the derivation of the Kalman filter,we can obtain the Kalman algorithm as follows: β(nln-1)=H(n)β(n-1),(0.5a) β(n)=β(nln-1)+K(n){X(n)-M(n)β(nln-1)},(0.5b) C(n)={I-K(n)M(n)}C(nln-1),(0.5c) K(n)=C(nln-1)M’(n){M(n)C(nln-1)M’(n)+U(n)}-1,(0.5d) C(nln-1)=H(n)C(n-1)H’(n)+v(n).(0.5e)We derive the maximum precision credibility model with the Kalman algorithm,the results are consistent with those obtained with the Bayesian estimation methods.Finally,we give the generalized Buhlmann-Straub model. Specific form is as follows:Assume that O is given,and X1,X2,…,Xnare independent random variables.As the risk of each insured person within the subset Xi is not independent and homogenous,X1,X2,…,Xn have the same conditional mean,but the variance of the group mean w_ll be greater than the individual variance divided by the number of individuals: E(Xi|Θ)=μ(Θ)；Vat(Xi|Θ)=γw(O)+ηv(O)/mi. We use the Kalman algorithm to derive it,the result is as follows: Where...