Approximation Of Surplus Reinsurance Ruin Probability

The paper mainly concerns the ruin probability of surplus reinsurance.First, we establish the following model as earnings process model of surplus rein-surance, by delining surplus reinsurance and calculating premium based on principle of pure premium: where c=λμ.Total claim number N(t) is made up of N1(t) and N2(t), N1(t) is the number of no reinsurance, N2(t) is the number of reinsurance happend, N1(t) obey the parameters for λp Poisson process, and N2(t) obey the parameters for λq Poisson process, where p=P(S≤m), and p+q=1.Theorem 1 For the surplus process U(t), the final ruin probability is where R is adjustment coefficient of earnings process U(t), is the only root of the following function g(r)= λp(Mx(r)-1)+λq(MY(r)-1)+λq(Mz(r)-1)-re= 0,The model is tedious and we can’t obtain obvious expression of ruin probability, so we define compensation amount paid by original insurer for the i time as And we can use the expected value principle to calculate premium. Then unit premuim rate of original insurer is where additional security premium rates of original insurer is θ=(?)/λμ-1, whe additional security premium rates of original insurer is η. Thus, earning process can be transformed for the following classic risk model: Hence, final ruin probability is where R is adjustment coefficient of earnings process U(t), is the only positive root of the following function: But, we still can’t obtain obvious expression of ruin probability at time.So use Brow-nian motion with drift as S1(t) diffusion approximation, instead ofS1(t), where{Wt,t≥0}is a standard Brow-nian motion, we can obtain a new earnings process:Theorem 2 For earnings process Utm, final ruin probability is ψDm(u)=e-RD(m)u where adjustment coefficient is the only positive root of the following function g(0)=0.Proposition 3 Let retention be then the following equation is minimum.