On ruin probability minimization under excess reinsurance

The problem of ruin probability minimization in the Cramer–Lundberg risk model under excess reinsurance is studied. Together with traditional maximization of the Lundberg characteristic coefficient $R$ is considered the problem of direct calculation of insurer's ruin probability $\psi_r(x)$ as an initial-capital function $x$ under the prescribed level of net-retention $r$. To solve this problem, we propose the excess variant of the Cramer integral equation which is an equivalent to the Hamilton–Jacobi–Bellman equation. The continuation method is used for solving this equation; by means of it is found the analytical solution to the Markov risk model. We demonstrated on a series of standard examples that with any admissible value of $x$ the ruin probability $\psi_x(r):=\psi_r(x)$ is usually a unimodal function $r$. A comparison of the analytic representation of ruin probability $\psi_r(x)$ with its asymptotic approximation with $x\rightarrow\infty$ was conducted.