Optimal investment and reinsurance on survival and growth problems for the risk model with common shock dependence

Abstract

This paper investigates goal-reaching problems regarding optimal investment and proportional reinsurance with two dependent classes of insurance business, where the two claim number processes are correlated through a common shock component. The optimization problems are formulated in a general form first, and then four criteria including maximum survival probability, minimum expected ruin penalty, minimum (maximum) expected time (reward) to reach a goal are fully discussed. By the technique of stochastic control theory and through the corresponding Hamilton–Jacobi–Bellman equation, the optimal results are derived and analyzed in different cases. In particular, when discussing the maximum survival probability with a target level U beyond the safe level (where ruin can be avoided with certainty once it is achieved), we construct ε-optimal (suboptimal) strategies to resolve the inaccessibility of the safe level caused by classical optimal strategies. Furthermore, numerical simulations and analysis are presented to illustrate the influence of typical parameters on the main results.