A pair of optimal reinsurance–investment strategies in the two-sided exit framework

In this paper, we derive and study a pair of optimal reinsurance–investment strategies under the two-sided exit framework which aims to (1) maximize the probability that the surplus reaches the target b before ruin occurs over the time horizon [0,e_λ] (where e_λ is an independent exponentially distributed random time); (2) minimize the probability that ruin occurs before the surplus reaches the target b over the time horizon [0,e_λ]. We assume the insurer can purchase proportional reinsurance and invest its wealth in a financial market consisting of a risk-free asset and a risky asset, where the dynamics of the latter is assumed to be correlated with the insurance surplus. By solving the associated Hamilton–Jacobi–Bellman (HJB) equation via a dual argument, an explicit expression for the optimal reinsurance–investment strategy is obtained. We find that the optimal strategy of objective (1) (objective (2) resp.) is always more aggressive (conservative resp.) than the strategy of minimizing the infinite-time ruin probability of Promislow and Young (2005). Due to the presence of the time factor e_λ, the optimal strategy under objective (1) or (2) may lead to more aggressive positions as the wealth level increases, a behavior which may be more consistent with industry practices.