Optimal control of capital injections by reinsurance in a diffusion approximation

In this paper we consider a diffusion approximation to a classical risk process, where the claims are reinsured by some reinsurance with deductible b ∈ [0,b?], where b = b? means “no reinsurance” and b = 0 means “full reinsurance”. The cedent can choose an adapted reinsurance strategy (b _t ) _{t ≥0}, i.?e. the deductible can be changed continuously. In addition, the cedent has to inject fresh capital in order to keep the surplus positive. The problem is to minimise the expected discounted cost over all admissible reinsurance strategies. We find an explicit expression for the value function and the optimal strategy using the Hamilton–Jacobi–Bellman approach. Some examples illustrate the method.