An optimization of a continuous time risk process

A continuous time risk process is considered, where the premium rate is constant and the claims form a compound Poisson process. We assume that an action is taken, either an investment to other business when the level of surplus reaches V>0$V>0$ or an injection of capital when the surplus goes below τ(0<ττ(0<τ<V). After assigning several costs related to managing the surplus, we obtain the long-run average cost per unit time. A numerical example is studied.