Optimal control of the surplus in an insurance policy

A classical continuous time surplus process is modified by adding two actions. If the level of the surplus goes below τ≥0τ≥0, we increase the level of the surplus up to initial level u>τu>τ by injecting capital to the surplus. Meanwhile, the excess amount of the surplus over V>uV>u is invested continuously to other business. After assigning several costs related to managing the surplus, we obtain the long-run average cost per unit time and illustrate a numerical example to show how to find an optimal investment policy minimizing the cost.