Optimal dividend policies for a class of growth-restricted diffusion processes under transaction costs and solvency constraints

In this paper, we consider a company where surplus follows a rather general diffusion process and whose objective is to maximize expected discounted dividend payments. With each dividend payment there are transaction costs and taxes and it is shown in [7] that under some reasonable assumptions, optimality is achieved by using a lump sum dividend barrier strategy, i.e. there is an upper barrier ū∗ and a lower barrier u∗ so that whenever surplus reaches ū∗, it is reduced to u∗ through a dividend payment. However, these optimal barriers may be unacceptably low from a solvency point of view. It is argued that in that case one should still we should look for a barrier strategy, but with barriers that satisfy a given constraint. We propose a solvency constraint similar to that in [6]; whenever dividends are paid out the probability of ruin within a fixed time T and with the same strategy in the future, should not exceed a predetermined level ε. It is shown how optimality can be achieved under this constraint, and numerical examples are given.