Methods for estimating optimal Dickson and Waters modification dividend barrier

In applications of collective risk theory, complete information for the distribution of individual claims amount is often unknown, but reliable estimates of its first few moments may be available. Dickson and Waters [Dickson, D.C.M. and Waters, H.R., (2004) Some optimal dividends problems, Astin Bulletin, 34, 49–74.] pointed out that shareholders should be liable to cover the deficit at ruin. Thus, they considered b the level of the barrier that maximizes the expectation of the difference between the discounted dividends until ruin and the discounted deficit at ruin. For such a situation, this paper develops methods for estimating the Dickson–Waters modification for the optimal dividend barrier b with the expectation of discounted penalty at ruin. In particular, two De Vylder approximations are explained, and the diffusion approximation for the expectation of discounted penalty at ruin is examined. For several claim amount distributions, the approximate values are compared numerically with exact values.