Optimal Dividends : Analysis with Brownian Motion

In the absence of dividends, the surplus of a company is modeled by a Wiener process (or Brownian motion) with positive drift. Now dividends are paid according to a barrier strategy: Whenever the (modified) surplus attains the level b, the ?overflow? is paid as dividends to shareholders. An explicit expression for the moment-generating function of the time of ruin is given. Let D denote the sum of the discounted dividends until ruin. Explicit expressions for the expectation and the moment-generating function of D are given; furthermore, the limiting distribution of D is determined when the variance parameter of the surplus process tends toward infinity. It is shown that the sum of the (undiscounted) dividends until ruin is a compound geometric random variable with exponentially distributed summands