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Optimal dividends: analysis with Brownian motion. (English) Zbl 1085.62122

Summary: 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.

The optimal level ${b}^{*}$ is the value of $b$ for which the expectation of $D$ is maximal. It is shown that ${b}^{*}$ is an increasing function of the variance parameter; as the variance parameter tends toward infinity, ${b}^{*}$ tends toward the ratio of the drift parameter and the valuation force of interest, which can be interpreted as the present value of a perpetuity. The leverage ratio is the expectation of $D$ divided by the initial surplus invested; it is observed that this leverage ratio is a decreasing function of the initial surplus. For $b={b}^{*}$, the expectation of $D$, considered as a function of the initial surplus, has the properties of a risk-averse utility function, as long as the initial surplus is less than ${b}^{*}$. The expected utility of $D$ is calculated for quadratic and exponential utility functions. In the appendix, the original discrete model of De Finetti is explained and a probabilistic identity is derived.

##### MSC:
 62P05 Applications of statistics to actuarial sciences and financial mathematics 60J70 Applications of Brownian motions and diffusion theory 91G50 Corporate finance