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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.

62P05 Applications of statistics to actuarial sciences and financial mathematics
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
91G50 Corporate finance (dividends, real options, etc.)
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