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Zbl 1085.62122
Gerber, Hans U.; Shiu, Elias S.W.
Optimal dividends: analysis with Brownian motion.
(English)
[J] N. Am. Actuar. J. 8, No. 1, 1-20 (2004). ISSN 1092-0277

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. \par 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 2000:
*62P05 Appl. of statistics to actuarial sciences and financial mathematics
60J70 Appl. of diffusion theory
91B28 Finance etc.

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