## 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 (population genetics, absorption problems, etc.) 91G50 Corporate finance (dividends, real options, etc.)
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