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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 (population genetics, absorption problems, etc.)
91G50 Corporate finance (dividends, real options, etc.)
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[1] Albrecher H., Computing 68 pp 289– (2002) · Zbl 1076.91521 · doi:10.1007/s00607-001-1447-4
[2] Asmussen Søren, Insurance: Mathematics & Economics 20 pp 1– (1997) · Zbl 1065.91529 · doi:10.1016/S0167-6687(96)00017-0
[3] Borch Karl., The Mathematical Theory of Insurance (1974)
[4] Borch Karl., Economics of Insurance (1990)
[5] Bühlmann Hans., Mathematical Methods in Risk Theory (1970) · Zbl 0209.23302
[6] Bühlmann Hans., Giornale dell’Istituto Italiano delgi Attuari 65 pp 1– (2002)
[7] M. Merce Claramunt, Bulletin of the Swiss Association of Actuaries pp 149– (2003)
[8] Cox D. R., The Theory of Stochastic Processes (1965) · Zbl 0149.12902
[9] De Finetti Bruno., Transactions of the XVth International Congress of Actuaries 2 pp 433– (1957)
[10] Feller William., An Introduction to Probability Theory and Its Applications, 2. ed. (1971) · Zbl 0077.12201
[11] Gerber Hans U., Operations Research 20 pp 37– (1972) · Zbl 0236.90079 · doi:10.1287/opre.20.1.37
[12] Gerber Hans U., An Introduction to Mathematical Risk Theory (1979) · Zbl 0431.62066
[13] Gerber Hans U., North American Actuarial Journal 2 (1) pp 48– (1998)
[14] Gerber Hans U., North American Actuarial Journal 7 (3) pp 37– (2003)
[15] Gerber Hans U., Metodi Statistici per la Finanza e le Assicurazioni pp 75– (2003)
[16] Grandell Jan., Aspects of Risk Theory (1991) · doi:10.1007/978-1-4613-9058-9
[17] J. Michael. Harrison, Brownian Motion and Stochastic Flow Systems (1985) · Zbl 0659.60112
[18] Højgaard Bjarne, Scandinavian Actuarial Journal pp 225– (2002) · Zbl 1039.91042 · doi:10.1080/03461230110106291
[19] Højgaard Bjarne, Mathematical Finance 9 pp 153– (1999) · Zbl 0999.91052 · doi:10.1111/1467-9965.00066
[20] Iglehart Donald L., Journal of Applied Probability 6 pp 285– (1969) · Zbl 0191.51202 · doi:10.2307/3211999
[21] Irbäck Johan., Scandinavian Actuarial Journal pp 97– (2003) · Zbl 1092.91043 · doi:10.1080/03461230110106345
[22] Jeanblanc-Picqué Monique, Russian Mathematical Surveys 20 pp 257– (1995) · Zbl 0878.90014 · doi:10.1070/RM1995v050n02ABEH002054
[23] Karlin Samuel, A First Course in Stochastic Processes, 2. ed. (1975)
[24] Klugman Stuart A., Loss Models: From Data to Decision (1998) · Zbl 0905.62104
[25] Lin X. Sheldon, Insurance: Mathematics & Economics 33 pp 551– (2003) · Zbl 1103.91369 · doi:10.1016/j.insmatheco.2003.08.004
[26] Miyasawa K., Journal of Operations Research Society of Japan 4 pp 95– (1962)
[27] Morill John., Naval Research Logistic Quarterly 13 pp 49– (1966) · Zbl 0149.16904 · doi:10.1002/nav.3800130105
[28] Paulsen Jostein, Insurance: Mathematics & Economics 20 pp 215– (1997) · Zbl 0894.90048 · doi:10.1016/S0167-6687(97)00011-5
[29] Panjer Harry H., Financial Economics: With Applications to Investments, Insurance, and Pensions (1998)
[30] Seal Hilary L., Stochastic Theory of a Risk Business (1969)
[31] Siegl Thomas, Insurance: Mathematics & Economics 24 pp 51– (1999) · Zbl 0944.91032 · doi:10.1016/S0167-6687(98)00037-7
[32] Takeuchi K., Journal of Operations Research Society of Japan 4 pp 114– (1962)
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