Leobacher, Gunther; Szölgyenyi, Michaela; Thonhauser, Stefan Bayesian dividend optimization and finite time ruin probabilities. (English) Zbl 1292.91182 Stoch. Models 30, No. 2, 216-249 (2014). The paper considers the valuation problem of a company, where the firm value is given by \[ X_t=x+\int_0^t \theta ds +\sigma B_t -L_t, \] \(x\) (\(>0\)) being the initial capital, \(\theta\) the constant unobservable drift, \(\sigma\) the constant known volatility, \((B_t)_{t\geq 0}\) a standard Brownian motion, and \((L_t)_{t\geq 0}\) the accumulated dividend process. Aim of the paper is the solution of the dividend maximization problem.First of all, since the drift parameter is supposed to be unknown, on the basis of filtering theory, an estimator is derived, so overcoming incomplete information. Then, the Hamilton-Jacobi-Bellman equation for the stochastic optimal control problem is derived and studied, and successively a numerical method for estimating the optimal dividend strategy is presented. The case of threshold strategies is analyzed; finally, the finite-time ruin probabilities are investigated. Reviewer: Emilia Di Lorenzo (Napoli) Cited in 5 Documents MSC: 91G50 Corporate finance (dividends, real options, etc.) 91B30 Risk theory, insurance (MSC2010) 60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 49L20 Dynamic programming in optimal control and differential games 93E11 Filtering in stochastic control theory 93E20 Optimal stochastic control 91G60 Numerical methods (including Monte Carlo methods) Keywords:dividend maximization; filtering theory; finite-time ruin probability; stochastic optimal control; viscosity solutions PDFBibTeX XMLCite \textit{G. Leobacher} et al., Stoch. Models 30, No. 2, 216--249 (2014; Zbl 1292.91182) Full Text: DOI arXiv References: [1] DOI: 10.1007/BF03191909 · Zbl 1187.93138 · doi:10.1007/BF03191909 [2] DOI: 10.1016/S0167-6687(96)00017-0 · Zbl 1065.91529 · doi:10.1016/S0167-6687(96)00017-0 [3] DOI: 10.1080/10920277.2009.10597549 · doi:10.1080/10920277.2009.10597549 [4] DOI: 10.1111/j.0960-1627.2005.00220.x · Zbl 1136.91016 · doi:10.1111/j.0960-1627.2005.00220.x [5] DOI: 10.1090/S0273-0979-1992-00266-5 · Zbl 0755.35015 · doi:10.1090/S0273-0979-1992-00266-5 [6] de Finetti B., Transactions of the XVth International Congress of Actuaries 2 pp 433– (1957) [7] DOI: 10.1007/s00780-006-0027-z · Zbl 1142.91052 · doi:10.1007/s00780-006-0027-z [8] DOI: 10.1080/15326349.2011.593408 · Zbl 1237.91127 · doi:10.1080/15326349.2011.593408 [9] Fleming W., Controlled Markov Processes and Viscosity Solutions (2006) · Zbl 1105.60005 [10] Gerber H., ASTIN Bull. 9 (1) pp 125– (1977) · doi:10.1017/S0515036100011454 [11] DOI: 10.1080/03461230601165201 · Zbl 1164.62080 · doi:10.1080/03461230601165201 [12] DOI: 10.1016/j.insmatheco.2003.12.001 · Zbl 1136.91481 · doi:10.1016/j.insmatheco.2003.12.001 [13] DOI: 10.1070/RM1995v050n02ABEH002054 · Zbl 0878.90014 · doi:10.1070/RM1995v050n02ABEH002054 [14] DOI: 10.1007/s00780-012-0174-3 · Zbl 1252.93135 · doi:10.1007/s00780-012-0174-3 [15] DOI: 10.1007/978-1-4612-0949-2 · doi:10.1007/978-1-4612-0949-2 [16] Karatzas I., Options Pricing, Interest Rates and Risk Management (2001) · Zbl 0699.90010 [17] DOI: 10.1007/978-1-4612-6051-6_2 · doi:10.1007/978-1-4612-6051-6_2 [18] DOI: 10.1007/978-1-4757-1665-8 · doi:10.1007/978-1-4757-1665-8 [19] Liptser R. S., Statistics of Random Processes II–Applications (2001) [20] DOI: 10.1007/978-3-540-89500-8 · Zbl 1165.93039 · doi:10.1007/978-3-540-89500-8 [21] DOI: 10.1016/0165-1889(95)00904-3 · Zbl 0875.90045 · doi:10.1016/0165-1889(95)00904-3 [22] DOI: 10.1239/jap/1118777176 · Zbl 1138.93428 · doi:10.1239/jap/1118777176 [23] DOI: 10.1007/s00780-004-0132-9 · Zbl 1063.91040 · doi:10.1007/s00780-004-0132-9 [24] Schmidli H., Stochastic Control in Insurance (2008) · Zbl 1133.93002 [25] DOI: 10.1137/0322005 · Zbl 0535.93071 · doi:10.1137/0322005 [26] DOI: 10.1016/j.insmatheco.2011.01.002 · Zbl 1218.91096 · doi:10.1016/j.insmatheco.2011.01.002 [27] Wheeden R., Measure and Integral: An Introduction to Real Analysis (1977) · Zbl 0362.26004 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.