Abstract
This paper studies the stochastic optimal control problem of finding optimal dividend policies of an insurance company in discrete time with the use of general Lipschitz payoff functions involving indicators of profitability and risk. To construct positional optimal controls and to evaluate the performance indicators, the dynamic programming method is validated. The convergence rate of the successive approximation method in finding generally unbounded Bellman functions is estimated. The Pareto-optimal set of the problem is numerically approximated by so-called barrier-proportional control strategies.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
B. de Finetti, “Su un’ impostazione alternativa della teoria collettiva del rischio,” in: Trans. XV-th Intern. Congress of Actuaries, 2, 433–443 (1957).
K. Borch, “The theory of risk (with discussion),” J. Roy. Statist. Soc. Ser. B, 29, 432–467 (1967).
H. U. Gerber, An Introduction to Mathematical Risk Theory, Huebner Foundation Monographs, Philadelphia (1979).
H. Schmidli, Stochastic Control in Insurance, Springer, London (2008).
B. Avanzi, “Strategies for dividend distribution: A review,” North Amer. Actuar. J., 13, No. 2, 217–251 (2009).
H. Albrecher and S. Thonhauser, “Optimality results for dividend problems in insurance,” Rev. R. Acad. Cien. Ser. A. Math., 103, No. 2, 295–320 (2009).
S. Asmussen and H. Albrecher, Ruin Probabilities, 2nd Edition, World Sci., London (2010).
S. E. Shreve, J. P. Lehoczky, and D. P. Gaver, “Optimal consumption for general diffusions with absorbing and deflecting barriers,” SIAM J. Control and Optim., 22, No. 1, 55–75 (1984).
A. B. Piunovsky, Optimal Control of Random Sequences in Problems with Constraints [in Russian], Nauchn. Kniga, Moscow (1996).
A. B. Piunovsky, Optimal Control of Random Sequences in Problems with Constraints, Springer, Dordrecht (1997).
S. P. Sethi, Optimal Consumption and Investment with Bankruptcy, Kluwer, Boston–Dordrecht–London (1997).
J. Paulsen, “Optimal dividend payouts for diffusions with solvency constraints,” Finance Stoch., 7, No. 4, 457–473 (2003).
C. Hipp, “Optimal dividend payment under a ruin constraint: Discrete time and state space,” Blatter der DGVFM (Blatter der Deutschen Gesellschaft fur Versicherungs-und Finanzmathematik e.V.), 26, No. 2, 255–264 (2003).
D. C. M. Dickson and H. R. Waters, “Some optimal dividend problems,” ASTIN Bulletin, 34, No. 1, 49–74 (2004).
H. U. Gerber, E. S. W. Shiu, and N. Smith, “Maximizing dividends without bankruptcy,” ASTIN Bull., 36, No. 1, 5–23 (2006).
S. Thonhauser and H. Albrecher, “Dividend maximization under consideration of the time value of ruin,” Insurance: Mathematics and Economics, 41, 163–184 (2007).
E. Bayraktar and V. Young, “Maximizing utility of consumption subject to a constraint on the probability of lifetime ruin,” Finance and Research Letters, 5, No. 4, 204–212 (2008).
B. V. Norkin, “On numerical solution of the stochastic optimal control problem for the dividend policy of an insurance company,” Computer Mathematics, No. 1, 131–139 (2014).
B. V. Norkin, “On stochastic optimal control of risk processes in discrete time,” Theory of Optimal Solutions, No. 1, 55–62 (2014).
B. V. Norkin, “On stochastic optimal control of discrete-time risk processes,” Problems of Control and Informatics, No. 5, 108–121 (2014).
B. V. Norkin, “Necessary and sufficient conditions of existence and uniqueness of solutions to integral equations of actuarial mathematics,” Cybernetics and Systems Analysis, 42, No. 5, 743–749 (2006).
B. V. Norkin, “On the solution of the basic integral equation of actuarial mathematics by the successive approximation method,” Ukrainian Mathematical Journal, 59, No, 12, 112–127 (2007).
B. V. Norkin, “Systems simulation analysis and optimization of insurance business,” Cybernetics and Systems Analysis,” 50, No. 2, 260–270 (2014).
H. U. Gerber and E. S. W. Shiu, “On the time value of ruin,” North Amer. Actuar. J., 2, 48–78 (1998).
Y. Ermoliev, T. Ermolieva, G. Fischer, et al., “Discounting, catastrophic risks management and vulnerability modeling,” Math. and Comput. in Simul., 79, 917–924 (2008).
H. U. Gerber, “Entscheidungskriterien fur den zusammengesetzten Poisso–Prozess,” Schweiz. Verein. Versicherungsmath. Mitt., 69, 185–228 (1969).
D. P. Bertsekas and S. E. Shreve, Stochastic Optimal Control: The Discrete-Time Case, Athena Sci., Belmont (Mass.) (1996).
J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis [Russian translation], Mir, Moscow (1988).
A. N. Shiryaev, Probability [in Russian], Nauka, Moscow (1980).
R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Springer, Berlin (1998).
H. Gerber, “The dilemma between dividends and safety and a generalization of the Lundberg–Cramer formulas,” Scand. Actuar. J., No. 1, 46–57 (1974).
H. Albrecher and R. Kainhofer, “Risk theory with a nonlinear dividend barrier,” Computing, 68, No. 4, 289–311 (2002).
K. J. Kushner and P. Dupuis, Numerical Methods for Stochastic Control Problems in Continuous Time, Springer, New York (1992).
B. V. Norkin, “On the identification of models of dynamic financial analysis of an insurance company,” Computer Mathematics, No. 2, 24–33 (2013).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 139–154, September–October, 2014.
Rights and permissions
About this article
Cite this article
Norkin, B.V. Stochastic Optimal Control of Risk Processes with Lipschitz Payoff Functions. Cybern Syst Anal 50, 774–787 (2014). https://doi.org/10.1007/s10559-014-9668-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10559-014-9668-7