Abstract
This paper is a survey of some classical contributions and recent progress in identifying optimal dividend payment strategies in the framework of collective risk theory. In particular, available mathematical tools are discussed and some challenges are described that occur under various objective functions and model assumptions. Finally, some open research problems in this field are stated.
Resumen
Este artículo es una revista de algunos resultados clásicos y avances recientes en la identificación de estrategias óptimas de pago de dividendos en el marco de la teoría de riesgo para modelos colectivos. En particular, describimos las herramientas matemáticas disponibles y discutimos algunos de los retos que se presentan bajo diferentes funciones objetivo y supuestos del modelo. Finalmente, presentamos problemas abiertos de investigación en esta área.
Similar content being viewed by others
References
Albrecher, H., Borst, S., Boxma, O. andResing, J., (2009). The tax identity in risk theory — a simple proof and an extension, Insurance Math. Econom., 44, 2, 304–306..
Albrecher, H., Claramunt, M. M. andMármol, M., (2005). On the distribution of dividend payments in a Sparre Andersen model with generalized Erlang(n) interclaim times,Insurance Math. Econom.,37(2), 324–334.
Albrecher, H. andHartinger, J., (2006). On the non-optimality of horizontal dividend barrier strategies in the Sparre Andersen model,Hermis J. Comp. Math. Appl.,7, 109–122.
Albrecher, H. andHartinger, J., (2007). A risk model with multi-layer dividend strategy,N. Am. Actuar. J.,11(2), 43–64.
Albrecher, H., Hartinger, J. andTichy, R. F., (2005). On the distribution of dividend payments and the discounted penalty function in a risk model with linear dividend barrier,Scand. Actuar. J.,2005(2), 103–126.
Albrecher, H. andHipp, C., (2007). Lundberg’s risk process with tax,Bl. DGVFM,28(1), 13–28.
Albrecher, H. andKainhofer, R., (2002). Risk theory with a nonlinear dividend barrier,Computing,68(4), 289–311.
Albrecher, H., Renaud, J.-F. andZhou, X., (2008). A Lévy insurance risk process with tax,J. Appl. Probab.,45(2), 363–375.
Albrecher, H. andThonhauser, S., (2008). Optimal dividend strategies for a risk process under force of interest,Insurance Math. Econom.,43(1), 134–149.
Albrecher, H. and Thonhauser, S., (2009). Some remarks on an impulse control problem in insurance,Preprint, University of Lausanne.
Alegre, A., Claramunt, M. and Mármol, M., (1999). Dividend policy and ruin probability,Proceedings of the Third Int. Congress on Insurance, Mathematics & Economics, London.
Asmussen, S., (2000).Ruin probabilities, World Scientific Publishing Co. Inc., River Edge.
Asmussen, S. andTaksar, M., (1997). Controlled diffusion models for optimal dividend pay-out,Insurance Math. Econom.,20(1), 1–15.
Avanzi, B., (2009). Strategies for dividend distribution: a review,N. Am. Actuar. J. to appear.
Avram, F., Palmowski, Z. andPistorius, M. R., (2007). On the optimal dividend problem for a spectrally negative Lévy process,Ann. Appl. Probab.,17(1), 156–180.
Azcue, P. andMuler, N., (2005). Optimal reinsurance and dividend distribution policies in the Cramér-Lundberg model,Math. Finance,15(2), 261–308.
Badescu, D. S., Drekic, S. andLandriault, D., (2007). On the analysis of a multi-threshold Markovian risk model,Scand. Act. J.,2007(4), 248–260.
Bäuerle, N., (2004). Approximation of optimal reinsurance and dividend payout policies,Math. Finance,14(1), 99–113.
Bayraktar, E. andYoung, V., (2008). Maximizing utility of consumption subject to a constraint on the probability of lifetime ruin,Finance and Research Letters,5, 4, 204–212.
Bayraktar, E. andYoung, V., (2008). Minimizing the probability of lifetime ruin under random consumption,N. Am. Actuar. J.,12, 4, 384–400.
Bayraktar, E. andYoung, V., (2008). Minimizing the probability of ruin when consumption is ratcheted,N. Am. Actuar. J.,12, 4, 428–442.
Bensoussan, A. andLions, J.-L., (1982).Applications of variational inequalities in stochastic control, volume12 ofStudies in Mathematics and its Applications. North-Holland Publishing Co., Amsterdam. Translated from the French.
Bensoussan, A. andLions, J.-L., (1984).Impulse control and quasivariational inequalities, Gauthier-Villars, Montrouge.
Bensoussan, A., Liu, R. H. andSethi, S. P., (2005). Optimality of an (s, S) policy with compound Poisson and diffusion demands: a quasi-variational inequalities approach,SIAM J. Control Optim.,44(5), 1650–1676.
Bertsekas, D. P. andShreve, S. E., (1978).Stochastic optimal control, Academic Press Inc., New York.
Bismut, J.-M., (1973). Conjugate convex functions in optimal stochastic control,J. Math. Anal. Appl.,44, 384–404.
Black, F. andScholes, M., (1973). The pricing of options and corporate liabilities,J. Political Econ.,81(3), 637–659.
Boguslavskaya, E., (2006).Optimization Problems in Financial Mathematics: Explicit Solutions for Diffusion Models, Ph. D. Thesis, University of Amsterdam.
Borch, K., (1967). The theory of risk. (With discussion),J. Roy. Statist. Soc. Ser. B,29, 432–467.
Borch, K., (1974).The Mathematical Theory of Insurance, Lexington, Massachussets.
Browne, S., (1995). Optimal investment policies for a firm with a random risk process: exponential utility and minimizing the probability of ruin,Math. Oper. Res.,20(4), 937–958.
Bühlmann, H., (1970).Mathematical Methods in Risk Theory, Springer-Verlag, Heidelberg.
Cadenillas, A., Choulli, T., Taksar, M. andZhang, L., (2006). Classical and impulse stochastic control for the optimization of the dividend and risk policies of an insurance firm,Math. Finance,16(1), 181–202.
Cadenillas, A., Sarkar, S. andZapatero, F., (2007). Optimal dividend policy with mean-reverting cash reservoir,Math. Finance,17(1), 81–109.
Cai, J., Gerber, H. U. andYang, H., (2006). Optimal dividends in an Ornstein-Uhlenbeck type model with credit and debit interest,N. Am. Actuar. J.,10(2), 94–119.
Crandall, M. G., Ishii, H. andLions, P.-L., (1992). User’s guide to viscosity solutions of second order partial differential equations,Bull. Amer. Math. Soc. (N.S.),27(1), 1–67.
Crandall, M. G. andLions, P.-L., (1983). Viscosity solutions of Hamilton-Jacobi equations,Trans. Amer. Math. Soc.,277(1), 1–42.
Davis, M. H. A., (1993).Markov models and optimization, Chapman & Hall London.
De Finetti, B., (1957). Su un’ impostazione alternativa dell teoria collettiva del risichio,Transactions of the XVth congress of actuaries, (II), 433–443.
Deangelo, H. andDeangelo, L., (2006). The irrelevance of the MM dividend irrelevance theorem,Journal of Financial Economics,79, 293–315.
Deangelo, H. andDeangelo, L., (2008). Reply to “the irrelevance of the MM dividend irrelevance theorem”,Journal of Financial Economics,87, 532–533.
Dickson, D. C. M. andWaters, H. R., (2004). Some optimal dividends problems,Astin Bull.,34(1), 49–74.
Fleming, W. H., (1977). Generalized solutions in optimal stochastic control, InDifferential games and control theory, II (Proc. 2nd Conf., Univ. Rhode Island, Kingston, R.I., 1976), 147–165. Lecture Notes in Pure and Appl. Math.,30. Dekker, New York.
Fleming, W. H. andSoner, H. M., (1993).Controlled Markov processes and viscosity solutions, Springer, New York.
Frankfurter, G. andWood, B., (2002). Dividend policy theories and their empirical tests,International Review of Financial Analysis,11(2), 111–138.
Frostig, E., (2005). On the expected time to ruin and the expected dividends when dividends are paid while the surplus is above a constant barrier,J. Appl. Probab.,42(3), 595–607.
Gaier, J. andGrandits, P., (2002). Ruin probabilities in the presence of regularly varying tails and optimal investment,Insurance Math. Econom.,30(2), 211–217.
Gaier, J., Grandits, P. andSchachermayer, W., (2003). Asymptotic ruin probabilities and optimal investment,Ann. Appl. Probab.,13(3), 1054–1076.
Garrido, J., (1989). Stochastic differential equations for compounded risk reserves,Insurance Math. Econom.,8(3), 165–173.
Gerber, H. U., (1969). Entscheidungskriterien fuer den zusammengesetzten Poisson-prozess,Schweiz. Aktuarver. Mitt., (1), 185–227.
Gerber, H. U., (1974). The dilemma between dividends and safety and a generalization of the Lundberg-Cramér formulas,Scand. Actuar. J.,1974(1), 46–57.
Gerber, H. U., (1981). On the probability of ruin in the presence of a linear dividend barrier,Scand. Actuar. J.,1981(2), 105–115.
Gerber, H. U., Lin, S. andYang, H., (2006). A note on the dividends-penalty identity and the optimal dividend barrier,Astin Bull.,36(2), 489–503.
Gerber, H. U. andShiu, E. S. W., (2004). Optimal dividends: analysis with Brownian motion,N. Am. Actuar. J.,8(1), 1–20.
Gerber, H. U. andShiu, E. S. W., (2006). On optimal dividends: from reflection to refraction,J. Comput. Appl. Math.,186(1), 4–22.
Gerber, H. U. andShiu, E. S. W., (2006). On optimal dividend strategies in the compound Poisson model,N. Am. Actuar. J.,10(2), 76–93.
Gerber, H. U., Shiu, E. S. W. andSmith, N., (2006). Maximizing dividends without bankruptcy,Astin Bull.,36(1), 5–23.
Gordon, M. J., (1959). Dividends, earnings and stock prices, Review of Economics and Statistics, 41, 99–105.
Grandell, J., (1991).Aspects of risk theory, Springer, New York.
Grandits, P., Hubalek, F., Schachermayer, W. andZigo, M., (2007). Optimal expected exponential utility of dividend payments in a brownian risk model, Scand.Actuar. J.,2007(2), 73–107.
Handley, J., (2008). Dividend policy: Reconciling DD with MM.Journal of Financial Economics,87, 528–531.
He, L. andLiang, Z., (2009). Optimal financing and dividend control of the insurance company with fixed and proportional transaction costs,Insurance Math. Econom.,44, 1, 88–94.
Hipp, C., (2003). Optimal dividend payment under a ruin constraint: discrete time and state space, working paper.
Hipp, C. andPlum, M., (2000). Optimal investment for insurers,Insurance Math. Econom.,27(2), 215–228.
Hipp, C. andPlum, M., (2003). Optimal investment for investors with state dependent income, and for insurers,Finance Stoch.,7(3), 299–321.
Hipp, C. andSchmidli, H., (2004). Asymptotics of ruin probabilities for controlled risk processes in the small claims case,Scand. Actuar. J., 2004 (5), 321–335.
Hipp, C. andM. Vogt, (2003). Optimal dynamic XL reinsurance,Astin Bull.,33 (2), 193–207.
Højgaard, B., (2002). Optimal dynamic premium control in non-life insurance: maximizing dividend payouts,Scand. Actuar. J.,2002(20), 225–245.
HØjgaard, B. andTaksar, M., (1999). Controlling risk exposure and dividends payout schemes: insurance company example,Math. Finance,9(2), 153–182.
HØjgaard, B. andTaksar, M., (2004). Optimal dynamic portfolio selection for a corporation with controllable risk and dividend distribution policy,Quant. Finance,4(3), 315–327.
Hubalek, F. andSchachermayer, W., (2004). Optimizing expected utility of dividend payments for a Brownian risk process and a peculiar nonlinear ODE,Insurance Math. Econom.,34(2), 193–225.
Iglehart, D. L., (1969). Diffusion approximations in collective risk theory,J. Appl. Probability,6, 285–292.
Irbäck, J., (2003). Asymptotic theory for a risk process with a high dividend barrier,Scand. Act. J.,2003(2), 97–118.
Jeanblanc-Picqué, M. andShiryaev, A. N., (1995). Optimization of the flow of dividends,Uspekhi Mat. Nauk,50(2(302)), 25–46.
Kerekesha, D. P., (2003). Exact solution of the risk equation with a piecewise-constant current reserve function,Teor. Îmovīr. Mat. Stat.,69, 57–62.
Kramkov, D. andSchachermayer, W., (1999). The asymptotic elasticity of utility functions and optimal investment in incomplete markets,Ann. Appl. Probab.,9(3), 904–950.
Kulenko, N. andSchmidli, H., (2008). Optimal dividend strategies in a Cramér-Lundberg model with capital injections,Insurance Math. Econom.,43(2), 270–278.
Kushner, H. J., (1999). Existence of optimal controls for variance control, InStochastic analysis, control, optimization and applications, Systems Control Found. Appl., 421–437. Birkhäuser Boston, Boston, MA.
Kyprianou,A. andLoeffen,R., (2009). Refracted Lévy processes,Annales de l’Institut Henri Poincaré, to appear.
Kyprianou A., Rivero, V. and Song, R., (2008). Convexity and smoothness of scale functions and de Finetti’s control problem, working paper.
Kyprianou, A. E. andPalmowski, Z., (2007). Distributional study of de Finetti’s dividend problem for a general Lévy insurance risk process,J. Appl. Probab.,44(2), 428–443.
Kyprianou, A. E. andSurya, B. A., (2007). Principles of smooth and continuous fit in the determination of endogenous bankruptcy levels,Finance Stoch.,11(1), 131–152.
Landriault, D., (2008). Randomized dividends in the compound binomial model with a general premium rate,Scand. Act. J.,2008(1), 1–15.
Leung, K. S., Kwok, Y. K. andLeung, S. Y., (2008). Finite-time dividend-ruin models,Insurance Math. Econom.,42(1), 154–162.
Li, S. andGarrido, J., (2004). On a class of renewal risk models with a constant dividend barrier,Insurance Math. Econom.,35(3), 691–701.
Lin, X. S. andPavlova, K. P., (2006). The compound Poisson risk model with a threshold dividend strategy,Insurance Math. Econom.,38(1), 57–80.
Lin, X. S. andSendova, K. P., (2008). The compound Poisson risk model with multiple thresholds,Insurance Math. Econom.,42(2), 617–627.
Lin, X. S., Willmot, G. E. andDrekic, S., (2003). The classical risk model with a constant dividend barrier: analysis of the Gerber-Shiu discounted penalty function,Insurance Math. Econom.,33(3), 551–566.
Loeffen, R., (2008). On optimality of the barrier strategy in de Finetti’s dividend problem for spectrally negative Lévy processes,Ann. Appl. Probab.,18(5), 1669–1680.
Loeffen, R., (2008). An optimal dividends problem with transaction costs for spectrally negative Lévy processes, to appear.
Loeffen, R., (2009). An optimal dividends problem with a terminal value for spectrally negative Lévy processes with a completel monotone jump density.J. Appl. Probab.,46(1), 85–98.
Lundberg, F., (1903).Approximerad framställning av sannolikhetsfunktionen. Återförsäkring av kollektivrisker, Akad. Afhandling. Almqvist o. Wiksell, Uppsala.
Merton, R. C., (1971). Optimum consumption and portfolio rules in a continuous-time model,J. Econom. Theory,3(14), 373–413.
Miller, M. H. andModigliani, F., (1961). Dividend policy, growth, and the valuation of shares,The Journal of Business,34(14), 411–433.
Mnif, M. andSulem, A., (2005). Optimal risk control and dividend policies under excess of loss reinsurance,Stochastics,77(5), 455–476.
Øksendal, B. andSulem, A., (2005).Applied stochastic control of jump diffusions, Springer, Berlin.
Paulsen, J., (2003). Optimal dividend payouts for diffusions with solvency constraints,Finance Stoch.,7(4), 457–473.
Paulsen, J., (2007). Optimal dividend payments until ruin of diffusion processes when payments are subject to both fixed and proportional costs,Adv. in Appl. Probab.,39(3), 669–689.
Paulsen, J., (2008). Optimal dividend payments and reinvestments of diffusion processes with both fixed and proportional costs,SIAM J. Control Optim.,47(5), 2201–2226.
Paulsen, J. andGjessing, H. K., (1997). Optimal choice of dividend barriers for a risk process with stochastic return on investments,Insurance Math. Econom.,20(3), 215–223.
Peskir, G. andShiryaev, A., (2006).Optimal stopping and free-boundary problems, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel.
Porteus, E. L., (1977). On optimal dividend, reinvestment, and liquidation policies for the firm,Operations Res.,25(5), 818–834.
Radner, R. andShepp, L., (1996). Risk vs. profit potential: a model for corporate strategy,J. Econom. Dynamics Control,20, 1373–1393.
Renaud, J.-F. andZhou, X., (2007). Distribution of the present value of dividend payments in a Lévy risk model,J. Appl. Probab.,44(2), 420–427.
Rogers, L. C. G. andWilliams, D., (1994).Diffusions, Markov processes, and martingales. Vol. 1, John Wiley & Sons Ltd., Chichester, second edition.
Rolski, T., Schmidli, H., Schmidt, V. andTeugels, J., (1999).Stochastic processes for insurance and finance, John Wiley & Sons Ltd., Chichester.
Schäl, M., (1998). On piecewise deterministic Markov control processes: control of jumps and of risk processes in insurance,Insurance Math. Econom.,22(1), 75–91.
Schmidli, H., (2001). Optimal proportional reinsurance policies in a dynamic setting,Scand. Actuar. J.,2001(1), 55–68.
Schmidli, H., (2002). On minimizing the ruin probability by investment and reinsurance,Ann. Appl. Probab.,12(3), 890–907.
Schmidli, H., (2004). Diffusion approximations, Teugels, J. L. and Sundt, B. (ed.),Encyclopedia of Actuarial Sciences, J. Wiley and Sons, Chichester,1, 519–522.
Schmidli, H., (2006). Optimisation in non-life insurance,Stoch. Models,22(4), 689–722.
Schmidli, H., (2008).Stochastic Control in Insurance, Springer, New York.
Shreve, S. E., Lehoczky, J. P. andGaver, D. P., (1984). Optimal consumption for general diffusions with absorbing and reflecting barriers,SIAM J. Control Optim.,22(1), 55–75.
Siegl, T. andTichy, R. F., (1996). Lösungsverfahren eines Risikomodells bei exponentiell fallender Schadensverteilung,Schweiz. Aktuarver. Mitt., (1), 95–118.
Soner, H. M., (1986). Optimal control with state-space constraint. II,SIAM J. Control Optim.,24(6), 1110–1122.
Sparre Andersen, E., (1957). On the collective theory of risk in the case of contagion between the claims,Transactions of the XVth Int. Congress of Actuaries, New York, (II), 219–229.
Taksar, M. I., (2000). Optimal risk and dividend distribution control models for an insurance company,Math. Methods Oper. Res.,51(1), 1–42.
Tan, J. andYang, X., (2006). The compound binomial model with randomized decisions on paying dividends,Insurance Math. Econom.,39(1), 1–18.
Thonhauser, S. andAlbrecher, H., (2007). Dividend maximization under consideration of the time value of ruin,Insurance Math. Econom.,41(1), 163–184.
Vath, V., Pham, H. andVilleneuve, S., (2008). A mixed singular/switching control problem for a dividend policy with reversible technology investment,Ann. Appl. Probab.,18(3), 1164–1200.
Wang, W. and Zhang, C., (2009). Optimal dividend strategies in the diffusion model with stochastic return on investments,J. of Syst. Science and Complexity, to appear.
Whittle, P., (1983).Optimization over time. Vol. II, John Wiley & Sons Ltd., Chichester.
Young, L. C., (1969).Lectures on the calculus of variations and optimal control theory, W. B. Saunders Co., Philadelphia.
Zhou, X., (2006). Classical risk model with a multi-layer premium rate, working paper,Concordia University.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Albrecher, H., Thonhauser, S. Optimality results for dividend problems in insurance. Rev. R. Acad. Cien. Serie A. Mat. 103, 295–320 (2009). https://doi.org/10.1007/BF03191909
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF03191909