×

General drawdown-based de Finetti optimization for spectrally negative Lévy risk processes. (English) Zbl 1396.91314

Summary: For spectrally negative Lévy risk processes, we consider a general version of de Finetti’s optimal dividend problem in which the ruin time is replaced with a general drawdown time from the running maximum in its value function. We identify a condition under which a barrier dividend strategy is optimal among all admissible strategies if the underlying process does not belong to a small class of compound Poisson processes with drift, for which the take-the-money-and-run dividend strategy is optimal. It generalizes the previous results on dividend optimization from ruin time based to drawdown time based. The associated drawdown functions are discussed in detail for examples of spectrally negative Lévy processes.

MSC:

91B30 Risk theory, insurance (MSC2010)
60G51 Processes with independent increments; Lévy processes
93E20 Optimal stochastic control
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Applebaum, D., Lévy Processes and Stochastic Calculus, (2009), Cambridge University Press · Zbl 1200.60001
[2] Avanzi, B.; Shen, J.; Wong, B., Optimal dividends and capital injections in the dual model with diffusion, ASTIN Bull., 41, 611-644, (2011) · Zbl 1242.91089
[3] Avanzi, B.; Pérez, J.-L; Wong, B.; Yamazaki, K., On optimal joint reflective and refractive dividend strategies in spectrally positive Lévy models, Insurance Math. Econom., 72, 148-162, (2017) · Zbl 1394.91185
[4] Avram, F.; Kyprianou, A. E.; Pistorius, M. R., Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options, Ann. Appl. Prob., 14, 215-238, (2004) · Zbl 1042.60023
[5] Avram, F.; Palmowski, Z.; Pistorius, M. R., On the optimal dividend problem for a spectrally negative Lévy process, Ann. Appl. Prob., 17, 156-180, (2007) · Zbl 1136.60032
[6] Avram, F.; Palmowski, Z.; Pistorius, M. R., On Gerber-Shiu functions and optimal dividend distribution for a Lévy risk process in the presence of a penalty function, Ann. Appl. Prob., 25, 1868-1935, (2015) · Zbl 1322.60055
[7] Avram, F.; Vu, N. L.; Zhou, X., On taxed spectrally negative Lévy processes with draw-down stopping, Insurance Math. Econom., 76, 69-74, (2017) · Zbl 1395.91245
[8] Azcue, P.; Muler, N., Optimal reinsurance and dividend distribution policies in the Cramér-Lundberg model, Math. Finance, 15, 261-308, (2005) · Zbl 1136.91016
[9] Azéma, J.; Yor, M., Séminaire de Probabilités, XIII, Une solution simple au problème de Skorokhod, 90-115, (1979), Springer: Springer, Berlin
[10] Bayraktar, E.; Kyprianou, A. E.; Yamazaki, K., On optimal dividends in the dual model, ASTIN Bull., 43, 359-372, (2013) · Zbl 1283.91192
[11] Bayraktar, E.; Kyprianou, A.; Yamazaki, K., Optimal dividends in the dual model under transaction costs, Insurance Math. Econom., 54, 133-143, (2014) · Zbl 1294.91071
[12] Bertoin, J., Lévy processes, (1996), Cambridge University Press
[13] Boguslavskaya, E. V., Optimization problems in financial mathematics: explicit solutions for diffusion models, (2006)
[14] Carr, P., First-order calculus and option pricing, J. Financial Eng., 1, (2014)
[15] De Finetti, B. D., Trans. 15th International Congress of Actuaries, Su un’impostazion alternativa dell teoria collecttiva del rischio, 433-443, (1957)
[16] Gerber, H. U., Entscheidungskriterien für den zusammengesetzten Poisson-prozess, Mitteilungen Vereinigung Schweiz. Versicherungsmath., 69, 185-227, (1969) · Zbl 0193.20501
[17] Højgaard, B.; Taksar, M., Controlling risk exposure and dividends payout schemes: insurance company example, Math. Finance, 9, 153-182, (1999) · Zbl 0999.91052
[18] Ikeda, N.; Watanabe, S., Stochastic Differential Equations and Diffusion Processes, (1981), North Holland: North Holland, Amsterdam · Zbl 0495.60005
[19] Jacod, J.; Shiryaev, A. N., Limit Theorems for Stochastic Processes, (2003), Springer: Springer, Berlin · Zbl 1018.60002
[20] Kuznetsov, A.; Kyprianou, A. E.; Rivero, V., Lévy Matters II, The theory of scale functions for spectrally negative Lévy processes, 97-186, (2012), Springer: Springer, Heidelberg · Zbl 1261.60047
[21] Kyprianou, A. E., Introductory Lectures on Fluctuations of Lévy Processes with Applications, (2006), Springer: Springer, Berlin · Zbl 1104.60001
[22] Kyprianou, A. E.; Palmowski, Z., Distributional study of de Finetti’s dividend problem for a general Lévy insurance risk process, J. Appl. Prob., 44, 428-443, (2007) · Zbl 1137.60047
[23] Kyprianou, A. E.; Loeffen, R.; Pérez, J.-L., Optimal control with absolutely continuous strategies for spectrally negative Lévy processes, J. Appl. Prob., 49, 150-166, (2012) · Zbl 1253.93001
[24] Lehoczky, J. P., Formulas for stopped diffusion processes with stopping times based on the maximum, Ann. Prob., 5, 601-607, (1977) · Zbl 0367.60093
[25] Li, B.; Vu, N. L.; Zhou, X., Exit problems for general draw-down times of spectrally negative Lévy processes, (2017)
[26] Loeffen, R. L., On optimality of the barrier strategy in de Finetti’s dividend problem for spectrally negative Lévy processes, Ann. Appl. Prob., 18, 1669-1680, (2008) · Zbl 1152.60344
[27] Loeffen, R. L., An optimal dividends problem with a terminal value for spectrally negative Lévy processes with a completely monotone jump density, J. Appl. Prob., 46, 85-98, (2009) · Zbl 1166.60051
[28] Loeffen, R. L., An optimal dividends problem with transaction costs for spectrally negative Lévy processes, Insurance Math. Econom., 45, 41-48, (2009) · Zbl 1231.91211
[29] Loeffen, R. L.; Renaud, J.-F., De Finetti’s optimal dividends problem with an affine penalty function at ruin, Insurance Math. Econom., 46, 98-108, (2010) · Zbl 1231.91212
[30] Pistorius, M. R., Séminaire de Probabilités XL, An excursion-theoretical approach to some boundary crossing problems and the Skorokhod embedding for reflected Lévy processes, 287-307, (2007), Springer: Springer, Berlin · Zbl 1126.60039
[31] Renaud, J.-F.; Zhou, X., Distribution of the present value of dividend payments in a Lévy risk model, J. Appl. Prob., 44, 420-427, (2007) · Zbl 1132.60041
[32] Shepp, L.; Shiryaev, A. N., The Russian option: reduced regret, Ann. Appl. Prob., 3, 631-640, (1993) · Zbl 0783.90011
[33] Shreve, S. E.; Lehoczky, J. P.; Gaver, D. P., Optimal consumption for general diffusions with absorbing and reflecting barriers, SIAM J. Control Optimization, 22, 55-75, (1984) · Zbl 0535.93071
[34] Taylor, H. M., A stopped Brownian motion formula, Ann. Prob., 3, 234-246, (1975) · Zbl 0303.60072
[35] Thonhauser, S.; Albrecher, H., Dividend maximization under consideration of the time value of ruin, Insurance Math. Econom., 41, 163-184, (2007) · Zbl 1119.91047
[36] Yao, D.; Yang, H.; Wang, R., Optimal dividend and capital injection problem in the dual model with proportional and fixed transaction costs, Europ. J. Operat. Res., 211, 568-576, (2011) · Zbl 1237.91143
[37] Yin, C.; Wen, Y., Optimal dividend problem with a terminal value for spectrally positive Lévy processes, Insurance Math. Econom., 53, 769-773, (2013) · Zbl 1290.91176
[38] Zhao, Y.; Chen, P.; Yang, H., Optimal periodic dividend and capital injection problem for spectrally positive Lévy processes, Insurance Math. Econom., 74, 135-146, (2017) · Zbl 1394.91243
[39] Zhao, Y.; Wang, R.; Yao, D.; Chen, P., Optimal dividends and capital injections in the dual model with a random time horizon, J. Optimization Theory Appl., 167, 272-295, (2015) · Zbl 1341.49021
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.