Hostname: page-component-848d4c4894-wg55d Total loading time: 0 Render date: 2024-05-22T13:43:48.776Z Has data issue: false hasContentIssue false

General drawdown-based de Finetti optimization for spectrally negative Lévy risk processes

Published online by Cambridge University Press:  26 July 2018

Wenyuan Wang*
Affiliation:
Xiamen University
Xiaowen Zhou*
Affiliation:
Concordia University
*
* Postal address: School of Mathematical Sciences, Xiamen University, Fujian, 361005, China. Email address: wwywang@xmu.edu.cn
** Postal address: Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. W., Montreal, H3G 1M8, Canada. Email address: xzhou@mathstat.concordia.ca

Abstract

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.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Applebaum, D. (2009). Lévy Processes and Stochastic Calculus, 2nd edn. Cambridge University Press. Google Scholar
[2]Avanzi, B., Shen, J. and Wong, B. (2011). Optimal dividends and capital injections in the dual model with diffusion. ASTIN Bull. 41, 611644. Google Scholar
[3]Avanzi, B., Pérez, J.-L, Wong, B. and Yamazaki, K. (2017). On optimal joint reflective and refractive dividend strategies in spectrally positive Lévy models. Insurance Math. Econom. 72, 148162. Google Scholar
[4]Avram, F., Kyprianou, A. E. and Pistorius, M. R. (2004). Exit problems for spectrally negative Lévy processes and applications to (Canadized) Russian options. Ann. Appl. Prob. 14, 215238. 10.1214/aoap/1075828052Google Scholar
[5]Avram, F., Palmowski, Z. and Pistorius, M. R. (2007). On the optimal dividend problem for a spectrally negative Lévy process. Ann. Appl. Prob. 17, 156180. Google Scholar
[6]Avram, F., Palmowski, Z. and Pistorius, M. R. (2015). On Gerber–Shiu functions and optimal dividend distribution for a Lévy risk process in the presence of a penalty function. Ann. Appl. Prob. 25, 18681935. Google Scholar
[7]Avram, F., Vu, N. L. and Zhou, X. (2017). On taxed spectrally negative Lévy processes with draw-down stopping. Insurance Math. Econom. 76, 6974. Google Scholar
[8]Azcue, P. and Muler, N. (2005). Optimal reinsurance and dividend distribution policies in the Cramér–Lundberg model. Math. Finance 15, 261308. Google Scholar
[9]Azéma, J. and Yor, M. (1979). Une solution simple au problème de Skorokhod. In Séminaire de Probabilités, XIII, Springer, Berlin, pp. 90115. 10.1007/BFb0070852Google Scholar
[10]Bayraktar, E., Kyprianou, A. E. and Yamazaki, K. (2013). On optimal dividends in the dual model. ASTIN Bull. 43, 359372. Google Scholar
[11]Bayraktar, E., Kyprianou, A. and Yamazaki, K. (2014). Optimal dividends in the dual model under transaction costs. Insurance Math. Econom. 54, 133143. Google Scholar
[12]Bertoin, J. (1996). Lévy processes. Cambridge University Press. Google Scholar
[13]Boguslavskaya, E. V. (2006). Optimization problems in financial mathematics: explicit solutions for diffusion models. Doctoral thesis. University of Amsterdam. Google Scholar
[14]Carr, P. (2014). First-order calculus and option pricing. J. Financial Eng. 1, 1450009. Google Scholar
[15]De Finetti, B. D. (1957). Su un'impostazion alternativa dell teoria collecttiva del rischio. In Trans. 15th International Congress of Actuaries, Vol. 2, pp. 433443. Google Scholar
[16]Gerber, H. U. (1969). Entscheidungskriterien für den zusammengesetzten Poisson-prozess. Mitteilungen Vereinigung Schweiz. Versicherungsmath. 69, 185227. Google Scholar
[17]Højgaard, B. and Taksar, M. (1999). Controlling risk exposure and dividends payout schemes: insurance company example. Math. Finance 9, 153182. Google Scholar
[18]Ikeda, N. and Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes. North Holland, Amsterdam. Google Scholar
[19]Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin. Google Scholar
[20]Kuznetsov, A., Kyprianou, A. E. and Rivero, V. (2012). The theory of scale functions for spectrally negative Lévy processes. In Lévy Matters II (Lecture Notes Math. 2061), Springer, Heidelberg, pp. 97186. 10.1007/978-3-642-31407-0_2Google Scholar
[21]Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin. Google Scholar
[22]Kyprianou, A. E. and Palmowski, Z. (2007). Distributional study of de Finetti's dividend problem for a general Lévy insurance risk process. J. Appl. Prob. 44, 428443. Google Scholar
[23]Kyprianou, A. E., Loeffen, R. and Pérez, J.-L. (2012). Optimal control with absolutely continuous strategies for spectrally negative Lévy processes. J. Appl. Prob. 49, 150166. Google Scholar
[24]Lehoczky, J. P. (1977). Formulas for stopped diffusion processes with stopping times based on the maximum. Ann. Prob. 5, 601607. Google Scholar
[25]Li, B., Vu, N. L. and Zhou, X. (2017). Exit problems for general draw-down times of spectrally negative Lévy processes. Submitted. Available at https://arxiv.org/abs/1702.07259. Google Scholar
[26]Loeffen, R. L. (2008). On optimality of the barrier strategy in de Finetti's dividend problem for spectrally negative Lévy processes. Ann. Appl. Prob. 18, 16691680. 10.1214/07-AAP504Google Scholar
[27]Loeffen, R. L. (2009). An optimal dividends problem with a terminal value for spectrally negative Lévy processes with a completely monotone jump density. J. Appl. Prob. 46, 8598. Google Scholar
[28]Loeffen, R. L. (2009). An optimal dividends problem with transaction costs for spectrally negative Lévy processes. Insurance Math. Econom. 45, 4148. Google Scholar
[29]Loeffen, R. L. and Renaud, J.-F. (2010). De Finetti's optimal dividends problem with an affine penalty function at ruin. Insurance Math. Econom. 46, 98108. Google Scholar
[30]Pistorius, M. R. (2007). An excursion-theoretical approach to some boundary crossing problems and the Skorokhod embedding for reflected Lévy processes. In Séminaire de Probabilités XL, Springer, Berlin, pp. 287307. Google Scholar
[31]Renaud, J.-F. and Zhou, X. (2007). Distribution of the present value of dividend payments in a Lévy risk model. J. Appl. Prob. 44, 420427. Google Scholar
[32]Shepp, L. and Shiryaev, A. N. (1993). The Russian option: reduced regret. Ann. Appl. Prob. 3, 631640. Google Scholar
[33]Shreve, S. E., Lehoczky, J. P. and Gaver, D. P. (1984). Optimal consumption for general diffusions with absorbing and reflecting barriers. SIAM J. Control Optimization 22, 5575. Google Scholar
[34]Taylor, H. M. (1975). A stopped Brownian motion formula. Ann. Prob. 3, 234246. Google Scholar
[35]Thonhauser, S. and Albrecher, H. (2007). Dividend maximization under consideration of the time value of ruin. Insurance Math. Econom. 41, 163184. Google Scholar
[36]Yao, D., Yang, H. and Wang, R. (2011). Optimal dividend and capital injection problem in the dual model with proportional and fixed transaction costs. Europ. J. Operat. Res. 211, 568576. Google Scholar
[37]Yin, C. and Wen, Y. (2013). Optimal dividend problem with a terminal value for spectrally positive Lévy processes. Insurance Math. Econom. 53, 769773. Google Scholar
[38]Zhao, Y., Chen, P. and Yang, H. (2017). Optimal periodic dividend and capital injection problem for spectrally positive Lévy processes. Insurance Math. Econom. 74, 135146. Google Scholar
[39]Zhao, Y., Wang, R., Yao, D. and Chen, P. (2015). Optimal dividends and capital injections in the dual model with a random time horizon. J. Optimization Theory Appl. 167, 272295. Google Scholar