Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-23T20:30:11.395Z Has data issue: false hasContentIssue false
  • English
  • Français

Moment-constrained optimal dividends: precommitment and consistent planning

Part of: Stochastic systems and control Markov processes Mathematical finance Game theory

Published online by Cambridge University Press:  06 June 2022

Sören Christensen  and
Kristoffer Lindensjö Open the ORCID record for Kristoffer Lindensjö [Opens in a new window]
Sören Christensen*
Affiliation:
Kiel University
Kristoffer Lindensjö*
Affiliation:
Stockholm University
*
*Postal address: Department of Mathematics, Christian-Albrechts-University Kiel, Heinrich-Hecht-Platz 6, 24118 Kiel, Germany. Email address: christensen@math.uni-kiel.de
**Postal address: Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden. Email address: kristoffer.lindensjo@math.su.se
Get access Rights & Permissions [Opens in a new window]

Abstract

A moment constraint that limits the number of dividends in an optimal dividend problem is suggested. This leads to a new type of time-inconsistent stochastic impulse control problem. First, the optimal solution in the precommitment sense is derived. Second, the problem is formulated as an intrapersonal sequential dynamic game in line with Strotz’s consistent planning. In particular, the notions of pure dividend strategies and a (strong) subgame-perfect Nash equilibrium are adapted. An equilibrium is derived using a smooth fit condition. The equilibrium is shown to be strong. The uncontrolled state process is a fairly general diffusion.

Keywords

Constrained stochastic controlstochastic impulse controlStrotz’s consistent planningsubgame-perfect Nash equilibriumtime-inconsistency

MSC classification

Primary: 93E20: Optimal stochastic control
Secondary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 91A15: Stochastic games 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)
Type
Original Article
Information
Advances in Applied Probability , Volume 54 , Issue 2 , June 2022 , pp. 404 - 432
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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

Albrecher, H. and Thonhauser, S. (2009). Optimality results for dividend problems in insurance. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A [Mat.] 103, 295–320.CrossRefGoogle Scholar
Albrecher, H. and Thonhauser, S. (2012). On optimal dividend strategies in insurance with a random time horizon. In Stochastic Processes, Finance and Control: A Festschrift in Honor of Robert J. Elliott, World Scientific, Singapore, pp. 157–179.CrossRefGoogle Scholar
Alvarez, L. H. and Lempa, J. (2008). On the optimal stochastic impulse control of linear diffusions. SIAM J. Control Optimization 47, 703–732.CrossRefGoogle Scholar
Ankirchner, S., Klein, M. and Kruse, T. (2019). A verification theorem for optimal stopping problems with expectation constraints. Appl. Math. Optimization 79, 145–177.CrossRefGoogle Scholar
Avanzi, B. (2009). Strategies for dividend distribution: a review. N. Amer. Actuarial J. 13, 217–251.CrossRefGoogle Scholar
Bai, L., Hunting, M. and Paulsen, J. (2012). Optimal dividend policies for a class of growth-restricted diffusion processes under transaction costs and solvency constraints. Finance Stoch. 16, 477–511.CrossRefGoogle Scholar
Bai, L. and Paulsen, J. (2010). Optimal dividend policies with transaction costs for a class of diffusion processes. SIAM J. Control Optimization 48, 4987–5008.CrossRefGoogle Scholar
Bai, L. and Paulsen, J. (2012). On non-trivial barrier solutions of the dividend problem for a diffusion under constant and proportional transaction costs. Stoch. Process. Appl. 122, 4005–4027.CrossRefGoogle Scholar
Bayraktar, E. and Miller, C. W. (2019). Distribution-constrained optimal stopping. Math. Finance 29, 368–406.CrossRefGoogle Scholar
Bayraktar, E. and Yao, S. (2017). Dynamic programming principles for optimal stopping with expectation constraint. Preprint. Available at https://arxiv.org/abs/1708.02192.Google Scholar
Beiglböck, M., Eder, M., Elgert, C. and Schmock, U. (2018). Geometry of distribution-constrained optimal stopping problems. Prob. Theory Relat. Fields 172, 71–101.CrossRefGoogle ScholarPubMed
Bensoussan, A., Wong, K. C. and Yam, S. C. P. (2018). A paradox in time-consistency in the mean–variance problem? Finance Stoch. 23, 173–207.CrossRefGoogle Scholar
Björk, T., Khapko, M., and Murgoci, A. (2017). On time-inconsistent stochastic control in continuous time. Finance Stoch. 21, 331–360.CrossRefGoogle Scholar
Björk, T. and Murgoci, A. (2014). A theory of Markovian time-inconsistent stochastic control in discrete time. Finance Stoch. 18, 545–592.CrossRefGoogle Scholar
Borodin, A. N. and Salminen, P. (2012). Handbook of Brownian Motion—Facts and Formulae. Birkhäuser, Basel.Google Scholar
Chen, S., Li, Z. and Zeng, Y. (2014). Optimal dividend strategies with time-inconsistent preferences. J. Econom. Dynamics Control 46, 150–172.CrossRefGoogle Scholar
Chen, S., Li, Z. and Zeng, Y. (2018). Optimal dividend strategy for a general diffusion process with time-inconsistent preferences and ruin penalty. SIAM J. Financial Math. 9, 274–314.CrossRefGoogle Scholar
Chen, S., Wang, X., Deng, Y. and Zeng, Y. (2016). Optimal dividend-financing strategies in a dual risk model with time-inconsistent preferences. Insurance Math. Econom. 67, 27–37.CrossRefGoogle Scholar
Chen, S., Zeng, Y. and Hao, Z. (2017). Optimal dividend strategies with time-inconsistent preferences and transaction costs in the Cramér–Lundberg model. Insurance Math. Econom. 74, 31–45.CrossRefGoogle Scholar
Chow, Y.-L., Yu, X. and Zhou, C. (2020). On dynamic programming principle for stochastic control under expectation constraints. J. Optimization Theory Appl. 185, 803–818.CrossRefGoogle Scholar
Christensen, S., Irle, A. and Ludwig, A. (2017). Optimal portfolio selection under vanishing fixed transaction costs. Adv. Appl. Prob. 49, 1116–1143.CrossRefGoogle Scholar
Christensen, S. and Lindensjö, K. (2018). On finding equilibrium stopping times for time-inconsistent Markovian problems. SIAM J. Control Optimization 56, 4228–4255.CrossRefGoogle Scholar
Christensen, S. and Lindensjö, K. (2020). On time-inconsistent stopping problems and mixed strategy stopping times. Stoch. Process. Appl. 130, 2886–2917.CrossRefGoogle Scholar
Christensen, S. and Lindensjö, K. (2020). Time-inconsistent stopping, myopic adjustment & equilibrium stability: with a mean–variance application. In Stochastic Modeling and Control (Banach Center Publications 122), eds Jakubowski, J., Nieweglowski, M., Rásonyi, M. and Stettner, L., Institute of Mathematics, Polish Academy of Sciences, Warsaw, pp. 53–76.Google Scholar
Christensen, S. and Wittlinger, M. (2012). Optimal relaxed portfolio strategies for growth rate maximization problems with transaction costs. Working paper. Available at https://arxiv.org/abs/1209.0305.Google Scholar
De Angelis, T. and Ekström, E. (2017). The dividend problem with a finite horizon. Ann. Appl. Prob. 27, 3525–3546.CrossRefGoogle Scholar
Dickson, D. C. M. and Drekic, S. (2006). Optimal dividends under a ruin probability constraint. Ann. Actuarial Sci. 1, 291–306.CrossRefGoogle Scholar
Ferrari, G. and Schuhmann, P. (2019). An optimal dividend problem with capital injections over a finite horizon. SIAM J. Control Optimization 57, 2686–2719.CrossRefGoogle Scholar
Hernandez, C., Junca, M. and Moreno-Franco, H. (2018). A time of ruin constrained optimal dividend problem for spectrally one-sided Lévy processes. Insurance Math. Econom. 79, 57–68.CrossRefGoogle Scholar
Hipp, C. (2003). Optimal dividend payment under a ruin constraint: discrete time and state space. Blätter DGVFM 26, 255–264.CrossRefGoogle Scholar
Huang, Y.-J. and Nguyen-Huu, A. (2018). Time-consistent stopping under decreasing impatience. Finance Stoch. 22, 69–95.CrossRefGoogle Scholar
Huang, Y.-J. and Zhou, Z. (2020). Strong and weak equilibria for time-inconsistent stochastic control in continuous time. Math. Operat. Res. 46, 428–451.CrossRefGoogle Scholar
Jeanblanc-Picqué, M. and Shiryaev, A. N. (1995). Optimization of the flow of dividends. Uspekhi Mat. Nauk 50, 25–46.Google Scholar
Li, Y., Li, Z. and Zeng, Y. (2016). Equilibrium dividend strategy with non-exponential discounting in a dual model. J. Optimization Theory Appl. 168, 699–722.CrossRefGoogle Scholar
Lindensjö, K. (2019). A regular equilibrium solves the extended HJB system. Operat. Res. Lett. 47, 427–432.CrossRefGoogle Scholar
Lindensjö, K. and Lindskog, F. (2020). Optimal dividends and capital injection under dividend restrictions. Math. Meth. Operat. Res. 92, 461–487.CrossRefGoogle Scholar
Nutz, M. and Zhang, Y. (2020). Conditional optimal stopping: a time-inconsistent optimization. Ann. Appl. Prob. 30, 1669–1692.CrossRefGoogle Scholar
Øksendal, B. (1999). Stochastic control problems where small intervention costs have big effects. Appl. Math. Optimization 40, 355–375.Google Scholar
Øksendal, B., Ubøe, J. and Zhang, T. (2002). Non-robustness of some impulse control problems with respect to intervention costs. Stoch. Anal. Appl. 20, 999–1026.CrossRefGoogle Scholar
Paulsen, J. (2003). Optimal dividend payouts for diffusions with solvency constraints. Finance Stoch. 7, 457–473.CrossRefGoogle Scholar
Paulsen, J. (2007). Optimal dividend payments until ruin of diffusion processes when payments are subject to both fixed and proportional costs. Adv. Appl. Prob. 39, 669–689.CrossRefGoogle Scholar
Pedersen, J. and Peskir, G. (2018). Constrained dynamic optimality and binomial terminal wealth. SIAM J. Control Optimization 56, 1342–1357.CrossRefGoogle Scholar
Pedersen, J. and Peskir, G. (2016). Optimal mean–variance selling strategies. Math. Financial Econom. 10, 203–220.CrossRefGoogle Scholar
Pedersen, J. L. and Peskir, G. (2017). Optimal mean–variance portfolio selection. Math. Financial Econom. 11, 137–160.CrossRefGoogle Scholar
Peskir, G. and Shiryaev, A. (2006). Optimal Stopping and Free-Boundary Problems. Birkhäuser, Basel.Google Scholar
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, 55–75.CrossRefGoogle Scholar
Strotz, R. (1955). Myopia and inconsistency in dynamic utility maximization. Rev. Econom. Stud. 23, 165–180.CrossRefGoogle Scholar
Zhao, Q. Wei, J. and Wang, R. (2014). On dividend strategies with non-exponential discounting. Insurance Math. Econom. 58, 1–13.CrossRefGoogle Scholar
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, 272–295.CrossRefGoogle Scholar
Zhao, Y.-X. and Yao, D.-J. (2015). Optimal dividend and capital injection problem with a random time horizon and a ruin penalty in the dual model. Appl. Math. J. Chin. Univ. 30, 325–339.CrossRefGoogle Scholar
Zhu, J., Siu, T. K. and Yang, H. (2019). Singular dividend optimization for a linear diffusion model with time-inconsistent preferences. Europ. J. Operat. Res. 285, 66–80.CrossRefGoogle Scholar