Hostname: page-component-745bb68f8f-b95js Total loading time: 0 Render date: 2025-01-10T22:02:31.800Z Has data issue: false hasContentIssue false
  • English
  • Français

On stochastic control under poissonian intervention: optimality of a barrier strategy in a general Lévy model

Part of: Stochastic processes Stochastic systems and control Operations research and management science

Published online by Cambridge University Press:  09 January 2025

Kei Noba Open the ORCID record for Kei Noba [Opens in a new window]  and
Kazutoshi Yamazaki Open the ORCID record for Kazutoshi Yamazaki [Opens in a new window]
Kei Noba*
Affiliation:
The Institute of Statistical Mathematics
Kazutoshi Yamazaki*
Affiliation:
The University of Queensland
*
*Postal address: Department of Fundamental Statistical Mathematics, The Institute of Statistical Mathematics, 10-3 Midori-cho, Tachikawa-shi, Tokyo 190-8562, Japan. Email address: knoba@ism.ac.jp
**Postal address: School of Mathematics and Physics, The University of Queensland, St Lucia, Brisbane, QLD 4072, Australia. Email address: k.yamazaki@uq.edu.au
Get access Rights & Permissions [Opens in a new window]

Abstract

We study a version of the stochastic control problem of minimizing the sum of running and controlling costs, where control opportunities are restricted to independent Poisson arrival times. Under a general setting driven by a general Lévy process, we show the optimality of a periodic barrier strategy, which moves the process upward to the barrier whenever it is observed to be below it. The convergence of the optimal solutions to those in the continuous-observation case is also shown.

Keywords

Stochastic controlinventory modelsperiodic observationsmathematical financeLévy processes

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes
Secondary: 93E20: Optimal stochastic control 90B05: Inventory, storage, reservoirs
Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 24
Check for updates
Copyright
© The Author(s), 2025. 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., Bäuerle, N. and Thonhauser, S. (2011). Optimal dividend-payout in random discrete time. Statist. Risk Model. 28, 251–276.CrossRefGoogle Scholar
Albrecher, H., Ivanovs, J. and Zhou, X. (2016). Exit identities for Lévy processes observed at Poisson arrival times. Bernoulli 22, 1364–1382.CrossRefGoogle Scholar
Avanzi, B., Cheung, E. C., Wong, B. and Woo, J. K. (2013). On a periodic dividend barrier strategy in the dual model with continuous monitoring of solvency. Insurance Math. Econom. 52, 98–113.CrossRefGoogle Scholar
Baurdoux, E. J. and Yamazaki, K. (2015). Optimality of doubly reflected Lévy processes in singular control. Stoch. Process. Appl. 125, 2727–2751.CrossRefGoogle Scholar
Benkherouf, L. and Bensoussan, A. (2009). Optimality of an (s, S) policy with compound Poisson and diffusion demands: a quasi-variational inequalities approach. SIAM J. Control Optimization 48, 756–762.CrossRefGoogle Scholar
Bensoussan, A. and Tapiero, C. S. (1982). Impulsive control in management: prospects and applications. J. Optimization Theory Appl. 37, 419–442.CrossRefGoogle Scholar
Bensoussan, A., Liu, R. H. and Sethi, S. P. (2005). Optimality of an (s, S) policy with compound Poisson and diffusion demands: a quasi-variational inequalities approach. SIAM J. Control Optimization 44, 1650–1676.CrossRefGoogle Scholar
Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
Biffis, E. and Kyprianou, A. E. (2010). A note on scale functions and the time value of ruin for Lévy insurance risk processes. Insurance Math. Econom. 46, 85–91.CrossRefGoogle Scholar
Cai, N. and Yang, X. (2018). International reserve management: a drift-switching reflected jump-diffusion model. Math. Finance 28, 409–446.CrossRefGoogle Scholar
Carr, P. (1998). Randomization and the American put. Rev. Financial Studies 11, 597–626.CrossRefGoogle Scholar
Carr, P., Geman, H., Madan, D. and Yor, M. (2002). The fine structure of asset returns: an empirical investigation. J. Business 75, 305–333.CrossRefGoogle Scholar
Cheung, E. C., Liu, H. and Willmot, G. E. (2018). Joint moments of the total discounted gains and losses in the renewal risk model with two-sided jumps. Appl. Math. Comput. 331, 358–377.CrossRefGoogle Scholar
Dong, H., Yin, C. and Dai, H. (2019). Spectrally negative Lévy risk model under Erlangized barrier strategy. J. Comput. Appl. Math. 351, 101–116.CrossRefGoogle Scholar
Dupuis, P. and Wang, H. (2002). Optimal stopping with random intervention times. Adv. Appl. Prob. 34, 141–157.CrossRefGoogle Scholar
Egami, M. and Yamazaki, K. (2014). Phase-type fitting of scale functions for spectrally negative Lévy processes. J. Comput. Appl. Math. 264, 1–22.CrossRefGoogle Scholar
Hernández-Hernández, D., Pérez, J. L. and Yamazaki, K. (2016). Optimality of refraction strategies for spectrally negative Lévy processes. SIAM J. Control Optimization 54, 1126–1156.CrossRefGoogle Scholar
Jeanblanc-Picqué, M. (1993). Impulse control method and exchange rate. Math. Finance 3, 161–177.CrossRefGoogle Scholar
Kyprianou, A. E. (2014). Fluctuations of Lévy Processes with Applications, second edition. Springer.CrossRefGoogle Scholar
Kyprianou, A. E. and Loeffen, R. L. (2010). Refracted Lévy processes. Ann. Inst. H. Poincaré Prob. Statist. 46, 24–44.CrossRefGoogle Scholar
Kyprianou, A. E. and Pistorius, M. R. (2003). Perpetual options and Canadization through fluctuation theory. Ann. Appl. Prob. 13, 1077–1098.CrossRefGoogle Scholar
Leung, T., Yamazaki, K. and Zhang, H. (2015). An analytic recursive method for optimal multiple stopping: Canadization and phase-type fitting. Int. J. Theor. Appl. Finance 18, 1550032.CrossRefGoogle Scholar
Lkabous, M. A. (2021). Poissonian occupation times of spectrally negative Lévy processes with applications. Scand. Actuarial J. 2021, 916–935.CrossRefGoogle Scholar
Mundaca, G. and Øksendal, B. (1998). Optimal stochastic intervention control with application to the exchange rate. J. Math. Econ. 2, 225–243.CrossRefGoogle Scholar
Noba, K. and Yamazaki, K. (2023). On singular control for Lévy processes. Math. Operat. Res. 48, 1213–1234.CrossRefGoogle Scholar
Noba, K., Pérez, J. L. and Yamazaki, K. (2023). Refraction strategies in stochastic control: optimality for a general Lévy process model. To appear in SIAM J. Control Optimization. Available at arXiv:2308.08183.Google Scholar
Palmowski, Z., Pérez, J. L. Surya, B. and Yamazaki, K. (2020). The Leland–Toft optimal capital structure model under Poisson observations. Finance Stoch. 24, 1035–1082.CrossRefGoogle Scholar
Palmowski, Z., Pérez, J. L. and Yamazaki, K. (2021). Double continuation regions for American options under Poisson exercise opportunities. Math. Finance 31, 722–771.CrossRefGoogle Scholar
Pérez, J. L. and Yamazaki, K. (2020). Optimality of hybrid continuous and periodic barrier strategies in the dual model. Appl. Math. Optimization 82, 105–133.CrossRefGoogle Scholar
Pérez, J. L., Yamazaki, K. and Bensoussan, A. (2020). Optimal periodic replenishment policies for spectrally positive Lévy demand processes. SIAM J. Control Optimization 58, 3428–3456.CrossRefGoogle Scholar
Pérez, J. L., Yamazaki, K. and Yu, X. (2018). On the bail-out optimal dividend problem. J. Optimization Theory Appl. 179, 553–568.CrossRefGoogle Scholar
Saarinen, H. (2023). On Poisson constrained control of linear diffusions. Doctoral Thesis, University of Turku, Finland.Google Scholar
Wang, H. (2001). Some control problems with random intervention times. Adv. Appl. Prob. 33, 404–422.CrossRefGoogle Scholar
Wang, W., Wang, N. and Chen, M. (2023). On a doubly reflected risk process with running maximum dependent reflecting barriers. J. Comput. Appl. Math. 422, 114880.CrossRefGoogle Scholar
Wang, Z., Lkabous, M. A. and Landriault, D. (2023). A refracted Lévy process with delayed dividend pullbacks. Scand. Actuarial J. 2023, 885–906.CrossRefGoogle Scholar
Yamazaki, K. (2015). Contraction options and optimal multiple-stopping in spectrally negative Lévy models. Appl. Math. Optimization 72, 147–185.CrossRefGoogle Scholar
Yamazaki, K. (2017). Inventory control for spectrally positive Lévy demand processes. Math. Operat. Res. 42, 212–237.CrossRefGoogle Scholar
Yang, C., Sendova, K. P. and Li, Z. (2020). Parisian ruin with a threshold dividend strategy under the dual Lévy risk model. Insurance Math. Econom. 90, 135–150.CrossRefGoogle Scholar
Zhang, Z. and Han, X. (2017). The compound Poisson risk model under a mixed dividend strategy. Appl. Math. Comput. 315, 1–12.Google Scholar
Zhao, Y., Chen, P. and Yang, H. (2017). Optimal periodic dividend and capital injection problem for spectrally positive Lévy processes. Insurance Math. Econom. 74, 135–146.CrossRefGoogle Scholar
Zhao, Y., Wang, R., Yao, D., 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