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De Finetti’s control problem with a concave bound on the control rate

Part of: Stochastic systems and control Markov processes

Published online by Cambridge University Press:  25 January 2024

Félix Locas  and
Jean-François Renaud
Félix Locas*
Affiliation:
Université du Québec à Montréal
Jean-François Renaud*
Affiliation:
Université du Québec à Montréal
*
*Postal address: Département de mathématiques, Université du Québec à Montréal (UQAM), 201 av. Président-Kennedy, Montréal (Québec) H2X 3Y7, Canada.
*Postal address: Département de mathématiques, Université du Québec à Montréal (UQAM), 201 av. Président-Kennedy, Montréal (Québec) H2X 3Y7, Canada.
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Abstract

We consider De Finetti’s control problem for absolutely continuous strategies with control rates bounded by a concave function and prove that a generalized mean-reverting strategy is optimal in a Brownian model. In order to solve this problem, we need to deal with a nonlinear Ornstein–Uhlenbeck process. Despite the level of generality of the bound imposed on the rate, an explicit expression for the value function is obtained up to the evaluation of two functions. This optimal control problem has, as special cases, those solved in Jeanblanc-Picqué and Shiryaev (1995) and Renaud and Simard (2021) when the control rate is bounded by a constant and a linear function, respectively.

Keywords

Stochastic controlabsolutely continuous strategiesdividend paymentsdiffusion modelBrownian motionnonlinear Ornstein–Uhlenbeck process

MSC classification

Primary: 60J99: None of the above, but in this section
Secondary: 60J65: Brownian motion 93E20: Optimal stochastic control
Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 17
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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References

Albrecher, H., Azcue, P. and Muler, N. (2022). Optimal ratcheting of dividends in a Brownian risk model. SIAM J. Fin. Math. 13, 657–701.Google Scholar
Albrecher, H. and Thonhauser, S. (2009). Optimality results for dividend problems in insurance. Rev. R. Acad. Cienc. Exactas Fs. Nat. Ser. A Mat. RACSAM 103, 295–320.Google Scholar
Alvarez, L. H. R. and Shepp, L. A. (1998). Optimal harvesting of stochastically fluctuating populations. J. Math. Biol. 37, 155–177.Google Scholar
Angoshtari, B., Bayraktar, E. and Young, V. R. (2019). Optimal dividend distribution under drawdown and ratcheting constraints on dividend rates. SIAM J. Fin. Math. 2, 547–577.Google Scholar
Asmussen, S. and Taksar, M. (1997). Controlled diffusion models for optimal dividend pay-out. Insurance Math. Econom. 20, 1–15.Google Scholar
Avanzi, B. and Wong, B. (2012). On a mean reverting dividend strategy with Brownian motion. Insurance Math. Econom. 51, 229–238.Google Scholar
Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian motion—Facts and Formulae, 2nd edn. Birkhäuser, Basel.Google Scholar
Choulli, T., Taksar, M. and Zhou, X. Y. (2004). Interplay between dividend rate and business constraints for a financial corporation. Ann. Appl. Prob. 14, 1810–1837.Google Scholar
de Finetti, B. (1957). Su un’ impostazione alternativa dell teoria collettiva del rischio. Transactions of the XVth International Congress of Actuaries. 2, 433–443.Google Scholar
Ekström, E. and Lindensjö, K. (2023). De Finetti’s control problem with competition. App. Math. Optim. 87, 1–26.Google Scholar
Jeanblanc-Picqué, M. and Shiryaev, A. N. (1995). Optimization of the flow of dividends. Uspekhi Mat. Nauk. 50, 25–46.Google Scholar
Kyprianou, A. E. (2014). Fluctuations of Lévy processes with applications: Introductory Lectures, 2nd edn. Springer, Heidelberg.Google Scholar
Kyprianou, A. E., Loeffen, R. L. and Pérez, J.-L. (2012). Optimal control with absolutely continuous strategies for spectrally negative Lévy processes. J. Appl. Prob. 49, 150–166.CrossRefGoogle Scholar
Øksendal, B. (2003). Stochastic Differential Equations: An Introduction with Applications, 5th edn. Springer, New York.Google Scholar
Renaud, J.-F. and Simard, C. (2021). A stochastic control problem with linearly bounded control rates in a Brownian model. SIAM J. Control Optim. 59, 3103–3117.CrossRefGoogle Scholar
Somasundaram, D. (2001). Ordinary Differential Equations: A First Course, 1st edn. Alpha Science International, Oxford.Google Scholar