Hostname: page-component-89b8bd64d-7zcd7 Total loading time: 0 Render date: 2026-05-10T10:13:44.432Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal proportional reinsurance to maximize an insurer’s exponential utility under unobservable drift

Part of: Existence theories Stochastic systems and control Stochastic processes

Published online by Cambridge University Press:  21 February 2023

Xiaoqing Liang Open the ORCID record for Xiaoqing Liang [Opens in a new window]  and
Virginia R. Young
Xiaoqing Liang*
Affiliation:
Hebei University of Technology
Virginia R. Young*
Affiliation:
University of Michigan
*
*Postal address: Departmnt of Statistics, School of Sciences, Hebei University of Technology, Tianjin 300401, P. R. China. Email: liangxiaoqing115@hotmail.com
**Postal address: Department of Mathematics, University of Michigan, Ann Arbor, Michigan, 48109. Email: vryoung@umich.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We study an optimal reinsurance problem for a diffusion model, in which the drift of the claim follows an Ornstein–Uhlenbeck process. The aim of the insurer is to maximize the expected exponential utility of its terminal wealth. We consider two cases: full information and partial information. Full information occurs when the insurer directly observes the drift; partial information occurs when the insurer observes only its claims. By applying stochastic control and by solving the corresponding Hamilton–Jacobi–Bellman equations, we find the value function and the optimal reinsurance strategy under both full and partial information. We determine a relationship between the value function and reinsurance strategy under full information with the value function and reinsurance strategy under partial information.

Keywords

Exponential utilitymean-reverting driftincomplete informationoptimal reinsurance

MSC classification

Primary: 93E20: Optimal stochastic control 93E11: Filtering
Secondary: 49J40: Variational methods including variational inequalities 60G15: Gaussian processes

Information

Type
Original Article
Information
Journal of Applied Probability , Volume 60 , Issue 3 , September 2023 , pp. 874 - 894
Check for updates
Copyright
© The Author(s), 2023. 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.)

Article purchase

Temporarily unavailable

References

Bäuerle, N. and Leimcke, G. (2021). Robust optimal investment and reinsurance problems with learning. Scand. Actuar. J. 2021, 82–109.CrossRefGoogle Scholar
Bäuerle, N. and Rieder, U. (2007). Portfolio optimization with jumps and unobservable intensity process. Math. Finance 17, 205–224.CrossRefGoogle Scholar
Bi, J., Liang, Z. and Yuen, K. C. (2019). Optimal mean-variance investment/reinsurance with common shock in a regime-switching market. Math. Meth. Operat. Res. 90, 109–135.CrossRefGoogle Scholar
Björk, T., Davis, M. H. A. and Landén, C. (2010). Optimal investment under partial information. Math. Meth. Operat. Res. 71, 371–399.10.1007/s00186-010-0301-xCrossRefGoogle Scholar
Brachetta, M. and Ceci, C. (2020). A BSDE-based approach for the optimal reinsurance problem under partial information. Insurance Math. Econom. 95, 1–16.CrossRefGoogle Scholar
Brendle, S. (2004). Portfolio selection under partial observation and constant absolute risk aversion. Available at https://ssrn.com/abstract=590362.Google Scholar
Brendle, S. (2006). Portfolio selection under incomplete information. Stoch. Process. Appl. 116, 701–723.CrossRefGoogle Scholar
Browne, S. (1995). Optimal investment policies for a firm with a random risk process: Exponential utility and minimizing the probability of ruin. Math. Operat. Res. 20, 937–958.CrossRefGoogle Scholar
Chen, L. and Shen, Y. (2018). On a new paradigm of optimal reinsurance: A stochastic Stackelberg differential game between an insurer and a reinsurer. ASTIN Bull. 48, 905–960.10.1017/asb.2018.3CrossRefGoogle Scholar
Crandall, M. G., Ishii, H. and Lions, P. L. (1992). User’s guide to viscosity solutions of second-order partial differential equations. Bull. Amer. Math. Soc. 27, 1–67.CrossRefGoogle Scholar
Cvitanić, J. and Karatzas, I. (1992). Convex duality in constrained portfolio optimization, Ann. Appl. Prob. 2, 767–818.CrossRefGoogle Scholar
Dassios, A. and Jang, J.-W. (2005). Kalman–Bucy filtering for linear systems driven by the Cox process with shot noise intensity and its application to the pricing of reinsurance contracts. J. Appl. Prob. 42, 93–107.CrossRefGoogle Scholar
Delong, Ł. (2013). Backward Stochastic Differential Equations with Jumps and Their Actuarial and Financial Applications. Springer, New York.CrossRefGoogle Scholar
Gennotte, G. (1986). Optimal portfolio choice under incomplete information. J. Finance 41, 733–749.CrossRefGoogle Scholar
Hipp, C. and Taksar, M. (2010). Optimal non-proportional reinsurance control. Insurance Math. Econom. 47, 246–254.CrossRefGoogle Scholar
Honda, T. (2003). Optimal portfolio choice for unobservable and regime-switching mean returns. J. Econom. Dynam. Control 28, 45–78.CrossRefGoogle Scholar
Irgens, C. and Paulsen, J. (2004). Optimal control of risk exposure, reinsurance and investments for insurance portfolios. Insurance Math. Econom. 35, 21–51.CrossRefGoogle Scholar
Lakner, P. (1995). Utility maximization with partial information. Stoch. Process. Appl. 56, 247–273.CrossRefGoogle Scholar
Lakner, P. (1998). Optimal trading strategy for an investor: The case of partial information. Stoch. Process. Appl. 76, 77–97.CrossRefGoogle Scholar
Li, D., Li, D. and Young, V. R. (2017). Optimality of excess-loss reinsurance under a mean-variance criterion. Insurance Math. Econom. 75, 82–89.CrossRefGoogle Scholar
Liang, Z. and Bayraktar, E. (2014). Optimal reinsurance and investment with unobservable claim size and intensity. Insurance Math. Econom. 55, 156–166.CrossRefGoogle Scholar
Liang, Z., Yuen, K. C. and Guo, J. (2011). Optimal proportional reinsurance and investment in a stock market with Ornstein–Uhlenbeck process. Insurance Math. Econom. 49, 207–215.10.1016/j.insmatheco.2011.04.005CrossRefGoogle Scholar
Liptser, R. S. and Shiryaev, A. N. (1997). Statistics of Random Processes Vol. I (Stoch. Model. Appl. Prob. 5). Springer, New York.Google Scholar
Peng, S. (1992). Stochastic Hamilton–Jacobi–Bellman equations. SIAM J. Control Optimization 30, 284–304.CrossRefGoogle Scholar
Peng, X. and Hu, Y. (2013). Optimal proportional reinsurance and investment under partial information. Insurance Math. Econom. 53, 416–428.CrossRefGoogle Scholar
Promislow, S. D. and Young, V. R. (2005). Minimizing the probability of ruin when claims follow Brownian motion with drift. N. Amer. Actuar. J. 9, 110–128.CrossRefGoogle Scholar
Putschögl, W. and Sass, J. (2011). Optimal investment under dynamic risk constraints and partial information. Quant. Finance 11, 1547–1564.CrossRefGoogle Scholar
Schmidli, H. (2002). On minimizing the ruin probability by investment and reinsurance. Ann. Appl. Prob. 12, 890–907.CrossRefGoogle Scholar
Sun, Z., Zhang, X. and Yuen, K. C. (2020). Mean-variance asset-liability management with affine diffusion factor process and a reinsurance option. Scand. Actuar. J. 2020, 218–244.CrossRefGoogle Scholar
Xiong, J., Xu, Z. and Zheng, J. (2021). Mean-variance portfolio selection under partial information with drift uncertainty. Quant. Finance 21, 1–13.CrossRefGoogle Scholar
Yong, J. and Zhou, X. (1999). Stochastic Controls: Hamiltonian Systems and HJB Equations (Stoch. Model. Appl. Prob. 43). Springer, New York.Google Scholar
Zhang, X. and Siu, T. K. (2009). Optimal investment and reinsurance of an insurer with model uncertainty. Insurance Math. Econom. 45, 81–88.CrossRefGoogle Scholar