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https://doi.org/10.62836/emi.v5i1.0005
Liang, X., Zhang, X., Wei, D., & Ma, R. (2026). Modeling and Numerical Solution of Optimal Investment and Reinsurance Problems in Ambiguity Markets. Economics & Management Information, 5(1), 0005. https://doi.org/10.62836/emi.v5i1.0005
Volume 5, Issue 1, March 2026
Copyright (c) 2026 Xuezhen Liang, Xinyue Zhang, Dongping Wei, Ruihong Ma
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This work is licensed under a Creative Commons Attribution 4.0 International License.

Modeling and Numerical Solution of Optimal Investment and Reinsurance Problems in Ambiguity Markets

In a framework of market incompleteness induced by ambiguity, this paper employs stochastic processes and stochastic analysis to formulate the decision-making problem concerning investment, consumption, and proportional reinsurance in an ambiguous market as a one-dimensional stochastic optimal control problem over a finite time horizon. Specifically, the insurer aims to maximize the utility of terminal wealth through dynamic optimal strategies, which is inherently a forward–backward stochastic differential equation system. The ambiguity in the model is captured by the Chen–Epstein multiple-priors framework, leading to a fully nonlinear Hamilton–Jacobi–Bellman equation that is generally analytically intractable. To address this, an implicit finite difference scheme is designed to numerically solve the value function as well as the optimal investment proportion, consumption strategy, and retention ratio. Furthermore, a systematic analysis is conducted to examine the quantitative impact of key market parameter variations on these optimal strategies. The findings provide theoretical support and numerical decision-making references for insurance institutions in asset allocation and risk management under complex and uncertain environments.

reinsurance ambiguity stochastic differential utility finite difference method HJB equation

References

  1. Merton RC. Optimal Consumption and Portfolio Rules in a Continuous-Time Model. Journal of Economic Theory 1971; 3: 373–413.
  2. Browne S. Optimal Investment Policies for a Firm with a Random Risk Process: Exponential Utility and Minimizing the Probability of Ruin. Mathematics of Operations Research 1995; 20: 937–958.
  3. Promislow DS, Young VR. Minimizing the Probability of Ruin When Claims Follow Brownian Motion with Drift. North American Actuarial Journal 2005; 9: 109–128.
  4. Li Q L, Tang X J. Robust optimal reinsurance and investment with inflation risk: A game-theoretic approach and explicit solutions. AIMS Mathematics 2026; 11(3): 7330–7352.
  5. Schmidli H. Optimal Proportional Reinsurance Policies in a Dynamic Setting. Scandinavian Actuarial Journal 2001; 14: 55–68.
  6. Cao Y, Wan N. Optimal Proportional Reinsurance and Investment Based on Hamilton- Jacobi-Bellman Equation. Insurance: Mathematics and Economics 2009; 45(2): 157–162.
  7. Chen Z, Epstein L. Ambiguity, Risk and Asset Return in Continuous Time. Econometrica 2002; 70: 1403–1443.
  8. Schied, A. Robust Optimal Control for a Consumption-Investment Problem. Mathematical Methods of Operations Research 2008; 67: 1–20.
  9. Maenhout PJ. Robust Portfolio Rules and Asset Pricing. The Review of Financial Studies 2004; 17: 951–983
  10. Zhang X, Siu TK. Optimal Investment and Reinsurance of an Insurer with Model Uncertainty. Insurance: Mathematics and Economics 2009; 45: 81–88.
  11. Lin X, Zhang CH, Siu TK. Stochastic Differential Portfolio Games for an Insuer in a Jump-Diffusion Risk Process. Mathematical Methods of Operations Research 2012; 75: 83–100.
  12. Boulbrachene, M. The Finite Element Approximation of Hamilton-Jacobi-Bellman Equations. Computers and Mathematics with Applications 2001; 41: 993–1007.
  13. Kushner HJ, Dupuis PG. Numerical Methods for Stochastic Control Problems in Continuous Time; Springer: New York, NY, USA, 2001; Volume 24.
  14. Grandell J. Aspects of Risk Theory; Springer: New York, NY, USA, 1991.
  15. Duffie D, Epstein L. Stochastic Differential Utility. Econometrica 1992; 60: 353–394.
  16. Fleming WH, Soner HM. Controlled Markov Processes and Viscosity Solutions, 2nd ed.; Springer: New York, NY, USA, 2006.
  17. Jiang LS. Mathematical Models and Methods of Option Pricing; Higher Education Press: Beijing, China, 2003.
  18. Courant R, Isaacson E, Rees M. On the Solution of Non-Linear Hyperbolic Differential Equations. Communications on Pure and Applied Mathematics 1952; 5: 243–255.
  19. Bi JN, Meng QB, Zhang YJ. Dynamic Mean-Variance and Optimal Reinsurance Problems under the No-Bankruptcy Constraint for an Insurer. Annals of Operations Research 2014; 212: 43–50.

Supporting Agencies

  1. Funding: This research received no external funding.