Optimal Risk Management Strategies in a General Perturbed Risk Process

Document Type : Original Article

Author
Abouzar Bazyari
Department of Statistics, Faculty of Intelligent Systems Engineering and Data Science, Persian Gulf University, Bushehr, Iran.
10.22034/jirss.2025.2063080.1115
Abstract
This paper investigates the optimal risk management strategies in a general compound Poisson risk model consisting of safety loading of insurer and reinsurance to minimize the infinite-time ruin probability. Its price process is perturbed by a geometric Brownian motion with the drift and volatility of risky asset. In addition, we allow this company to buy proportional reinsurance to reduce the underlying risk and invest its surplus in a risky asset whose price is driven by correlated Brownian motions. We focus on the possibility of an insurance company utilizing optimal controls and study the optimization problem of minimizing the infinite-time ruin probability in a financial market. For the diffusion approximation of risk model, we obtain an analytic expression for the minimum nfinite-time ruin probability and the corresponding optimal controls by using the martingale approach. Since it is not easy to derive the explicit expression for the infinite-time ruin probability of perturbed risk model, we obtain the optimal strategies to maximize the Lundberg exponent when the claim amounts are identically distributed and have an exponentially decaying tail. Moreover, we study the effect of investment on the ruin probability in both perturbed risk models. Finally, some numerical examples are conducted to illustrate the effects of model parameters on the optimal risk management strategies and on the financial market.
Keywords
Diffusion Risk Process
Infinite-time Ruin Probability
Optimization Problem
Proportional Reinsurance
Risky Asset
Subjects
Asmussen, S. and Albrecher, H. (2010), Ruin Probabilities, World Scientific, London.
Bai, L. H. and Guo, J. Y. (2008), Optimal proportional reinsurance and investment with multiple risky assets and no-shorting constraint, Insurance: Mathematics and Economics, 42, 968–975.
Bazyari, A. (2025), Optimal risk management strategies in a dynamic setting under the force of interest, Journal of Industrial and Management Optimization, 21(5), 3985–4002.
Browne, S. (1995), Optimal investment policies for a firm with a random risk process: Exponential utility and minimization of the probability of ruin, Mathematics of Operations Research, 20, 937–958.
Chen, S., Li, Z., and Li, K. (2010), Optimal investment-reinsurance policy for an insurance company with VaR constraint, Insurance: Mathematics and Economics, 47(2), 144–153.
Grandell, J. (1991), Aspects of Risk Theory, Springer, New York, http://dx.doi.org/10.1007/978-1-4613-9058-9.
Hald, M. and Schmidli, H. (2004), On the maximization of the adjustment coefficient under proportional reinsurance, ASTIN Bulletin, 34, 75–83.
Han, X., Liang, Z., Yuan, Y. and Zhang, D. (2022), Optimal per-loss reinsurance and investment to minimize the probability of drawdown, Journal of Industrial and Management Optimization, 18, 4011–4041.
Hipp, C. and Plum, M. (2000), Optimal investment for insurers, Insurance: Mathematics and Economics, 27, 215–228.
Merton, R. C. (1971), Optimum consumption and portfolio rules in a continuous-time model, Journal of Economic Theory, 3(4), 373–413.
Liang, Z. and Bayraktar, E. (2014), Optimal reinsurance and investment with unobservable claim size and intensity, Insurance: Mathematics and Economics, 55, 156–166.
Liang, Z. and Long, M. (2015), Minimization of absolute ruin probability under negative correlation assumption, Insurance: Mathematics and Economics, 65, 247–258.
Liang, X. and Young, V. (2018), Minimizing the probability of ruin: Two riskless assets with transaction costs and proportional reinsurance, Statistics and Probability Letters, 140, 167–175.
Schmidli, H. (2001), Optimal proportional reinsurance policies in a dynamic setting, Scandinavian Actuarial Journal, 1, 55–68. 
Wang, W. and Peng, X. (2017), Reinsurer’s optimal reinsurance strategy with upper and lower premium constraints under distortion risk measures, Journal of Computational and Applied Mathematics, 355, 142–160.
Xu, L., Wang, M. and Zhang, B. (2018), Minimizing Lundberg inequality for ruin probability under correlated risk model by investment and reinsurance, Journal of Inequalities and Applications, 2018:244, https://doi.org/10.1186/s13660-018-1838-0.
Yi, B., Li, Z. F., Viens, F. G. and Zeng, Y. (2013), Robust optimal control for an insurer with reinsurance and investment under Heston’s stochastic volatility model, Insurance: Mathematics and Economics, 53, 601–614.
Zhang, X. and Siu, T. K. (2012), On optimal proportional reinsurance and investment in a Markovian regime-switching economy, Acta Mathematica Sinica (English Series), 28, 67–82.
Zhang, X., Hui, M. and Zeng, Y. (2016), Optimal investment and reinsurance strategies for insurers with generalized mean-variance premium principle and no-short selling, Insurance: Mathematics and Economics, 67, 125–132.
Zhang, N., Jin, Z., Qian, L. and Wang, R. (2018), Optimal quota-share reinsurance based on the mutual benefit of insurer and reinsurer, Journal of Computational and Applied Mathematics, 342, 337–351.
Zeng, Y., Li, D. P. and Gu, A. L. (2016), Robust equilibrium reinsurance-investment strategy for a mean-variance insurer in a model with jumps, Insurance: Mathematics and Economics, 66, 138–152. 
Journal of the Iranian Statistical Society
Volume 24, Issue 1
June 2025
Pages 117-138

  • Receive Date 08 June 2025
  • Revise Date 08 September 2025
  • Accept Date 01 October 2025