×

Approximation of a class of non-zero-sum investment and reinsurance games for regime-switching jump-diffusion models. (English) Zbl 1426.91205

Summary: This work develops an approximation procedure for a class of non-zero-sum stochastic differential investment and reinsurance games between two insurance companies. Both proportional reinsurance and excess-of loss reinsurance policies are considered. We develop numerical algorithms to obtain the approximation to the Nash equilibrium by adopting the Markov chain approximation methodology. We establish the convergence of the approximation sequences and the approximation to the value functions. Numerical examples are presented to illustrate the applicability of the algorithms.

MSC:

91G05 Actuarial mathematics
91A15 Stochastic games, stochastic differential games
91A23 Differential games (aspects of game theory)
60J75 Jump processes (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] F. Lundberg, Approximerad Framställning av Sannolikehetsfunktionen, Aterförsäkering av Kollektivrisker, Almqvist & Wiksell, Uppsala. Akad. Afhandling, Almqvist o. Wiksell, Uppsala, 1903.; F. Lundberg, Approximerad Framställning av Sannolikehetsfunktionen, Aterförsäkering av Kollektivrisker, Almqvist & Wiksell, Uppsala. Akad. Afhandling, Almqvist o. Wiksell, Uppsala, 1903.
[2] Asmussen, S.; Høgaard, B.; Taksar, M., Optimal risk control and dividend distribution policies. example of excess-of loss reinsurance for an insurance corporation, Finance Stoch., 4.3, 299-324 (2000) · Zbl 0958.91026
[3] Choulli, T.; Taksar, M.; Zhou, X. Y., Excess-of loss reinsurance for a company with debt liability and constraints on risk reduction, Quant. Finance, 1, 573-596 (2001) · Zbl 1405.91251
[4] Hald, M.; Schimidli, H., On the maximization of the adjustment coefficient under proportional reinsurance, Astin Bull., 34, 75-83 (2004) · Zbl 1095.91033
[5] Browne, S., Optimal investment policies for a firm with a random risk process: exponential utility and minimizing the probability of ruin, Math. Oper. Res., 20, 4, 937-958 (1995) · Zbl 0846.90012
[6] Chen, S.; Li, Z.; Li, K., Optimal investment-reinsurance policy for an insurance company with var constraint, Insurance Math. Econom., 47, 2, 144-153 (2010) · Zbl 1231.91155
[7] Zhang, X.; Siu, T. K., On optimal proportional reinsurance and investment in a markovian regime-switching economy, Acta Math. Sin., 28, 67-82 (2012) · Zbl 1258.91115
[8] Zhang, X.; Meng, H.; Zeng, Y., Optimal investment and reinsurance strategies for insurers with generalized mean – variance premium principle and no-short selling, Insurance Math. Econom., 67, 125-132 (2016) · Zbl 1348.91193
[9] Bensoussan, A.; Frehse, J., Stochastic games for N players, J. Optim. Theory Appl., 105, 3, 543-565 (2000) · Zbl 0977.91006
[10] Zeng, X., Stochastic differential reinsurance games, J. Appl. Probab., 47, 2, 335-349 (2010) · Zbl 1222.93238
[11] Liu, J.; Yiu, K. F.C., Optimal stochastic differential games with var constraints, Discrete Contin. Dyn. Syst. Ser. B, 18, 7, 1889-1907 (2013) · Zbl 1273.93177
[12] Bensoussan, A.; Siu, C. C.; Yam, S. C.P.; Yang, H., A class of non-zero-sum stochastic differential investment and reinsurance games, Automatica, 50, 8, 2025-2037 (2014) · Zbl 1297.93180
[13] L. Chen, Y. Shen, On a new paradigm of optimal reinsurance: A stochastic Stackelberg differential game between an insurer and a reinsurer, Working paper, 2017.; L. Chen, Y. Shen, On a new paradigm of optimal reinsurance: A stochastic Stackelberg differential game between an insurer and a reinsurer, Working paper, 2017.
[14] Taksar, M.; Zeng, X., Optimal non-proportional reinsurance control and stochastic differential games, Insurance Math. Econom., 48, 1, 64-71 (2011) · Zbl 1218.91064
[15] Pun, C. S.; Wong, H. Y., Robust non-zero-sum stochastic differential reinsurance game, Insurance Math. Econom., 68, 169-177 (2016) · Zbl 1369.91094
[16] Yan, M.; Peng, F.; Zhang, S., A reinsurance and investment game between two insurance companies with the different opinions about some extra informaton, Insurance Math. Econom., 75, 58-70 (2017) · Zbl 1394.91239
[17] Wei, J.; Yang, H.; Wang, R., Classical and impulse control for the optimization of dividend and proportional reinsurance policies with regime switching, J. Optim. Theory Appl., 147, 2 (2010) · Zbl 1203.91118
[18] Sotomayor, L.; Cadenillas, A., Classical, singular, and impulse stochastic control for the optimal dividend policy when there is regime switching, Insurance Math. Econom., 48, 3, 344-354 (2011) · Zbl 1218.91096
[19] Zhu, J., Dividend optimization for a regime-switching diffusion model with restricted dividend rates, Astin Bull., 44, 459-494 (2014) · Zbl 1291.91137
[20] Jin, Z.; Wang, Y.; Yin, G., Numerical solutions of quantile hedging for guaranteed minimum death benefits under a regime-switching-jump-diffusion formulation, J. Comput. Appl. Math., 235, 2842-2860 (2011) · Zbl 1229.91358
[21] Yin, G.; Zhu, C., Hybrid Switching Diffusions: Properties and Applications (2010), Springer: Springer New York · Zbl 1279.60007
[22] Bensoussan, A.; Z. Yan, Z.; Yin, G., Threshold-type policies for real options using regime-switching models, SIAM J. Financial Math., 3, 667-689 (2012) · Zbl 1255.91444
[23] Bensoussan A. Hoe, S.; Yan, Z.; Yin, G., Real options with competition and regime switching, Math. Finance, 27, 224-250 (2017) · Zbl 1414.91403
[24] Kushner, H.; Dupuis, P., (Numerical Methods for Stochastic Control Problems in Continuous Time. Numerical Methods for Stochastic Control Problems in Continuous Time, Stochastic Modelling and Applied Probability, vol. 24 (2001), Springer: Springer New York) · Zbl 0968.93005
[25] Jin, Z.; Yin, G.; Wu, F., Optimal reinsurance strategies in regime-switching jump diffusion models: stochastic differential game formulation and numerical methods, Insurance Math. Econom., 53, 733-746 (2013) · Zbl 1290.91090
[26] Meng, H.; Siu, T. K.; Yang, H., Optimal dividens with debts and nonlinear insurance risk processes, Insurance Math. Econom., 53, 110-121 (2013) · Zbl 1284.91564
[27] Meng, H.; Li, S.; Jin, Z., A reinsurance game between two insurance companies with nonlinear risk processes, Insurance Math. Econom., 62, 91-97 (2015) · Zbl 1318.91120
[28] Espinosa, G.; Touzi, N., Optimal investment under relative performance concerns, Math. Finance, 25, 2, 221-257 (2015) · Zbl 1403.91310
[29] Yin, G.; Jin, H.; Z., Jin., Numerical methods for porfolio selection with bounded constrains, J. Comput. Appl. Math., 233, 564-581 (2009) · Zbl 1180.91276
[30] Yin, G.; Zhang, Q.; Badowski, G., Dicrete-time singularly perturbed markov chains: aggregation, occupation measures, and switching diffusion limit, Adv. Appl. Probab., 35, 449-476 (2003) · Zbl 1048.60058
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.