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Optimal insurance strategies: a hybrid deep learning Markov chain approximation approach. (English) Zbl 1447.91129

Summary: This paper studies deep learning approaches to find optimal reinsurance and dividend strategies for insurance companies. Due to the randomness of the financial ruin time to terminate the control processes, a Markov chain approximation-based iterative deep learning algorithm is developed to study this type of infinite-horizon optimal control problems. The optimal controls are approximated as deep neural networks in both cases of regular and singular types of dividend strategies. The framework of Markov chain approximation plays a key role in building the iterative equations and initialization of the algorithm. We implement our method to classic dividend and reinsurance problems and compare the learning results with existing analytical solutions. The feasibility of our method for complicated problems has been demonstrated by applying to an optimal dividend, reinsurance and investment problem under a high-dimensional diffusive model with jumps and regime switching.

MSC:

91G05 Actuarial mathematics
60J28 Applications of continuous-time Markov processes on discrete state spaces
68T07 Artificial neural networks and deep learning

Software:

Adam
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References:

[1] Albrecher, H., Beirlant, J. and Teugels, J.L. (2017) Reinsurance: Actuarial and Statistical Aspects. West Sussex: Wiley. · Zbl 1376.91004
[2] Aleandri, M. (2018) Modeling dynamic policyholder behaviour through machine learning techniques. Working paper.
[3] Arrow, K. (1963) Uncertainty and the welfare economics of medical care. American Economic Review, 53, 941-973.
[4] Asmussen, S. and Taksar, M. (1997) Controlled diffusion models for optimal dividend pay-out. Insurance: Mathematics and Economics, 20, 1-15. · Zbl 1065.91529
[5] Bachouch, A., Huré, C., Langrené, N. and Pham, H. (2018) Deep neural networks algorithms for stochastic control problems on finite horizon, part 2: Numerical applications. arXiv preprint arXiv:1812.05916.
[6] Bellman, R.E. (1961) Adaptive Control Processes: A Guided Tour.Princeton, NJ: Princeton University Press. · Zbl 0103.12901
[7] Borch, K. (1960) Reciprocal reinsurance treaties. ASTIN Bulletin, 1(4), 170-191.
[8] Borch, K. (1962) Equilibrium in a reinsurance market. Econometrica, 30, 424-444. · Zbl 0119.36504
[9] Carmona, R. and Lauriŕe, M. (2019) Convergence analysis of machine learning algorithms for the numerical solution of mean field control and games: II-The finite horizon case. arXiv preprint arXiv:1908.01613.
[10] De Finetti, B. (1957) Su unimpostazione alternativa della teoria collettiva del rischio. Transactions of the XVth International Congress of Actuaries, 2, 433-443.
[11] E, W., Han, J. and Jentzen, A. (2017) Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations. Communications in Mathematics and Statistics, 5(4), 349-380. · Zbl 1382.65016
[12] Fahrenwaldt, M.A., Weber, S. and Weske, K. (2018) Pricing of cyber insurance contracts in a network model. ASTIN Bulletin, 48(3), 1175-1218. · Zbl 1416.91175
[13] Fecamp, S., Mikael, J. and Warin, X. (2019) Risk management with machine-learning-based algorithms. arXiv preprint arXiv:1902.05287.
[14] Gerber, H.U. (1972) Games of economic survival with discrete and continuous income processes. Operations Research, 20(1), 37-45. · Zbl 0236.90079
[15] Han, J. and E, W. (2016). Deep learning approximation for stochastic control problems. arXiv preprint arXiv:1611.07422.
[16] Højgaard, B.H. and Taksar, M. (1999) Controlling risk exposure and dividends payout schemes: insurance company example. Mathematical Finance, 9(2), 153-182 · Zbl 0999.91052
[17] Huré, C., Pham, H., Bachouch, A. and Langrené, N. (2018) Deep neural networks algorithms for stochastic control problems on finite horizon, part I: Convergence analysis. arXiv preprint arXiv:1812.04300.
[18] Jin, Z., Yang, H. and Yin, G. (2013) Numerical methods for optimal dividend payment and investment strategies of regime-switching jump diffusion models with capital injections. Automatica, 49(8), 2317-2329. · Zbl 1364.93863
[19] Jin, Z., Yang, H. and Yin, G. (2018) Approximation of optimal ergodic dividend and reinsurance strategies using controlled Markov chains. IET Control Theory & Applications, 12(16), 2194-2204.
[20] Kingma, D.P. and Ba, J. (2014) Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980.
[21] Kushner, H. and Dupuis, P. (2001) Numerical Methods for Stochstic Control Problems in Continuous Time, Stochastic Modelling and Applied Probability, Vol. 24, 2nd edn. New York: Springer. · Zbl 0968.93005
[22] Kushner, H. and Yin, G. (2003) Stochastic Approximation and Recursive Algorithms and Applications, Stochastic Modelling and Applied Probability, Vol. 35, 2nd edn. New York: Springer. · Zbl 1026.62084
[23] Pereira, M., Wang, Z. and Theodorou, E.A. (2019) Neural network architectures for stochastic control using the nonlinear Feynman-Kac lemma. arXiv preprint arXiv:1902.03986.
[24] Van Staden, P.M., Dang, D.M. and Forsyth, P.A. (2018) Time-consistent mean-variance portfolio optimization: A numerical impulse control approach. Insurance: Mathematics and Economics, 83, 9-28. · Zbl 1417.91558
[25] Wei, J., Yang, H. and Wang, R. (2010) Classical and impulse control for the optimization of dividend and proportional reinsurance policies with regime switching. Journal of Optimization Theory and Applications, 1, 358-377. · Zbl 1203.91118
[26] Wüthrich, M.V. (2018a). Machine learning in individual claims reserving. Scandinavian Actuarial Journal, 6, 465-480. · Zbl 1416.91225
[27] Wüthrich, M.V. (2018b). Neural networks applied to chain-ladder reserving. European Actuarial Journal, 8(2), 407-436. · Zbl 1422.91381
[28] Wüthrich, M.V. and Buser, C. (2019) Data analytics for non-life insurance pricing. Swiss Finance Institute Research Paper, 16-68.
[29] Yang, H. and Zhang, L. (2005) Optimal investment for insurer with jump-diffusion risk process. Insurance: Mathematics and Economics, 37(3), 615-634. · Zbl 1129.91020
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