Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-25T22:07:19.818Z Has data issue: false hasContentIssue false
  • English
  • Français

OPTIMAL INSURANCE STRATEGIES: A HYBRID DEEP LEARNING MARKOV CHAIN APPROXIMATION APPROACH

Published online by Cambridge University Press:  06 May 2020

Xiang Cheng ,
Zhuo Jin  and
Hailiang Yang
Xiang Cheng
Affiliation:
Department of Economics, Centre for Actuarial Studies The University of Melbourne VIC 3010, Australia E-mail: [email protected]
Zhuo Jin
Affiliation:
Department of Economics, Centre for Actuarial Studies The University of Melbourne VIC 3010, Australia E-mail: [email protected]
Hailiang Yang*
Affiliation:
Department of Statistics and Actuarial Science The University of Hong KongHong Kong E-mail:[email protected]
*
E-mail:[email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Keywords

Neural networksdeep learningMarkov chain approximationreinsurance strategies.
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 50 , Issue 2 , May 2020 , pp. 449 - 477
Copyright
© Astin Bulletin 2020

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.)

References

Albrecher, H., Beirlant, J. and Teugels, J.L. (2017) Reinsurance: Actuarial and Statistical Aspects. West Sussex: Wiley.10.1002/9781119412540CrossRefGoogle Scholar
Aleandri, M. (2018) Modeling dynamic policyholder behaviour through machine learning techniques. Working paper.Google Scholar
Arrow, K. (1963) Uncertainty and the welfare economics of medical care. American Economic Review, 53, 941–973.Google Scholar
Asmussen, S. and Taksar, M. (1997) Controlled diffusion models for optimal dividend pay-out. Insurance: Mathematics and Economics, 20, 1–15.Google Scholar
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.Google Scholar
Bellman, R.E. (1961) Adaptive Control Processes: A Guided Tour. Princeton, NJ: Princeton University Press.10.1515/9781400874668CrossRefGoogle Scholar
Borch, K. (1960) Reciprocal reinsurance treaties. ASTIN Bulletin, 1(4), 170–191.CrossRefGoogle Scholar
Borch, K. (1962) Equilibrium in a reinsurance market. Econometrica, 30, 424–444.CrossRefGoogle Scholar
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.Google Scholar
De Finetti, B. (1957) Su unimpostazione alternativa della teoria collettiva del rischio. Transactions of the XVth International Congress of Actuaries, 2, 433–443.Google Scholar
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), 349–380.10.1007/s40304-017-0117-6CrossRefGoogle Scholar
Fahrenwaldt, M.A., Weber, S. and Weske, K. (2018) Pricing of cyber insurance contracts in a network model. ASTIN Bulletin, 48(3), 1175–1218.CrossRefGoogle Scholar
Fecamp, S., Mikael, J. and Warin, X. (2019) Risk management with machine-learning-based algorithms. arXiv preprint arXiv:1902.05287.Google Scholar
Gerber, H.U. (1972) Games of economic survival with discrete and continuous income processes. Operations Research, 20(1), 37–45.CrossRefGoogle Scholar
Han, J. and E, W. (2016). Deep learning approximation for stochastic control problems. arXiv preprint arXiv:1611.07422.Google Scholar
Højgaard, B.H. and Taksar, M. (1999) Controlling risk exposure and dividends payout schemes: insurance company example. Mathematical Finance, 9(2), 153–18210.1111/1467-9965.00066CrossRefGoogle Scholar
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.Google Scholar
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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Kingma, D.P. and Ba, J. (2014) Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980.Google Scholar
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.CrossRefGoogle Scholar
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.Google Scholar
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.Google Scholar
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.Google Scholar
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.CrossRefGoogle Scholar
Wüthrich, M.V. (2018a). Machine learning in individual claims reserving. Scandinavian Actuarial Journal, 6, 465–480.CrossRefGoogle Scholar
Wüthrich, M.V. (2018b). Neural networks applied to chain-ladder reserving. European Actuarial Journal, 8(2), 407–436.CrossRefGoogle Scholar
Wüthrich, M.V. and Buser, C. (2019) Data analytics for non-life insurance pricing. Swiss Finance Institute Research Paper, 16–68.Google Scholar
Yang, H. and Zhang, L. (2005) Optimal investment for insurer with jump-diffusion risk process. Insurance: Mathematics and Economics, 37(3), 615–634.Google Scholar