IDEAS home Printed from https://ideas.repec.org/a/gam/jrisks/v9y2021i6p108-d568047.html
   My bibliography  Save this article

Merton Investment Problems in Finance and Insurance for the Hawkes-Based Models

Author

Listed:
  • Anatoliy Swishchuk

    (Department of Mathematics & Statistics, University of Calgary, Calgary, AB T2N 1N4, Canada)

Abstract

We show how to solve Merton optimal investment stochastic control problem for Hawkes-based models in finance and insurance (Propositions 1 and 2), i.e., for a wealth portfolio X ( t ) consisting of a bond and a stock price described by general compound Hawkes process (GCHP), and for a capital R ( t ) (risk process) of an insurance company with the amount of claims described by the risk model based on GCHP. The main approach in both cases is to use functional central limit theorem for the GCHP to approximate it with a diffusion process. Then we construct and solve Hamilton–Jacobi–Bellman (HJB) equation for the expected utility function. The novelty of the results consists of the new Hawkes-based models and in the new optimal investment results in finance and insurance for those models.

Suggested Citation

  • Anatoliy Swishchuk, 2021. "Merton Investment Problems in Finance and Insurance for the Hawkes-Based Models," Risks, MDPI, vol. 9(6), pages 1-13, June.
    • Handle: RePEc:gam:jrisks:v:9:y:2021:i:6:p:108-:d:568047
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-9091/9/6/108/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-9091/9/6/108/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Anatoliy Swishchuk & Aiden Huffman, 2020. "General Compound Hawkes Processes in Limit Order Books," Risks, MDPI, vol. 8(1), pages 1-25, March.
    2. Cox, John C. & Huang, Chi-fu, 1989. "Optimal consumption and portfolio policies when asset prices follow a diffusion process," Journal of Economic Theory, Elsevier, vol. 49(1), pages 33-83, October.
    3. Henri Loubergé, 1998. "Risk and Insurance Economics 25 Years After," The Geneva Papers on Risk and Insurance - Issues and Practice, Palgrave Macmillan;The Geneva Association, vol. 23(4), pages 540-567, October.
    4. Browne, S., 1995. "Optimal Investment Policies for a Firm with a Random Risk Process: Exponential Utility and Minimizing the Probability of Ruin," Papers 95-08, Columbia - Graduate School of Business.
    5. Merton, Robert C., 1971. "Optimum consumption and portfolio rules in a continuous-time model," Journal of Economic Theory, Elsevier, vol. 3(4), pages 373-413, December.
    6. Sid Browne, 1995. "Optimal Investment Policies for a Firm With a Random Risk Process: Exponential Utility and Minimizing the Probability of Ruin," Mathematics of Operations Research, INFORMS, vol. 20(4), pages 937-958, November.
    7. Ioannis Karatzas & John P. Lehoczky & Suresh P. Sethi & Steven E. Shreve, 1986. "Explicit Solution of a General Consumption/Investment Problem," Mathematics of Operations Research, INFORMS, vol. 11(2), pages 261-294, May.
    8. Patrick J. Laub & Thomas Taimre & Philip K. Pollett, 2015. "Hawkes Processes," Papers 1507.02822, arXiv.org.
    9. Stéphane Goutte & David Guerreiro & Bilel Sanhaji & Sophie Saglio & Julien Chevallier, 2019. "Financial Mathematics, Volatility and Covariance Modelling," Post-Print halshs-02183052, HAL.
    10. Bjork, Tomas, 2009. "Arbitrage Theory in Continuous Time," OUP Catalogue, Oxford University Press, edition 3, number 9780199574742.
    11. Emmanuel Bacry & Sylvain Delattre & Marc Hoffmann & Jean-François Muzy, 2013. "Some limit theorems for Hawkes processes and application to financial statistics," Post-Print hal-01313994, HAL.
    12. Ehrlich, Isaac & Becker, Gary S, 1972. "Market Insurance, Self-Insurance, and Self-Protection," Journal of Political Economy, University of Chicago Press, vol. 80(4), pages 623-648, July-Aug..
    13. Dufresne, Francois & Gerber, Hans U., 1991. "Risk theory for the compound Poisson process that is perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 10(1), pages 51-59, March.
    14. Anatoliy Swishchuk, 2017. "Risk Model Based on General Compound Hawkes Process," Papers 1706.09038, arXiv.org.
    15. Bacry, E. & Delattre, S. & Hoffmann, M. & Muzy, J.F., 2013. "Some limit theorems for Hawkes processes and application to financial statistics," Stochastic Processes and their Applications, Elsevier, vol. 123(7), pages 2475-2499.
    16. Stanley R. Pliska, 1986. "A Stochastic Calculus Model of Continuous Trading: Optimal Portfolios," Mathematics of Operations Research, INFORMS, vol. 11(2), pages 371-382, May.
    17. Michael I. Taksar, 2000. "Optimal risk and dividend distribution control models for an insurance company," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 51(1), pages 1-42, February.
    18. Qiyue He & Anatoliy Swishchuk, 2019. "Quantitative and Comparative Analyses of Limit Order Books with General Compound Hawkes Processes," Risks, MDPI, vol. 7(4), pages 1-21, November.
    19. Merton, Robert C, 1969. "Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case," The Review of Economics and Statistics, MIT Press, vol. 51(3), pages 247-257, August.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Souhail Chebbi & Senda Ounaies, 2023. "Optimal Investment of Merton Model for Multiple Investors with Frictions," Mathematics, MDPI, vol. 11(13), pages 1-10, June.
    2. Qi Guo & Anatoliy Swishchuk & Bruno R'emillard, 2022. "Multivariate Hawkes-based Models in LOB: European, Spread and Basket Option Pricing," Papers 2209.07621, arXiv.org.
    3. Myles Sjogren & Timothy DeLise, 2021. "General Compound Hawkes Processes for Mid-Price Prediction," Papers 2110.07075, arXiv.org.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Anatoliy Swishchuk, 2021. "Merton Investment Problems in Finance and Insurance for the Hawkes-based Models," Papers 2104.02694, arXiv.org, revised May 2021.
    2. Zhu, Huiming & Deng, Chao & Yue, Shengjie & Deng, Yingchun, 2015. "Optimal reinsurance and investment problem for an insurer with counterparty risk," Insurance: Mathematics and Economics, Elsevier, vol. 61(C), pages 242-254.
    3. Zhao, Hui & Rong, Ximin, 2012. "Portfolio selection problem with multiple risky assets under the constant elasticity of variance model," Insurance: Mathematics and Economics, Elsevier, vol. 50(1), pages 179-190.
    4. Christensen, Bent Jesper & Parra-Alvarez, Juan Carlos & Serrano, Rafael, 2021. "Optimal control of investment, premium and deductible for a non-life insurance company," Insurance: Mathematics and Economics, Elsevier, vol. 101(PB), pages 384-405.
    5. Li, Zhongfei & Yao, Jing & Li, Duan, 2010. "Behavior patterns of investment strategies under Roy's safety-first principle," The Quarterly Review of Economics and Finance, Elsevier, vol. 50(2), pages 167-179, May.
    6. Bilel Jarraya & Abdelfettah Bouri, 2013. "A Theoretical Assessment on Optimal Asset Allocations in Insurance Industry," International Journal of Finance & Banking Studies, Center for the Strategic Studies in Business and Finance, vol. 2(4), pages 30-44, October.
    7. Zhuo Jin & Zuo Quan Xu & Bin Zou, 2020. "A Perturbation Approach to Optimal Investment, Liability Ratio, and Dividend Strategies," Papers 2012.06703, arXiv.org, revised May 2021.
    8. Myles Sjogren & Timothy DeLise, 2021. "General Compound Hawkes Processes for Mid-Price Prediction," Papers 2110.07075, arXiv.org.
    9. Sanjiv R. Das & Daniel Ostrov & Anand Radhakrishnan & Deep Srivastav, 2020. "Dynamic portfolio allocation in goals-based wealth management," Computational Management Science, Springer, vol. 17(4), pages 613-640, December.
    10. Yu, Bosco Wing-Tong & Pang, Wan Kai & Troutt, Marvin D. & Hou, Shui Hung, 2009. "Objective comparisons of the optimal portfolios corresponding to different utility functions," European Journal of Operational Research, Elsevier, vol. 199(2), pages 604-610, December.
    11. Jouini, Elyes, 2001. "Arbitrage and control problems in finance: A presentation," Journal of Mathematical Economics, Elsevier, vol. 35(2), pages 167-183, April.
    12. Bin Zou & Abel Cadenillas, 2017. "Optimal Investment and Liability Ratio Policies in a Multidimensional Regime Switching Model," Risks, MDPI, vol. 5(1), pages 1-22, January.
    13. Sona Kilianova & Daniel Sevcovic, 2013. "Transformation Method for Solving Hamilton-Jacobi-Bellman Equation for Constrained Dynamic Stochastic Optimal Allocation Problem," Papers 1307.3672, arXiv.org, revised Jul 2013.
    14. Bin Zou & Abel Cadenillas, 2014. "Optimal Investment and Risk Control Problem for an Insurer: Expected Utility Maximization," Papers 1402.3560, arXiv.org, revised Mar 2014.
    15. Dong, Yinghui & Zheng, Harry, 2019. "Optimal investment of DC pension plan under short-selling constraints and portfolio insurance," Insurance: Mathematics and Economics, Elsevier, vol. 85(C), pages 47-59.
    16. Marcos Escobar-Anel & Michel Kschonnek & Rudi Zagst, 2022. "Portfolio optimization: not necessarily concave utility and constraints on wealth and allocation," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 95(1), pages 101-140, February.
    17. Di Giacinto, Marina & Federico, Salvatore & Gozzi, Fausto & Vigna, Elena, 2014. "Income drawdown option with minimum guarantee," European Journal of Operational Research, Elsevier, vol. 234(3), pages 610-624.
    18. Frank Seifried, 2010. "Optimal investment with deferred capital gains taxes," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 71(1), pages 181-199, February.
    19. Suresh M. Sundaresan, 2000. "Continuous‐Time Methods in Finance: A Review and an Assessment," Journal of Finance, American Finance Association, vol. 55(4), pages 1569-1622, August.
    20. van Bilsen, Servaas & Laeven, Roger J.A., 2020. "Dynamic consumption and portfolio choice under prospect theory," Insurance: Mathematics and Economics, Elsevier, vol. 91(C), pages 224-237.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jrisks:v:9:y:2021:i:6:p:108-:d:568047. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.