IDEAS home Printed from https://ideas.repec.org/p/hal/wpaper/hal-02974961.html
   My bibliography  Save this paper

A Continuous-Time Model of Self-Protection

Author

Listed:
  • Sarah Bensalem
  • Nicolás Hernández Santibáñez

    (UCHILE - Universidad de Chile = University of Chile [Santiago])

  • Nabil Kazi-Tani

Abstract

This paper deals with an optimal linear insurance demand model, where the protection buyer can also exert time-dynamic costly prevention effort to reduce her risk exposure. This is expressed as a stochastic control problem, that consists in maximizing an exponential utility of a terminal wealth. We assume that the effort reduces the intensity of the jump arrival process, and we interpret this as dynamic self-protection. We solve the problem using a dynamic programming principle approach, and we provide a representation of the certainty equivalent of the buyer as the solution to an SDE. Using this representation, we prove that an exponential utility maximizer has an incentive to modify her effort dynamically only in the presence of a terminal reimbursement in the contract. Otherwise, the dynamic effort is actually constant, for a class of Compound Poisson loss processes. If there is no terminal reimbursement, we solve the problem explicitly and we identify the dynamic certainty equivalent of the protection buyer. This shows in particular that the Lévy property is preserved under exponential utility maximization. We also characterize the constant effort as a the unique minimizer of an explicit Hamiltonian, from which we can determine the optimal effort in particular cases. Finally, after studying the dependence of the SDE associated to the insurance buyer on the linear insurance contract parameter, we prove the existence of an optimal linear cover, that is not necessarily zero or full insurance.

Suggested Citation

  • Sarah Bensalem & Nicolás Hernández Santibáñez & Nabil Kazi-Tani, 2022. "A Continuous-Time Model of Self-Protection," Working Papers hal-02974961, HAL.
  • Handle: RePEc:hal:wpaper:hal-02974961
    Note: View the original document on HAL open archive server: https://hal.science/hal-02974961v2
    as

    Download full text from publisher

    File URL: https://hal.science/hal-02974961v2/document
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Christian Gollier & James Hammitt & Nicolas Treich, 2013. "Risk and choice: A research saga," Journal of Risk and Uncertainty, Springer, vol. 47(2), pages 129-145, October.
    2. Courbage, Christophe & Rey, Béatrice & Treich, Nicolas, 2013. "Prevention and precaution," IDEI Working Papers 805, Institut d'Économie Industrielle (IDEI), Toulouse.
    3. Bruno Biais & Thomas Mariotti & Jean-Charles Rochet & StÈphane Villeneuve, 2010. "Large Risks, Limited Liability, and Dynamic Moral Hazard," Econometrica, Econometric Society, vol. 78(1), pages 73-118, January.
    4. Nicolás Hernández Santibáñez & Dylan Possamaï & Chao Zhou, 2020. "Bank Monitoring Incentives Under Moral Hazard and Adverse Selection," Journal of Optimization Theory and Applications, Springer, vol. 184(3), pages 988-1035, March.
    5. Claude Henry, 1974. "Investment decisions under uncertainty: The "irreversibility effect"," ULB Institutional Repository 2013/327343, ULB -- Universite Libre de Bruxelles.
    6. 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..
    7. Epstein, Larry G, 1980. "Decision Making and the Temporal Resolution of Uncertainty," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 21(2), pages 269-283, June.
    8. Yuliy Sannikov, 2008. "A Continuous-Time Version of the Principal-Agent Problem," Review of Economic Studies, Oxford University Press, vol. 75(3), pages 957-984.
    9. Antonis Papapantoleon & Dylan Possamai & Alexandros Saplaouras, 2016. "Existence and uniqueness results for BSDEs with jumps: the whole nine yards," Papers 1607.04214, arXiv.org, revised Nov 2018.
    10. Henry, Claude, 1974. "Investment Decisions Under Uncertainty: The "Irreversibility Effect."," American Economic Review, American Economic Association, vol. 64(6), pages 1006-1012, December.
    11. Jakša Cvitanić & Dylan Possamaï & Nizar Touzi, 2018. "Dynamic programming approach to principal–agent problems," Finance and Stochastics, Springer, vol. 22(1), pages 1-37, January.
    12. Georges Dionne (ed.), 2013. "Handbook of Insurance," Springer Books, Springer, edition 2, number 978-1-4614-0155-1, June.
    13. repec:hal:spmain:info:hdl:2441/2iocb3a66198q8ovlu1gfkqie7 is not listed on IDEAS
    14. Morlais, Marie-Amelie, 2010. "A new existence result for quadratic BSDEs with jumps with application to the utility maximization problem," Stochastic Processes and their Applications, Elsevier, vol. 120(10), pages 1966-1995, September.
    15. L. Eeckhoudt & C. Gollier & H. Schlesinger, 2005. "Economic and financial decisions under risk," Post-Print hal-00325882, HAL.
    16. Bensalem, Sarah & Santibáñez, Nicolás Hernández & Kazi-Tani, Nabil, 2020. "Prevention efforts, insurance demand and price incentives under coherent risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 93(C), pages 369-386.
    17. Griselda Deelstra & Guillaume Plantin, 2014. "Risk Theory and Reinsurance," Post-Print hal-03256838, HAL.
    Full references (including those not matched with items on IDEAS)

    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. Sarah Bensalem & Nicolás Hernández-Santibáñez & Nabil Kazi-Tani, 2023. "A continuous-time model of self-protection," Finance and Stochastics, Springer, vol. 27(2), pages 503-537, April.
    2. Bensalem, Sarah & Santibáñez, Nicolás Hernández & Kazi-Tani, Nabil, 2020. "Prevention efforts, insurance demand and price incentives under coherent risk measures," Insurance: Mathematics and Economics, Elsevier, vol. 93(C), pages 369-386.
    3. Sarah Bensalem & Nicolás Hernández Santibáñez & Nabil Kazi-Tani, 2019. "Prevention efforts, insurance demand and price incentives under coherent risk measures," Working Papers hal-01983433, HAL.
    4. Dionne, Georges & Harrington, Scott, 2017. "Insurance and Insurance Markets," Working Papers 17-2, HEC Montreal, Canada Research Chair in Risk Management.
    5. Meglena Jeleva & Stéphane Rossignol, 2019. "Optimists, Pessimists, and the Precautionary Principle," Environmental & Resource Economics, Springer;European Association of Environmental and Resource Economists, vol. 74(1), pages 367-396, September.
    6. Nicolás Hernández Santibáñez & Dylan Possamaï & Chao Zhou, 2020. "Bank Monitoring Incentives Under Moral Hazard and Adverse Selection," Journal of Optimization Theory and Applications, Springer, vol. 184(3), pages 988-1035, March.
    7. Sarah Bensalem, 2020. "Self-insurance and Non-concave Distortion Risk Measures," Working Papers hal-02936349, HAL.
    8. Dylan Possamai & Nizar Touzi, 2020. "Is there a Golden Parachute in Sannikov's principal-agent problem?," Papers 2007.05529, arXiv.org, revised Oct 2022.
    9. Courbage, Christophe & Rey, Béatrice & Treich, Nicolas, 2013. "Prevention and precaution," IDEI Working Papers 805, Institut d'Économie Industrielle (IDEI), Toulouse.
    10. Christian Gollier & James Hammitt & Nicolas Treich, 2013. "Risk and choice: A research saga," Journal of Risk and Uncertainty, Springer, vol. 47(2), pages 129-145, October.
    11. Attanasi, Giuseppe Marco & Montesano, Aldo, 2010. "Testing Value vs Waiting Value in Environmental Decisions under Uncertainty," TSE Working Papers 10-154, Toulouse School of Economics (TSE).
    12. Jessica Martin & Stéphane Villeneuve, 2021. "A Class of Explicit optimal contracts in the face of shutdown," Working Papers hal-03124102, HAL.
    13. Gabriela Zeller & Matthias Scherer, 2023. "Risk mitigation services in cyber insurance: optimal contract design and price structure," The Geneva Papers on Risk and Insurance - Issues and Practice, Palgrave Macmillan;The Geneva Association, vol. 48(2), pages 502-547, April.
    14. Lontzek, Thomas S. & Narita, Daiju, 2009. "The effect of uncertainty on decision making about climate change mitigation: a numerical approach of stochastic control," Kiel Working Papers 1539, Kiel Institute for the World Economy (IfW Kiel).
    15. Heinzel Christoph & Richard Peter, 2021. "Precautionary motives with multiple instruments," Working Papers SMART 21-09, INRAE UMR SMART.
    16. Martin, Jessica & Villeneuve, Stéphane, 2021. "A Class of Explicit optimal contracts in the face of shutdown," TSE Working Papers 21-1183, Toulouse School of Economics (TSE), revised Apr 2022.
    17. Ingham, Alan & Ma, Jie & Ulph, Alistair, 2007. "Climate change, mitigation and adaptation with uncertainty and learning," Energy Policy, Elsevier, vol. 35(11), pages 5354-5369, November.
    18. Giovanni Immordino, 2001. "Choosing between traditional and innovative technologies: the case of scientific uncertainty," CSEF Working Papers 74, Centre for Studies in Economics and Finance (CSEF), University of Naples, Italy.
    19. Narain, Urvashi & Hanemann, Michael & Fisher, Anthony C., 2002. "Uncertainty, Learning, and the Irreversibility Effect," CUDARE Working Papers 198691, University of California, Berkeley, Department of Agricultural and Resource Economics.
    20. Caroline Orset, 2014. "Innovation and the precautionary principle," Economics of Innovation and New Technology, Taylor & Francis Journals, vol. 23(8), pages 780-801, November.

    More about this item

    Keywords

    Self-Protection; Prevention effort; Dynamic programming; Continuation utility; Backward SDEs;
    All these keywords.

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    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:hal:wpaper:hal-02974961. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: . General contact details of provider: https://hal.archives-ouvertes.fr/ .

    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: CCSD (email available below). General contact details of provider: https://hal.archives-ouvertes.fr/ .

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

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.