IDEAS home Printed from https://ideas.repec.org/p/cfi/fseres/cf507.html
   My bibliography  Save this paper

Supplementary file for "Sup-inf/inf-sup problem on choice of a probability measure by FBSDE approach (Forthcoming in IEEE Transactions on Automatic Control)"

Author

Listed:
  • Taiga Saito

    (Graduate School of Economics, The University of Tokyo)

  • Akihiko Takahashi

    (Graduate School of Economics, The University of Tokyo)

Abstract

This paper presents a problem on model uncertainties in stochastic control, in which an agent assumes a best case scenario on one risk and at the same time a worst case scenario on another risk. Particularly, the agent maximizes its view on a Brownian motion, simultaneously minimizing its view on another Brownian motion in choice of a probability measure. This selection method of a probability measure generalizes an approach to model uncertainties in which one considers the worst case scenarios for the views on Brownian motions, such as in the robust control. Specifically, we newly formulate and solve this problem based on a backward stochastic differential equation (BSDE) approach as a sup-inf (resp., inf-sup) optimal control problem on choice of a probability measure with the control domains dependent on stochastic processes. Concretely, we show that under certain conditions, the sup-inf and inf-sup problems are equivalent and these are solved by finding a solution of a BSDE with a stochastic Lipschitz driver. Then, we investigate two cases in which the optimal probability measure is explicitly obtained. The expression of the optimal probability measure includes signs of the diffusion terms of the value process, which are hard to determine in general. In these cases, we show two methods of determining the signs: the first one is by comparison theorems, and the second one is to predetermine the signs a priori and confirm them afterwards by explicitly solving the corresponding equations.

Suggested Citation

  • Taiga Saito & Akihiko Takahashi, 2021. "Supplementary file for "Sup-inf/inf-sup problem on choice of a probability measure by FBSDE approach (Forthcoming in IEEE Transactions on Automatic Control)"," CARF F-Series CARF-F-507, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
  • Handle: RePEc:cfi:fseres:cf507
    as

    Download full text from publisher

    File URL: https://www.carf.e.u-tokyo.ac.jp/admin/wp-content/uploads/2021/02/F507.pdf
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Hisashi Nakamura & Wataru Nozawa & Akihiko Takahashi, 2009. "Macroeconomic Implications of Term Structures of Interest Rates Under Stochastic Differential Utility with Non-Unitary EIS," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 16(3), pages 231-263, September.
    2. Buckdahn, Rainer & Hu, Ying & Li, Juan, 2011. "Stochastic representation for solutions of Isaacs’ type integral–partial differential equations," Stochastic Processes and their Applications, Elsevier, vol. 121(12), pages 2715-2750.
    3. Kiyohiko G. Nishimura & Seisho Sato & Akihiko Takahashi, 2019. "Term Structure Models During the Global Financial Crisis: A Parsimonious Text Mining Approach," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 26(3), pages 297-337, September.
    4. Lars Peter Hansen & Thomas J Sargent, 2014. "Robust Control and Model Uncertainty," World Scientific Book Chapters, in: UNCERTAINTY WITHIN ECONOMIC MODELS, chapter 5, pages 145-154, World Scientific Publishing Co. Pte. Ltd..
    5. Zengjing Chen & Larry Epstein, 2002. "Ambiguity, Risk, and Asset Returns in Continuous Time," Econometrica, Econometric Society, vol. 70(4), pages 1403-1443, July.
    6. Hisashi Nakamura & Keita Nakayama & Akihiko Takahashi, 2008. "Term Structure of Interest Rates under Recursive Preferences in Continuous Time," CIRJE F-Series CIRJE-F-540, CIRJE, Faculty of Economics, University of Tokyo.
    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. Taiga Saito & Akihiko Takahashi, 2021. "Supplementary File for "Sup-Inf/Inf-Sup Problem on Choice of a Probability Measure by FBSDE Approach"," CIRJE F-Series CIRJE-F-1160, CIRJE, Faculty of Economics, University of Tokyo.
    2. Taiga Saito & Akihiko Takahashi, 2019. "A novel approach to asset pricing with choice of probability measures," CARF F-Series CARF-F-471, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Jan 2021.
    3. Taiga Saito & Akihiko Takahashi, 2019. "A Novel Approach to Asset Pricing with Choice of Probability Measures," CIRJE F-Series CIRJE-F-1131, CIRJE, Faculty of Economics, University of Tokyo.
    4. Keisuke Kizaki & Taiga Saito & Akihiko Takahashi, 2024. "Multi-agent Equilibrium Model with Heterogeneous Views on Fundamental Risks in Incomplete Market," CIRJE F-Series CIRJE-F-1224, CIRJE, Faculty of Economics, University of Tokyo.
    5. Keisuke Kizaki & Taiga Saito & Akihiko Takahashi, 2023. "Multi-agent Robust Optimal Investment Problem in Incomplete Market," CARF F-Series CARF-F-575, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    6. Keisuke Kizaki & Taiga Saito & Akihiko Takahashi, 2022. "Multi-agent Robust Optimal Investment Problem in Incomplete Market," CIRJE F-Series CIRJE-F-1198, CIRJE, Faculty of Economics, University of Tokyo.
    7. Hansen, Lars Peter & Sargent, Thomas J., 2022. "Structured ambiguity and model misspecification," Journal of Economic Theory, Elsevier, vol. 199(C).
    8. Aït-Sahalia, Yacine & Matthys, Felix, 2019. "Robust consumption and portfolio policies when asset prices can jump," Journal of Economic Theory, Elsevier, vol. 179(C), pages 1-56.
    9. Maillet, Bertrand & Tokpavi, Sessi & Vaucher, Benoit, 2015. "Global minimum variance portfolio optimisation under some model risk: A robust regression-based approach," European Journal of Operational Research, Elsevier, vol. 244(1), pages 289-299.
    10. Yehuda Izhakian, 2012. "Ambiguity Measurement," Working Papers 12-01, New York University, Leonard N. Stern School of Business, Department of Economics.
    11. Raman Uppal & Tan Wang, 2003. "Model Misspecification and Underdiversification," Journal of Finance, American Finance Association, vol. 58(6), pages 2465-2486, December.
    12. Hui Chen & Nengjiu Ju & Jianjun Miao, 2014. "Dynamic Asset Allocation with Ambiguous Return Predictability," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 17(4), pages 799-823, October.
    13. Dirk Becherer & Klebert Kentia, 2017. "Good Deal Hedging and Valuation under Combined Uncertainty about Drift and Volatility," Papers 1704.02505, arXiv.org.
    14. Souta Nakatani & Kiyohiko G. Nishimura & Taiga Saito & Akihiko Takahashi, 2020. "Interest Rate Model with Investor Attitude and Text Mining (Published in IEEE Access)," CARF F-Series CARF-F-479, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    15. repec:pri:metric:wp051_2013_hansen_scheinkman_stochastic-compounding-and-uncertain-valuati is not listed on IDEAS
    16. Ortoleva, Pietro, 2010. "Status quo bias, multiple priors and uncertainty aversion," Games and Economic Behavior, Elsevier, vol. 69(2), pages 411-424, July.
    17. Jianjun Miao, 2009. "Ambiguity, Risk and Portfolio Choice under Incomplete Information," Annals of Economics and Finance, Society for AEF, vol. 10(2), pages 257-279, November.
    18. Shuang Li & Haijun Wang, 2023. "Robust irreversible investment strategy with ambiguity to jump and diffusion risk," International Review of Finance, International Review of Finance Ltd., vol. 23(3), pages 645-665, September.
    19. Izhakian, Yehuda, 2020. "A theoretical foundation of ambiguity measurement," Journal of Economic Theory, Elsevier, vol. 187(C).
    20. Itzhak Ben-David & John R. Graham & Campbell R. Harvey, 2007. "Managerial Overconfidence and Corporate Policies," NBER Working Papers 13711, National Bureau of Economic Research, Inc.
    21. Mingli Zheng & Chong Wang & Chaozheng Li, 2016. "Insurance Contracts with Adverse Selection When the Insurer Has Ambiguity about the Composition of the Consumers," Annals of Economics and Finance, Society for AEF, vol. 17(1), pages 179-206, May.

    More about this item

    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:cfi:fseres:cf507. 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: the person in charge (email available below). General contact details of provider: https://edirc.repec.org/data/catokjp.html .

    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.