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Asymptotics of the probability minimizing a "down-side" risk

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  • Hiroaki Hata
  • Hideo Nagai
  • Shuenn-Jyi Sheu

Abstract

We consider a long-term optimal investment problem where an investor tries to minimize the probability of falling below a target growth rate. From a mathematical viewpoint, this is a large deviation control problem. This problem will be shown to relate to a risk-sensitive stochastic control problem for a sufficiently large time horizon. Indeed, in our theorem we state a duality in the relation between the above two problems. Furthermore, under a multidimensional linear Gaussian model we obtain explicit solutions for the primal problem.

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  • Hiroaki Hata & Hideo Nagai & Shuenn-Jyi Sheu, 2010. "Asymptotics of the probability minimizing a "down-side" risk," Papers 1001.2131, arXiv.org.
    • Handle: RePEc:arx:papers:1001.2131
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    References listed on IDEAS

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    1. Michael Stutzer, 2011. "Portfolio choice with endogenous utility: a large deviations approach," World Scientific Book Chapters, in: Leonard C MacLean & Edward O Thorp & William T Ziemba (ed.), THE KELLY CAPITAL GROWTH INVESTMENT CRITERION THEORY and PRACTICE, chapter 43, pages 619-640, World Scientific Publishing Co. Pte. Ltd..
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    Cited by:

    1. Rongju Zhang & Nicolas Langren'e & Yu Tian & Zili Zhu & Fima Klebaner & Kais Hamza, 2017. "Skewed target range strategy for multiperiod portfolio optimization using a two-stage least squares Monte Carlo method," Papers 1704.00416, arXiv.org, revised Jun 2019.
    2. Hideo Nagai, 2011. "Asymptotics of the probability of minimizing 'down-side' risk under partial information," Quantitative Finance, Taylor & Francis Journals, vol. 11(5), pages 789-803.
    3. Ichihara, Naoyuki, 2012. "Large time asymptotic problems for optimal stochastic control with superlinear cost," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1248-1275.
    4. Hiroaki Hata, 2011. "“Down-Side Risk” Probability Minimization Problem with Cox-Ingersoll-Ross’s Interest Rates," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 18(1), pages 69-87, March.
    5. Huyen Pham, 2014. "Long time asymptotics for optimal investment," Papers 1408.6455, arXiv.org.
    6. Rongju Zhang & Nicolas Langrené & Yu Tian & Zili Zhu & Fima Klebaner & Kais Hamza, 2019. "Skewed target range strategy for multiperiod portfolio optimization using a two-stage least squares Monte Carlo method," Post-Print hal-02909342, HAL.
    7. Anatolii A. Puhalskii & Michael Jay Stutzer, 2016. "On minimising a portfolio's shortfall probability," Papers 1602.02192, arXiv.org, revised May 2017.
    8. Tadashi Hayashi & Jun Sekine, 2011. "Risk-sensitive Portfolio Optimization with Two-factor Having a Memory Effect," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 18(4), pages 385-403, November.
    9. Watanabe, Yûsuke, 2013. "Asymptotic analysis for a downside risk minimization problem under partial information," Stochastic Processes and their Applications, Elsevier, vol. 123(3), pages 1046-1082.
    10. Huyen Pham, 2014. "Long time asymptotics for optimal investment," Working Papers hal-01058657, HAL.

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