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Global adapted solution of one-dimensional backward stochastic Riccati equations, with application to the mean-variance hedging

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  • Kohlmann, Michael
  • Tang, Shanjian

Abstract

Backward stochastic Riccati equations are motivated by the solution of general linear quadratic optimal stochastic control problems with random coefficients, and the solution has been open in the general case. One distinguishing difficult feature is that the drift contains a quadratic term of the second unknown variable. In this paper, we obtain the global existence and uniqueness result for a general one-dimensional backward stochastic Riccati equation. This solves the one-dimensional case of Bismut-Peng's problem which was initially proposed by Bismut (Lecture Notes in Math. 649 (1978) 180). We use an approximation technique by constructing a sequence of monotone drifts and then passing to the limit. We make full use of the special structure of the underlying Riccati equation. The singular case is also discussed. Finally, the above results are applied to solve the mean-variance hedging problem with general random market conditions.

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  • Kohlmann, Michael & Tang, Shanjian, 2002. "Global adapted solution of one-dimensional backward stochastic Riccati equations, with application to the mean-variance hedging," Stochastic Processes and their Applications, Elsevier, vol. 97(2), pages 255-288, February.
    • Handle: RePEc:eee:spapps:v:97:y:2002:i:2:p:255-288
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    1. Lepeltier, J. P. & San Martin, J., 1997. "Backward stochastic differential equations with continuous coefficient," Statistics & Probability Letters, Elsevier, vol. 32(4), pages 425-430, April.
    2. Stein, Elias M & Stein, Jeremy C, 1991. "Stock Price Distributions with Stochastic Volatility: An Analytic Approach," The Review of Financial Studies, Society for Financial Studies, vol. 4(4), pages 727-752.
    3. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    4. Christian Gourieroux & Jean Paul Laurent & Huyên Pham, 1998. "Mean‐Variance Hedging and Numéraire," Mathematical Finance, Wiley Blackwell, vol. 8(3), pages 179-200, July.
    5. Martin Schweizer & HuyËn Pham & (*), Thorsten RheinlÄnder, 1998. "Mean-variance hedging for continuous processes: New proofs and examples," Finance and Stochastics, Springer, vol. 2(2), pages 173-198.
    6. Hull, John C & White, Alan D, 1987. "The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
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    Cited by:

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    2. Masaaki Fujii & Akihiko Takahashi, 2013. "Making Mean-Variance Hedging Implementable in a Partially Observable Market -with supplementary contents for stochastic interest rates-," CARF F-Series CARF-F-332, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    3. Andrew E. B. Lim, 2004. "Quadratic Hedging and Mean-Variance Portfolio Selection with Random Parameters in an Incomplete Market," Mathematics of Operations Research, INFORMS, vol. 29(1), pages 132-161, February.
    4. Xiaofei Shi & Daran Xu & Zhanhao Zhang, 2023. "Deep learning algorithms for hedging with frictions," Digital Finance, Springer, vol. 5(1), pages 113-147, March.
    5. Mitsui, Ken-ichi & Tabata, Yoshio, 2008. "A stochastic linear-quadratic problem with Lévy processes and its application to finance," Stochastic Processes and their Applications, Elsevier, vol. 118(1), pages 120-152, January.
    6. Masaaki Fujii & Akihiko Takahashi, 2013. "Making Mean-Variance Hedging Implementable in a Partially Observable Market -with supplementary contents for stochastic interest rates-," CIRJE F-Series CIRJE-F-891, CIRJE, Faculty of Economics, University of Tokyo.
    7. Masaaki Fujii & Akihiko Takahashi, 2014. "Optimal Hedging for Fund & Insurance Managers with Partially Observable Investment Flows," CARF F-Series CARF-F-338, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    8. Shaolin Ji & Hanqing Jin & Xiaomin Shi, 2017. "Mean-variance portfolio selection with nonlinear wealth dynamics and random coefficients," Papers 1705.06141, arXiv.org, revised Nov 2022.
    9. Masaaki Fujii & Akihiko Takahashi, 2014. "Optimal Hedging for Fund & Insurance Managers with Partially Observable Investment Flows," CIRJE F-Series CIRJE-F-914, CIRJE, Faculty of Economics, University of Tokyo.
    10. Xiaofei Shi & Daran Xu & Zhanhao Zhang, 2021. "Deep Learning Algorithms for Hedging with Frictions," Papers 2111.01931, arXiv.org, revised Dec 2022.
    11. Ying Hu & Xiaomin Shi & Zuo Quan Xu, 2022. "Non-homogeneous stochastic LQ control with regime switching and random coefficients," Papers 2201.01433, arXiv.org, revised Jul 2023.
    12. Masaaki Fujii & Akihiko Takahashi, 2014. "Optimal Hedging for Fund & Insurance Managers with Partially Observable Investment Flows," CARF F-Series CARF-F-348, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    13. Li, Bo & Huang, Tian, 2022. "Control variable parameterization and optimization method for stochastic linear quadratic models," Chaos, Solitons & Fractals, Elsevier, vol. 154(C).
    14. Johannes Muhle-Karbe & Marcel Nutz & Xiaowei Tan, 2019. "Asset Pricing with Heterogeneous Beliefs and Illiquidity," Papers 1905.05730, arXiv.org, revised Mar 2020.
    15. Bruno Bouchard & Masaaki Fukasawa & Martin Herdegen & Johannes Muhle-Karbe, 2018. "Equilibrium Returns with Transaction Costs," Post-Print hal-01569408, HAL.
    16. Masaaki Fujii & Akihiko Takahashi, 2014. "Optimal Hedging for Fund & Insurance Managers with Partially Observable Investment Flows," Papers 1401.2314, arXiv.org, revised Jul 2014.
    17. Lukas Gonon & Johannes Muhle-Karbe & Xiaofei Shi, 2019. "Asset Pricing with General Transaction Costs: Theory and Numerics," Papers 1905.05027, arXiv.org, revised Apr 2020.
    18. Johannes Muhle-Karbe & Xiaofei Shi & Chen Yang, 2020. "An Equilibrium Model for the Cross-Section of Liquidity Premia," Papers 2011.13625, arXiv.org.
    19. Bruno Bouchard & Masaaki Fukasawa & Martin Herdegen & Johannes Muhle-Karbe, 2018. "Equilibrium returns with transaction costs," Finance and Stochastics, Springer, vol. 22(3), pages 569-601, July.
    20. Chonghu Guan & Xiaomin Shi & Zuo Quan Xu, 2022. "Continuous-time Markowitz's mean-variance model under different borrowing and saving rates," Papers 2201.00914, arXiv.org, revised May 2023.
    21. Martin Herdegen & Johannes Muhle-Karbe & Dylan Possamai, 2019. "Equilibrium Asset Pricing with Transaction Costs," Papers 1901.10989, arXiv.org, revised Sep 2020.

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