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Time-consistent mean-variance portfolio selection in discrete and continuous time


  • Christoph Czichowsky



It is well known that mean-variance portfolio selection is a time-inconsistent optimal control problem in the sense that it does not satisfy Bellman’s optimality principle and therefore the usual dynamic programming approach fails. We develop a time-consistent formulation of this problem, which is based on a local notion of optimality called local mean-variance efficiency, in a general semimartingale setting. We start in discrete time, where the formulation is straightforward, and then find the natural extension to continuous time. This complements and generalises the formulation by Basak and Chabakauri (2010) and the corresponding example in Björk and Murgoci (2010), where the treatment and the notion of optimality rely on an underlying Markovian framework. We justify the continuous-time formulation by showing that it coincides with the continuous-time limit of the discrete-time formulation. The proof of this convergence is based on a global description of the locally optimal strategy in terms of the structure condition and the Föllmer–Schweizer decomposition of the mean-variance trade-off. As a by-product, this also gives new convergence results for the Föllmer–Schweizer decomposition, i.e., for locally risk-minimising strategies. Copyright Springer-Verlag 2013

Suggested Citation

  • Christoph Czichowsky, 2013. "Time-consistent mean-variance portfolio selection in discrete and continuous time," Finance and Stochastics, Springer, vol. 17(2), pages 227-271, April.
  • Handle: RePEc:spr:finsto:v:17:y:2013:i:2:p:227-271
    DOI: 10.1007/s00780-012-0189-9

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    References listed on IDEAS

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    Cited by:

    1. Li, Yongwu & Qiao, Han & Wang, Shouyang & Zhang, Ling, 2015. "Time-consistent investment strategy under partial information," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 187-197.
    2. Sigrid Källblad, 2017. "Risk- and ambiguity-averse portfolio optimization with quasiconcave utility functionals," Finance and Stochastics, Springer, vol. 21(2), pages 397-425, April.
    3. Li, Bin & Li, Danping & Xiong, Dewen, 2016. "Alpha-robust mean-variance reinsurance-investment strategy," Journal of Economic Dynamics and Control, Elsevier, vol. 70(C), pages 101-123.
    4. Martijn Pistorius & Mitja Stadje, 2016. "On Dynamic Deviation Measures and Continuous-Time Portfolio Optimisation," Papers 1604.08037,
    5. repec:spr:fuzodm:v:17:y:2018:i:2:d:10.1007_s10700-017-9266-z is not listed on IDEAS
    6. Zhao, Qian & Shen, Yang & Wei, Jiaqin, 2014. "Consumption–investment strategies with non-exponential discounting and logarithmic utility," European Journal of Operational Research, Elsevier, vol. 238(3), pages 824-835.
    7. Zhou, Zhongbao & Xiao, Helu & Yin, Jialing & Zeng, Ximei & Lin, Ling, 2016. "Pre-commitment vs. time-consistent strategies for the generalized multi-period portfolio optimization with stochastic cash flows," Insurance: Mathematics and Economics, Elsevier, vol. 68(C), pages 187-202.
    8. Cong, F. & Oosterlee, C.W., 2016. "On pre-commitment aspects of a time-consistent strategy for a mean-variance investor," Journal of Economic Dynamics and Control, Elsevier, vol. 70(C), pages 178-193.
    9. Xiangyu Cui & Duan Li & Xun Li, 2014. "Mean-Variance Policy for Discrete-time Cone Constrained Markets: The Consistency in Efficiency and Minimum-Variance Signed Supermartingale Measure," Papers 1403.0718,
    10. Tomas Björk & Mariana Khapko & Agatha Murgoci, 2017. "On time-inconsistent stochastic control in continuous time," Finance and Stochastics, Springer, vol. 21(2), pages 331-360, April.
    11. Tomas Björk & Agatha Murgoci, 2014. "A theory of Markovian time-inconsistent stochastic control in discrete time," Finance and Stochastics, Springer, vol. 18(3), pages 545-592, July.

    More about this item


    Mean-variance criterion; Markowitz problem; Portfolio optimisation; Time consistency; Time-inconsistent optimal control; Local risk minimisation; Föllmer–Schweizer decomposition; Convergence of optimal trading strategies; 91G10; 93E20; 60G48; G11; C61;

    JEL classification:

    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions
    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis


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