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On stochastic control problems with higher-order moments

Author

Listed:
  • Yike Wang
  • Jingzhen Liu
  • Alain Bensoussan
  • Ka-Fai Cedric Yiu
  • Jiaqin Wei

Abstract

In this paper, we focus on a class of time-inconsistent stochastic control problems, where the objective function includes the mean and several higher-order central moments of the terminal value of state. To tackle the time-inconsistency, we seek both the closed-loop and the open-loop Nash equilibrium controls as time-consistent solutions. We establish a partial differential equation (PDE) system for deriving a closed-loop Nash equilibrium control, which does not include the equilibrium value function and is different from the extended Hamilton-Jacobi-Bellman (HJB) equations as in Bj\"ork et al. (Finance Stoch. 21: 331-360, 2017). We show that our PDE system is equivalent to the extended HJB equations that seems difficult to be solved for our higher-order moment problems. In deriving an open-loop Nash equilibrium control, due to the non-separable higher-order moments in the objective function, we make some moment estimates in addition to the standard perturbation argument for developing a maximum principle. Then, the problem is reduced to solving a flow of forward-backward stochastic differential equations. In particular, we investigate linear controlled dynamics and some objective functions affine in the mean. The closed-loop and the open-loop Nash equilibrium controls are identical, which are independent of the state value, random path and the preference on the odd-order central moments. By sending the highest order of central moments to infinity, we obtain the time-consistent solutions to some control problems whose objective functions include some penalty functions for deviation.

Suggested Citation

  • Yike Wang & Jingzhen Liu & Alain Bensoussan & Ka-Fai Cedric Yiu & Jiaqin Wei, 2024. "On stochastic control problems with higher-order moments," Papers 2412.13521, arXiv.org, revised Jan 2025.
    • Handle: RePEc:arx:papers:2412.13521
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    References listed on IDEAS

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    1. Joro, Tarja & Na, Paul, 2006. "Portfolio performance evaluation in a mean-variance-skewness framework," European Journal of Operational Research, Elsevier, vol. 175(1), pages 446-461, November.
    2. Kraus, Alan & Litzenberger, Robert H, 1976. "Skewness Preference and the Valuation of Risk Assets," Journal of Finance, American Finance Association, vol. 31(4), pages 1085-1100, September.
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    Cited by:

    1. Carole Bernard & Jinghui Chen & Steven Vanduffel, 2025. "Higher moments under dependence uncertainty with applications in insurance," Papers 2508.16600, arXiv.org.
    2. Yue Cao & Zongxia Liang & Sheng Wang & Xiang Yu, 2025. "Equilibrium Portfolio Selection under Utility-Variance Analysis of Log Returns in Incomplete Markets," Papers 2511.05861, arXiv.org, revised Nov 2025.

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