On stochastic control problems with higher-order moments

Paper: arXiv 2412.13521

Authors: 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 Björk 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.

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 1.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a highly sophisticated theoretical framework for time-inconsistent stochastic control problems involving higher-order central moments, deriving new PDE systems and FBSDE flows. However, it is purely mathematical with no empirical backtesting, financial datasets, or applied numerical validation.

Research Flowchart

  flowchart TD
    A[Goal: Time-consistent stochastic control for higher-order central moments] --> B[Inputs: Controlled dynamics & objective function with mean/higher-order moments]
    
    B --> C[Closed-loop Equilibrium Analysis]
    B --> D[Open-loop Equilibrium Analysis]
    
    C --> E[Derive novel PDE system<br>Equivalent to extended HJB]
    D --> F[Apply perturbation & moment estimates<br>Solve flow of FBSDEs]
    
    E --> G[Findings: Identical controls independent of state, path, & odd moments; extended to infinite-order penalty problems]
    F --> G

Cited References (76)

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1 of 76 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.