Equilibrium strategies for stochastic control problems with higher-order moments and applications to portfolio selection

Paper: arXiv 2504.04113

Authors: Yike Wang, Jingzhen Liu, Jiaqin Wei

Abstract

In this paper we derive a novel characterization result for time-consistent stochastic control problems with higher-order moments, originally formulated by Wang et al. [SIAM J. Control. Optim., 63 (2025), 1560–1589], and newly explore many solvable instances including a mean-variance-excess kurtosis portfolio selection problem. By improving an asymptotic result of the variational process for the uniform boundedness and integrability properties, we obtain both the sufficiency and necessity of an equilibrium condition for an open-loop Nash equilibrium control (ONEC). This condition is simply formulated by the diagonal processes of a flow of backward stochastic differential equations (BSDEs) whose data do not necessarily satisfy the usual square-integrability condition. In particular, for linear controlled dynamics with deterministic parameters, we show that the ONEC can be derived by solving a polynomial algebraic equation under a class of nonlinear objective functions. Interestingly, the mean-variance equilibrium strategy is an ONEC for our general higher-order moment problem if and only if a homogeneity condition holds. Additionally, in the case with random parameters, we characterize the ONEC by finitely many BSDEs with a recurrence relation. As an intuitive illustration, the solution to the mean-variance-skewness problems is given by a quadratic BSDE.

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper offers a rigorous theoretical advancement in time-consistent stochastic control under higher-order moments using flows of BSDEs and variational arguments. However, it is purely mathematical with no empirical data analysis, backtests, or computational implementations provided.

Research Flowchart

  flowchart TD
    A["<b>Research Goal</b><br/>Characterize Open-Loop Nash Equilibrium Controls (ONEC) for time-consistent stochastic control with higher-order moments"] --> B["<b>Methodology & Inputs</b><br/>Controlled stochastic dynamics + Variational process with improved asymptotic boundedness"]
    
    B --> C["<b>Core Derivation</b><br/>Establish necessary & sufficient conditions via diagonal processes of a BSDE flow"]
    
    C --> D1["<b>Deterministic Parameters</b><br/>Reduce ONEC derivation to solving polynomial algebraic equations"]
    C --> D2["<b>Random Parameters</b><br/>Characterize ONEC via finitely many recursive BSDEs (e.g., Quadratic BSDE for Mean-Variance-Skewness)"]
    
    D1 --> E["<b>Key Findings & Outcomes</b><br/>1. Solved Mean-Variance-Excess Kurtosis portfolio problem<br/>2. Proved MV strategy is an ONEC for higher-order problems iff a homogeneity condition holds"]
    D2 --> E

Cited References (47)

1. A non-exponential discounting time-inconsistent stochastic optimal control problem for jump-diffusion (2019) — link
2. Time-inconsistent stochastic optimal control problems: a backward stochastic partial differential equations approach (2020) — link
3. On time-inconsistent stochastic control in continuous time (2017) — link
4. A theory of M arkovian time-inconsistent stochastic control in discrete time (2014) — link
5. Mean Variance Portfolio Optimization with State Dependent Risk Aversion (2014) — link
6. An optimal control approach to portfolio optimization with conditioning information (2010) — link
7. L$^P$ solutions of backward stochastic differential equations (2003) — link
8. Mean-Variance-Skewness Portfolio Performance Gauging: A General Shortage Function and Dual Approach (2007) — link

2 of 47 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.