Paper: arXiv 2511.05861

Authors: Yue Cao, Zongxia Liang, Sheng Wang, Xiang Yu

Abstract

This paper investigates a time-inconsistent portfolio selection problem in the incomplete mar ket model, integrating expected utility maximization with risk control. The objective functional balances the expected utility and variance on log returns, giving rise to time inconsistency and motivating the search of a time-consistent equilibrium strategy. We characterize the equilibrium via a coupled quadratic backward stochastic differential equation (BSDE) system and establish the existence theory in two special cases: (i)the two Brownian motions driven the price dynamics and the factor process are independent with $ρ= 0$; (ii) the trading strategy is constrained to be bounded. For the general case with correlation coefficient $ρ\neq 0$, we introduce the notion of an approximate time-consistent equilibrium. Employing the solution structure from the equilibrium in the case $ρ= 0$, we can construct an approximate time-consistent equilibrium in the general case with an error of order $O(ρ^2)$. Numerical examples and financial insights are also presented based on deep learning algorithms.

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 5.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper relies heavily on advanced stochastic control theory, backward stochastic differential equations (BSDEs), and theoretical approximations (O(ρ²) error), indicating high mathematical complexity. While it mentions deep learning algorithms for numerical examples, the primary focus is theoretical existence and approximation results without providing code, backtests, or statistical metrics typical of high empirical rigor.

Research Flowchart

  flowchart TD
  A["Research Goal:<br>Time-consistent equilibrium strategy<br>for portfolio selection with utility-variance analysis"] --> B["Methodology: Coupled Quadratic BSDE System"]
  
  B --> C["Case Analysis 1:<br>ρ = 0 (Independent Brownian motions)<br>Full Equilibrium Established"]
  
  B --> D["Case Analysis 2:<br>Constrained Bounded Strategy<br>Equilibrium Established"]
  
  B --> E["General Case: ρ ≠ 0<br>Approximate Time-consistent Equilibrium<br>Error Order O(ρ²)"]
  
  C --> F["Computational Process:<br>Deep Learning Algorithms<br>for BSDE Solvers"]
  D --> F
  E --> F
  
  F --> G["Key Findings/Outcomes:<br>1. Numerical Solutions for Equilibrium Strategies<br>2. Financial Insights on Risk Control<br>3. Practical Implementation via Deep Learning"]