Equilibrium Portfolio Selection under Utility-Variance Analysis of Log Returns in Incomplete Markets

Battle Count: 66

The paper directly addresses portfolio selection under realistic market conditions (incomplete markets with stochastic factors). The utility-variance framework on log returns is highly relevant for practical portfolio management. The time-consistent equilibrium concept addresses the behavioral issue of strategy re-optimization. The numerical results provide actionable insights on how investment ratios change with correlation, risk aversion, and variance weighting. However, the theoretical complexity and the restriction to one risky asset somewhat limit immediate practical implementation.

The paper involves highly advanced mathematics: coupled quadratic BSDE systems, perturbation methods, game-theoretic equilibrium concepts, PDE theory (quasi-linear parabolic equations), and deep learning for numerical solutions. Implementing the full theoretical framework requires expertise in stochastic analysis, BSDE theory, and deep learning. The numerical scheme requires careful handling of the Euler discretization, neural network training with batch normalization, and multi-stage learning rate schedules. The approximate equilibrium construction for ρ≠0 adds additional complexity.

The paper provides detailed mathematical formulations, parameter settings for numerical experiments (r=0.017, σ=0.15, X₀=0.273, λ=0.27, ν=0.065, ζ=1, γ=0.1, T=2), neural network architecture (1 input, 2 hidden layers of 11 dimensions, 1 output), and training details (Adam optimizer, batch size 512, 5000 epochs, four-stage learning rate schedule). However, no code repository is provided, and the deep learning implementation details (exact initialization, random seeds) are not fully specified.