S-shaped Utility Maximization with VaR Constraint and Partial Information

Paper: arXiv 2506.10103

Authors: Dongmei Zhu, Ashley Davey, Harry Zheng

Abstract

We study S-shaped utility maximisation with VaR constraint and unobservable drift coefficient. Using the Bayesian filter, the concavification principle, and the change of measure, we give a semi-closed integral representation for the dual value function and find a critical wealth level that determines if the constrained problem admits a unique optimal solution and Lagrange multiplier or is infeasible. We also propose three algorithms (Lagrange, simulation, deep neural network) to solve the problem and compare their performances with numerical examples.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical formulation combining Bayesian filtering, dynamic dual control, and concavification principles under partial information. However, its empirical rigor is limited primarily to synthetic numerical simulations and numerical algorithm comparisons rather than real-world historical market backtests.

Research Flowchart

  flowchart TD
    A[Goal: S-Shaped Utility Maximization with VaR Constraint & Unobservable Drift] --> B[Inputs: Asset Prices, Partial Info, Risk Tolerances & VaR Limit]
    B --> C[Bayesian Filter & Change of Measure<br/><i>Transform to Full Info Model</i>]
    C --> D[Concavification Principle & Dual Theory<br/><i>Derive Semi-Closed Dual Value Function</i>]
    D --> E[Check Critical Wealth Threshold<br/><i>Determine Problem Feasibility</i>]
    E --> F[Algorithm Execution & Benchmarking<br/><i>Lagrange, Simulation, DNN</i>]
    F --> G[Outcomes: Optimal Investment Strategy & Algorithm Performance Comparison]