Dynamic Mean-Variance Asset Allocation in General Incomplete Markets A Nonlocal BSDE-based Feedback Control Approach

Paper: arXiv 2412.18498

Authors: Qian Lei, Chi Seng Pun, Jingxiang Tang

Abstract

This paper studies dynamic mean-variance (MV) asset allocation problems in general incomplete markets. Besides of the conventional MV objective on portfolio’s terminal wealth, our framework can accommodate running MV objectives with general (non-exponential) discounting factors while in general, any time-dependent preferences. We attempt the problem with a game-theoretic framework while decompose the equilibrium control policies into two parts: the first part is a myopic strategy characterized by a linear Volterra integral equation of the second kind and the second part reveals the hedging demand governed by a system of nonlocal backward stochastic differential equations. We manage to establish the well-posedness of the solutions to the two aforementioned equations in tailored Bananch spaces by the fixed-point theorem. It allows us to devise a numerical scheme for solving for the equilibrium control policy with guarantee and to conclude that the dynamic (equilibrium) mean-variance policy in general settings is well-defined. Our probabilistic approach allows us to consider a board range of stochastic factor models, such as the Chan–Karolyi–Longstaff–Sanders (CKLS) model. For which, we verify all technical assumptions and provide a sound numerical scheme. Numerical examples are provided to illustrate our framework.

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical framework for time-inconsistent dynamic mean-variance problems in incomplete markets using nonlocal BSDEs and Volterra integral equations. However, it is fundamentally a theoretical applied mathematics contribution with minimal empirical application, relying primarily on synthetic numerical illustrations.

Research Flowchart

  flowchart TD
    A[Research Goal: Dynamic MV Asset Allocation in Incomplete Markets] --> B[Input: Market Model & Preferences<br/>Multi-Asset, CKLS, General Discounting]
    B --> C[Methodology: Game-Theoretic Framework]
    C --> D[Policy Decomposition]
    D --> E[1. Myopic Strategy<br/>Linear Volterra Integral Eq.]
    D --> F[2. Hedging Demand<br/>Nonlocal System of BSDEs]
    E & F --> G[Fixed-Point Theorem in Banach Spaces<br/>Well-Posedness & Numerical Scheme]
    G --> H[Outcomes: Guaranteed Well-Defined<br/>Equilibrium MV Policy & Numerical Proof]