On the mean-variance problem through the lens of multivariate fake stationary affine Volterra dynamics

Paper: arXiv 2604.01300

Authors: Emmanuel Gnabeyeu

Abstract

We investigate the continuous-time Markowitz mean-variance portfolio selection problem within a multivariate class of fake stationary affine Volterra models. In this non-Markovian and non-semimartingale market framework with unbounded random coefficients, the classical stochastic control approach cannot be directly applied to the associated optimization task. Instead, the problem is tackled using a stochastic factor solution to a Riccati backward stochastic differential equation (BSDE). The optimal feedback control is characterized by means of this equation, whose explicit solutions is derived in terms of multi-dimensional Riccati-Volterra equations. Specifically, we obtain analytical closed-form expressions for the optimal portfolio policies as well as the mean-variance efficient frontier, both of which depend on the solution to the associated multivariate Riccati-Volterra system. To illustrate our results, numerical experiments based on a two dimensional fake stationary rough Heston model highlight the impact of rough volatilities and stochastic correlations on the optimal Markowitz strategies.

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 3.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a highly sophisticated mathematical framework utilizing BSDEs and Riccati-Volterra equations to solve the mean-variance problem in non-semimartingale markets. While theoretically strong and novel in its continuous-time optimization approach, it lacks extensive empirical backtesting, relying primarily on stylized numerical simulations of a two-dimensional rough Heston model.

Research Flowchart

  flowchart TD
    A[Research Goal: Solve Markowitz Mean-Variance Problem under Multivariate Fake Stationary Affine Volterra Dynamics] --> B[Model Setup: Non-Markovian/Non-Semimartingale Market with Unbounded Random Coefficients]
    B --> C[Methodology: Formulate Riccati Backward Stochastic Differential Equation BSDE]
    C --> D[Analytical Solution: Derive Multi-Dimensional Riccati-Volterra System]
    D --> E[Outcomes: Closed-Form Optimal Portfolio Strategies & Efficient Frontier]
    E --> F[Numerical Experiments: 2D Fake Stationary Rough Heston Model]
    F --> G[Key Insight: Quantified Impact of Rough Volatility & Stochastic Correlation on Strategies]