Paper: arXiv 2603.11046

Authors: Emmanuel Gnabeyeu

Abstract

This paper is concerned with Merton’s portfolio optimization problem in a Volterra stochastic environment described by a multivariate fake stationary Volterra–Heston model. Due to the non-Markovianity and non-semimartingality of the underlying processes, the classical stochastic control approach cannot be directly applied in this setting. Instead, the problem is tackled using a stochastic factor solution to a Riccati backward stochastic differential equation (BSDE). Our approach is inspired by the martingale optimality principle combined with a suitable verification argument. The resulting optimal strategies for Merton’s problems are derived in semi-closed form depending on the solutions to time-dependent multivariate Riccati-Volterra equations, while the optimal value is expressed using the solution to this original Riccati BSDE. Numerical results on a two dimensional fake stationary rough Heston model illustrate the impact of stationary rough volatilities on the optimal Merton strategies.

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 formulation of utility maximization under multivariate fake stationary affine Volterra models using stochastic factor solutions to Riccati BSDEs. While the mathematical derivation and theoretical contributions are highly advanced, the paper focuses primarily on analytical proofs and offers limited empirical backtesting beyond brief numerical illustrations.

Research Flowchart

  flowchart TD
    A["Goal: Solve Merton Portfolio Optimization in Multivariate Volterra-Heston Model"] --> B["Inputs: Non-Markovian Asset Prices & Fake Stationary Rough Volatilities"]
    B --> C["Apply Martingale Optimality Principle & Verification Argument"]
    C --> D["Map Problem to Stochastic Factor Solution via Riccati BSDE"]
    D --> E["Compute Semi-Closed Form Strategies via Time-Dependent Riccati-Volterra Equations"]
    E --> F["Numerical Simulation: 2D Fake Stationary Rough Heston Model"]
    F --> G["Outcomes: Optimal Value Function & Impact of Rough Volatility on Strategies"]