Paper: arXiv 2605.00688

Authors: Sigui Brice Dro, Emmanuel Gnabeyeu

Abstract

This paper is concerned with portfolio selection for an investor with exponential, power, and logarithmic utility in multi-asset financial markets allowing jumps. We investigate the classical Merton’s portfolio optimization problem in a Volterra stochastic environment described by a multivariate Volterra–Heston model with jumps driven by an independent Poisson random measure. Owing to the non-Markovian and non-semimartingale nature of the model, classical stochastic control techniques are not directly applicable. Instead, the problem is tackled using the martingale optimality principle by constructing a family of supermartingale processes characterized via solutions to an original Riccati backward stochastic differential equation with jumps (Riccati BSDEJ).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 BSDEJ. Numerical experiments on a two-dimensional rough Heston model illustrate the impact of both path roughness and jumps components on the value function and optimal strategies in the Merton problem.

Complexity vs Empirical Score

  • Math Complexity: 9.2/10
  • Empirical Rigor: 3.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents rigorous and advanced mathematical proofs constructing a family of supermartingales and original Riccati BSDEJs to solve Merton’s problem under non-Markovian rough Volterra dynamics. However, its empirical rigor is limited primarily to synthetic numerical simulations on a two-dimensional model rather than backtesting against historical market data.

Research Flowchart

  flowchart TD
    A["Research Goal: Optimal Merton Portfolio Optimization under Non-Markovian Volterra-Heston Model with Jumps"] --> B["Inputs: Multi-asset market model with roughness parameters, jump distributions & investor utility (Exp, Power, Log)"]
    B --> C["Methodology: Apply Martingale Optimality Principle to bypass non-semimartingale limitations"]
    C --> D["Computation: Formulate & solve original Riccati Backward Stochastic Differential Equation with Jumps (BSDEJ)"]
    D --> E["Computation: Solve time-dependent multivariate Riccati-Volterra integral equations"]
    E --> F["Outcomes: Semi-closed form optimal trading strategies & closed-form value functions"]
    F --> G["Findings: Roughness (H) and jump components significantly alter optimal asset allocations and utility value"]