Kernel Learning for Mean-Variance Trading Strategies

ArXiv ID: 2507.10701 “View on arXiv”

Authors: Owen Futter, Nicola Muca Cirone, Blanka Horvath

Abstract

In this article, we develop a kernel-based framework for constructing dynamic, pathdependent trading strategies under a mean-variance optimisation criterion. Building on the theoretical results of (Muca Cirone and Salvi, 2025), we parameterise trading strategies as functions in a reproducing kernel Hilbert space (RKHS), enabling a flexible and non-Markovian approach to optimal portfolio problems. We compare this with the signature-based framework of (Futter, Horvath, Wiese, 2023) and demonstrate that both significantly outperform classical Markovian methods when the asset dynamics or predictive signals exhibit temporal dependencies for both synthetic and market-data examples. Using kernels in this context provides significant modelling flexibility, as the choice of feature embedding can range from randomised signatures to the final layers of neural network architectures. Crucially, our framework retains closed-form solutions and provides an alternative to gradient-based optimisation.

Keywords: Reproducing Kernel Hilbert Space (RKHS), Mean-variance optimization, Path-dependent strategies, Signature-based frameworks, Non-Markovian optimization

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 8.0/10
• Quadrant: Holy Grail
• Why: The paper employs advanced mathematics including reproducing kernel Hilbert spaces, kernel trick, and path-dependent PDEs for optimization, while also demonstrating robust empirical validation with synthetic data and market data backtests that explicitly compare against baseline methods.

  flowchart TD
    A["Research Goal: Develop a kernel-based framework for dynamic,<br>path-dependent trading strategies under mean-variance optimisation"] --> B{"Methodology"};
    B --> C["Parameterise strategies as functions in<br>Reproducing Kernel Hilbert Space RKHS"];
    C --> D["Compare with Signature-based<br>framework of Futter et al."];
    D --> E["Computational Process:<br>Derive closed-form solutions vs gradient-based methods"];
    
    subgraph Data/Inputs
        F["Synthetic Asset Dynamics<br>Market Data with Temporal Dependencies"]
    end
    
    E <--> F;
    F --> G["Key Findings/Outcomes"];
    
    G --> H["RKHS & Signatures significantly<br>outperform classical Markovian methods"];
    G --> I["Kernels provide modelling flexibility<br>via feature embeddings e.g., randomised signatures, NN layers"];
    G --> J["Framework retains closed-form solutions<br>alternative to gradient-based optimisation"];