Low-dimensional approximations of the conditional law of Volterra processes: a non-positive curvature approach

ArXiv ID: 2405.20094 “View on arXiv”

Authors: Unknown

Abstract

Predicting the conditional evolution of Volterra processes with stochastic volatility is a crucial challenge in mathematical finance. While deep neural network models offer promise in approximating the conditional law of such processes, their effectiveness is hindered by the curse of dimensionality caused by the infinite dimensionality and non-smooth nature of these problems. To address this, we propose a two-step solution. Firstly, we develop a stable dimension reduction technique, projecting the law of a reasonably broad class of Volterra process onto a low-dimensional statistical manifold of non-positive sectional curvature. Next, we introduce a sequentially deep learning model tailored to the manifold’s geometry, which we show can approximate the projected conditional law of the Volterra process. Our model leverages an auxiliary hypernetwork to dynamically update its internal parameters, allowing it to encode non-stationary dynamics of the Volterra process, and it can be interpreted as a gating mechanism in a mixture of expert models where each expert is specialized at a specific point in time. Our hypernetwork further allows us to achieve approximation rates that would seemingly only be possible with very large networks.

Keywords: Volterra Processes, Stochastic Volatility, Dimensionality Reduction, Deep Learning, Mathematical Finance

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 4.0/10
• Quadrant: Lab Rats
• Why: The paper presents dense, advanced mathematics including Riemannian geometry, non-positive curvature, and universal approximation theorems for Volterra processes, with minimal empirical validation limited to an ablation study without backtests or real financial data.

  flowchart TD
    A["Research Goal<br>Predict conditional evolution<br>of Volterra processes"] --> B{"Key Challenge<br>Curse of dimensionality<br>from infinite, non-smooth paths"}
    B --> C["Step 1: Dimensionality Reduction<br>Project law to low-dim manifold<br>of non-positive curvature"]
    C --> D["Step 2: Sequential DL Model<br>Geometry-aware architecture"]
    D --> E["Hypernetwork & Gating<br>Dynamically updates parameters<br>Mixture of experts for time"]
    E --> F["Computational Process<br>Train model on manifold<br>to approximate conditional law"]
    F --> G["Key Findings/Outcomes<br>Stable reduction<br>High approximation rates<br>without large networks"]