Paper: arXiv 2309.03966

Abstract

This paper introduces FourNet, a novel single-layer feed-forward neural network (FFNN) method designed to approximate transition densities for which closed-form expressions of their Fourier transforms, i.e. characteristic functions, are available. A unique feature of FourNet lies in its use of a Gaussian activation function, enabling exact Fourier and inverse Fourier transformations and drawing analogies with the Gaussian mixture model. We mathematically establish FourNet’s capacity to approximate transition densities in the $L_2$-sense arbitrarily well with finite number of neurons. The parameters of FourNet are learned by minimizing a loss function derived from the known characteristic function and the Fourier transform of the FFNN, complemented by a strategic sampling approach to enhance training. We derive practical bounds for the $L_2$ estimation error and the potential pointwise loss of nonnegativity in FourNet for $d$-dimensions ($d\ge 1$), highlighting its robustness and applicability in practical settings. FourNet’s accuracy and versatility are demonstrated through a wide range of dynamics common in quantitative finance, including Lévy processes and the Heston stochastic volatility models-including those augmented with the self-exciting Queue-Hawkes jump process.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 6.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper demonstrates high mathematical complexity through rigorous $L_2$-analysis, universal approximation theorems, and multi-dimensional error bounds, while maintaining strong empirical rigor with validated demonstrations across diverse financial models (Lévy, Heston, Queue-Hawkes) and practical error controls for real-world applications.

Research Flowchart

  flowchart TD
  A["Research Goal"] --> B["FourNet Architecture"]
  A --> C["Input Data"]
  subgraph B ["Neural Network Structure"]
      B1["Gaussian Activation"]
      B2["Fourier Transform"]
  end
  subgraph C ["Financial Models"]
      C1["Levy Processes"]
      C2["Heston Model"]
      C3["Queue-Hawkes Jumps"]
  end
  B --> D["Training Process"]
  C --> D
  D --> E["Loss Function"]
  D --> F["Strategic Sampling"]
  E --> G["Minimization"]
  F --> G
  G --> H["Key Outcomes"]
  subgraph H ["Findings"]
      H1["Arbitrary L2 Approximation"]
      H2["Pointwise Error Bounds"]
      H3["Preserves Nonnegativity"]
  end