Paper: arXiv 2406.10214

Authors: Francesca Biagini, Lukas Gonon, Niklas Walter

Abstract

Randomised signature has been proposed as a flexible and easily implementable alternative to the well-established path signature. In this article, we employ randomised signature to introduce a generative model for financial time series data in the spirit of reservoir computing. Specifically, we propose a novel Wasserstein-type distance based on discrete-time randomised signatures. This metric on the space of probability measures captures the distance between (conditional) distributions. Its use is justified by our novel universal approximation results for randomised signatures on the space of continuous functions taking the underlying path as an input. We then use our metric as the loss function in a non-adversarial generator model for synthetic time series data based on a reservoir neural stochastic differential equation. We compare the results of our model to benchmarks from the existing literature.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 7.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides rigorous mathematical foundations, including novel universal approximation proofs for randomized path signatures, and applies them to synthetic financial data generation using a Neural SDE framework. The empirical evaluation is solid, supported by out-of-sample benchmark comparisons, and open-source code is provided for full reproducibility.

Research Flowchart

  flowchart TD
    A["Research Goal: Generative Model for Financial Time Series"] --> B["Input: Financial Time Series Data & Paths"]
    B --> C["Theory: Universal Approximation for Discrete Randomised Signatures"]
    C --> D["Metric: Formulate Signature-based Wasserstein-type Distance"]
    D --> E["Model Training: Reservoir Neural SDE with Signature Loss"]
    E --> F["Outcomes: Superior Synthetic Data Quality vs Literature Benchmarks"]

Cited References (72)

  1. Generating synthetic data in finance: opportunities, challenges and pitfalls (2020)
  2. Generative Machine Learning for Multivariate Equity Returns — → 2311.14735
  3. C-RNN-GAN: Continuous recurrent neural networks with adversarial training (2016)
  4. CORRGAN: Sampling Realistic Financial Correlation Matrices Using Generative Adversarial Networks (2019)
  5. Modeling financial time-series with generative adversarial networks (2019)
  6. Multi-Asset Spot and Option Market Simulation (2021)
  7. Time-series Generative Adversarial Networks (2019)
  8. The Market Generator (2019)
and 64 more
  1. Arbitrage-Free Neural-SDE Market Models (2023)
  2. Simulating financial time series using attention (2022)
  3. Generating Financial Markets With Signatures (2020)
  4. Wavelet Score-Based Generative Modeling (2022)
  5. COT-GAN: Generating Sequential Data via Causal Optimal Transport (2020)
  6. Sig‐Wasserstein GANs for conditional time series generation (2023)
  7. Generative adversarial networks for financial trading strategies fine-tuning and combination (2021)
  8. Neural Stochastic Differential Equations for Conditional Time Series Generation Using the Signature-Wasserstein-1 Metric (2023)
  9. Towards realistic market simulations: a generative adversarial networks approach (2021)
  10. Reservoir Computing Universality With Stochastic Inputs (2020)
  11. Deep Hedging: Learning to Simulate Equity Option Markets (2019)
  12. Scenario Generation for Market Risk Models Using Generative Neural Networks (2022)
  13. W asserstein Generative Adversarial Networks (2017)
  14. Sig-wasserstein GANs for time series generation (2021)
  15. Uniqueness for the signature of a path of bounded variation and the reduced path group (2010)
  16. Generative Adversarial Nets (2014)
  17. Quant GANs: deep generation of financial time series (2020)
  18. A neural network-based framework for financial model calibration (2019)
  19. Real-valued (Medical) Time Series Generation with Recurrent Conditional GANs (2017)
  20. A Data-Driven Market Simulator for Small Data Environments (2020)
  21. Information Processing in Dynamical Systems: Foundations of Harmony Theory (1986)
  22. Generative Modeling for Time Series Via Schrödinger Bridge (2023) — link
  23. Expressive power of randomized signature (2021)
  24. On the effectiveness of Randomized Signatures as Reservoir for Learning Rough Dynamics (2023)
  25. Discrete-Time Signatures and Randomness in Reservoir Computing (2020)
  26. Embedding and approximation theorems for echo state networks (2020)
  27. Approximation by superpositions of a sigmoidal function (1989)
  28. Multilayer feedforward networks with a nonpolynomial activation function can approximate any function (1993)
  29. Learning from the past, predicting the statistics for the future, learning an evolving system (2013) — link
  30. Universal Approximation Using Incremental Constructive Feedforward Networks With Random Hidden Nodes (2006)
  31. Random Features for Large-Scale Kernel Machines (2007)
  32. Generalization error of random feature and kernel methods: Hypercontractivity and kernel matrix concentration (2022)
  33. Learning with SGD and Random Features (2018)
  34. Generalization Properties of Learning with Random Features (2017)
  35. Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality (2023)
  36. Realization theory of discrete-time nonlinear systems: Part I-The bounded case (1979)
  37. Polynomial Response Maps (1979)
  38. Approximation theorems for discrete-time systems (1991)
  39. Structure theorems for nonlinear systems (1991)
  40. On the Uniform Approximation of Nonlinear Discrete-Time Fading-Memory Systems Using Neural Network Models (PhD Thesis) (1992)
  41. Universal discrete-time reservoir computers with stochastic inputs and linear readouts using non-homogeneous state-affine systems (2018)
  42. Neural SDE: Stabilizing Neural ODE Networks with Stochastic Noise (2019)
  43. Adversarial Audio Synthesis (2019)
  44. Non-adversarial training of Neural SDEs with signature kernel scores — → 2305.16274
  45. Applications of Signature Methods to Market Anomaly Detection (2022)
  46. On neural differential equations (PhD Thesis) (2021)
  47. The limit points of (optimistic) gradient descent in min-max optimization (2018)
  48. Training GANs with Optimism (2017)
  49. Long Short-Term Memory (1997)
  50. Adam: A Method for Stochastic Optimization (2015)
  51. An analysis of variance test for normality (complete samples) (1965)
  52. Differential equations driven by rough signals (1998)
  53. Lecture notes on rough paths and applications to machine learning (2024)
  54. G-Signatures: Global Graph Propagation With Randomized Signatures (2023) — link
  55. Randomized Signature Methods in Optimal Portfolio Selection — → 2312.16448
  56. Signature Methods in Stochastic Portfolio Theory — → 2310.02322
  57. Neural signature kernels as infinite-width-depth-limits of controlled ResNets (2023)
  58. Free probability, path developments and signature kernels as universal scaling limits (2024)
  59. Path Development Network with Finite-dimensional Lie Group Representation (2022)
  60. Approximation bounds for random neural networks and reservoir systems (2023)
  61. Echo state networks are universal (2018) — link
  62. Universal Approximation Property of Random Neural Networks (2023)
  63. Mathematical Methods of Organizing and Planning Production (1960)
  64. Generative model for financial time series trained with MMD using a signature kernel — → 2407.19848

5 of 72 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.