Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Universal randomised signatures for generative time series modelling

Francesca Biagini, Lukas Gonon, Niklas Walter

TL;DR

The paper tackles realistic synthetic financial time-series generation by establishing universal approximation results for a reservoir composed of the randomised path signature and introducing a Wasserstein-type metric RS-W_1 built on linear readouts of the reservoir. It then constructs a non-adversarial generator using random feature neural networks, guided by RS-W_1, and extends to a conditional setting with the C-RS-W_1 metric to incorporate past information. Empirical results across Brownian motion, autoregressive processes, and real data (S&P 500 and FOREX) demonstrate that RS-W_1 and its conditional variant yield improved covariance and autocorrelation structure, better marginal distributions, and competitive or superior performance relative to Sig-W_1 baselines and LSTM generators. The work provides practical training guidelines, evaluative metrics, and code scaffolding, offering a scalable alternative to GAN-based approaches for synthetic financial time-series generation with a principled statistical foundation.

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.

Universal randomised signatures for generative time series modelling

TL;DR

The paper tackles realistic synthetic financial time-series generation by establishing universal approximation results for a reservoir composed of the randomised path signature and introducing a Wasserstein-type metric RS-W_1 built on linear readouts of the reservoir. It then constructs a non-adversarial generator using random feature neural networks, guided by RS-W_1, and extends to a conditional setting with the C-RS-W_1 metric to incorporate past information. Empirical results across Brownian motion, autoregressive processes, and real data (S&P 500 and FOREX) demonstrate that RS-W_1 and its conditional variant yield improved covariance and autocorrelation structure, better marginal distributions, and competitive or superior performance relative to Sig-W_1 baselines and LSTM generators. The work provides practical training guidelines, evaluative metrics, and code scaffolding, offering a scalable alternative to GAN-based approaches for synthetic financial time-series generation with a principled statistical foundation.

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.
Paper Structure (23 sections, 5 theorems, 68 equations, 5 figures, 10 tables, 1 algorithm)

This paper contains 23 sections, 5 theorems, 68 equations, 5 figures, 10 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Randomised signature of a datastream
  3. Universal approximation properties
  4. Time series generation using the randomised signature
  5. The randomised signature Wasserstein-1 (RS-$W_1$) metric
  6. Numerical results
  7. Brownian motion with drift
  8. Autoregressive process
  9. S&P 500 return data
  10. FOREX return data
  11. Conditional time series generation using the randomised signature
  12. The conditional randomised signature Wasserstein-1 (C-RS-$W_1$) metric
  13. Practical implementation
  14. Empirical results
  15. Brownian motion
  16. ...and 8 more sections

Key Result

Lemma 7

Following the Sampling Scheme samplingscheme1, for every $t=1,\dots,T$ there exists almost surely a unique continuous injective function $G_t:(\mathbb{R}^{d})^t\to (\mathbb{R}^{d})^T$ such that

Figures (5)

  • Figure 1: $50$ randomly selected trajectories from both generated and real out-of-sample samples of a Brownian motion with drift $\mu=0$ and variance $\sigma=1$.
  • Figure 2: $50$ randomly selected generated future trajectories based on a past path of a Brownian motion with drift $\mu=0$ and variance $\sigma=1$.
  • Figure 3: KDEs for generated and test data of values of a AR(1)-process with $\varphi=0.1$ at two different timepoints.
  • Figure 4: KDEs for generated and test data of S&P 500 log-returns.
  • Figure 5: KDEs for generated and test data of FOREX EUR/USD log-returns.

Theorems & Definitions (9)

  • Definition 1: Randomised signature
  • Definition 2: Random feature neural network
  • Remark 3
  • Remark 5
  • Lemma 7
  • Proposition 8: Universality of random feature neural networks
  • Proposition 9
  • Lemma 11
  • Proposition 12