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Trajectory Generator Matching for Time Series

T. Jahn, J. Chemseddine, P. Hagemann, C. Wald, G. Steidl

TL;DR

This work addresses generating time series from irregular observations by extending trajectory flow matching to models with both diffusion and jump dynamics. It stabilizes Brownian-bridge constructions and parameterizes jump densities with Gaussian-like forms to obtain analytic KL losses, enabling simulation-free training on irregular grids. Memory-based conditioning is introduced to recover joint distributions rather than mere marginals, and the framework integrates drift and jump dynamics via a Markov superposition principle. Empirical results on synthetic and finance-inspired data demonstrate that combining Jump, SDE, and memory-based conditioning yields strong performance under irregular sampling, with notable gains from the Markov superposition approach.

Abstract

Accurately modeling time-continuous stochastic processes from irregular observations remains a significant challenge. In this paper, we leverage ideas from generative modeling of image data to push the boundary of time series generation. For this, we find new generators of SDEs and jump processes, inspired by trajectory flow matching, that have the marginal distributions of the time series of interest. Specifically, we can handle discontinuities of the underlying processes by parameterizing the jump kernel densities by scaled Gaussians that allow for closed form formulas of the corresponding Kullback-Leibler divergence in the loss. Unlike most other approaches, we are able to handle irregularly sampled time series.

Trajectory Generator Matching for Time Series

TL;DR

This work addresses generating time series from irregular observations by extending trajectory flow matching to models with both diffusion and jump dynamics. It stabilizes Brownian-bridge constructions and parameterizes jump densities with Gaussian-like forms to obtain analytic KL losses, enabling simulation-free training on irregular grids. Memory-based conditioning is introduced to recover joint distributions rather than mere marginals, and the framework integrates drift and jump dynamics via a Markov superposition principle. Empirical results on synthetic and finance-inspired data demonstrate that combining Jump, SDE, and memory-based conditioning yields strong performance under irregular sampling, with notable gains from the Markov superposition approach.

Abstract

Accurately modeling time-continuous stochastic processes from irregular observations remains a significant challenge. In this paper, we leverage ideas from generative modeling of image data to push the boundary of time series generation. For this, we find new generators of SDEs and jump processes, inspired by trajectory flow matching, that have the marginal distributions of the time series of interest. Specifically, we can handle discontinuities of the underlying processes by parameterizing the jump kernel densities by scaled Gaussians that allow for closed form formulas of the corresponding Kullback-Leibler divergence in the loss. Unlike most other approaches, we are able to handle irregularly sampled time series.

Paper Structure

This paper contains 19 sections, 6 theorems, 58 equations, 13 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Problem Setting
  4. Markov Processes via Generators
  5. Learning the Drift and the Jump Kernel
  6. Learning Irregular Time Series
  7. Numerical Experiments
  8. Verification of Theoretical Results
  9. Synthetic Datasets
  10. Limitations and Conclusions
  11. Additional Definitions
  12. Proofs
  13. Proof of Proposition \ref{['th1']}
  14. Proof of Remark \ref{['prop:post_x_0']}.
  15. Proof of Proposition \ref{['prop2']}.
  16. ...and 4 more sections

Key Result

Proposition 2

For arbitrary fixed $x_0,x_1 \in \mathbb{R}^d$, set $m_t \coloneqq (1-t)x_0 + t x_1$ and $\tau_t\coloneqq \eta^2 t(1-t)+\rho^2$ with $\eta > 0$. Let $P_t = P_t(\cdot,x_0,x_1) = \mathcal{N}(m_t,\tau_t)$ with density $p_t$. Then each of the following generators $\mathcal{L}_t^{{\rm Diff}} = \mathcal{L

Figures (13)

  • Figure 1: Difference between analytical means and standard deviations against one dimensional integration.
  • Figure 2: Results on the synthetic dataset using a $50\%$ subsampling rate. From Left to Right: Ground truth, TFM, JUMP, SDE method, Markov superposition with $\alpha=0.90$.
  • Figure 3: Results on the synthetic Black-Scholes dataset using a $50\%$ subsampling rate. From Left to Right: Ground truth, TFM, JUMP, SDE method, Markov superposition with $\alpha=0.35$.
  • Figure 4: Ground truth data
  • Figure 5: Results on the synthetic dataset using a $10\%$ subsampling rate. From Left to Right: TFM, JUMP, SDEM. The best $\alpha$ for the Markov superposition of the SDEM and the JUMP is zero, which is why it coincides with the SDEM
  • ...and 8 more figures

Theorems & Definitions (11)

  • Remark 1
  • Proposition 2
  • Remark 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Remark 7
  • Lemma 8
  • proof
  • Proposition 9
  • ...and 1 more