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Schrödinger bridges with jumps for time series generation

Stefano De Marco, Huyên Pham, Davide Zanni

TL;DR

Numerical experiments show that incorporating jumps substantially improves the realism of the generated data, in particular by capturing abrupt movements, heavy tails, and regime changes that diffusion-only models fail to reproduce.

Abstract

We study generative modeling for time series using entropic optimal transport and the Schrödinger bridge (SB) framework, with a focus on applications in finance and energy modeling. Extending the diffusion-based approach of Hamdouche, Henry-Labordère, Pham, 2023, we introduce a jump-diffusion Schrödinger bridge model that allows for discontinuities in the generative dynamics. Starting from a Schrödinger bridge entropy minimization problem, we reformulate the task as a stochastic control problem whose solution characterizes the optimal controlled jump-diffusion process. When sampled on a fixed time grid, this process generates synthetic time series matching the joint distributions of the observed data. The model is fully data-driven, as both the drift and the jump intensity are learned directly from the data. We propose practical algorithms for training, sampling, and hyperparameter calibration. Numerical experiments on simulated and real datasets, including financial and energy time series, show that incorporating jumps substantially improves the realism of the generated data, in particular by capturing abrupt movements, heavy tails, and regime changes that diffusion-only models fail to reproduce. Comparisons with state-of-the-art generative models highlight the benefits and limitations of the proposed approach.

Schrödinger bridges with jumps for time series generation

TL;DR

Numerical experiments show that incorporating jumps substantially improves the realism of the generated data, in particular by capturing abrupt movements, heavy tails, and regime changes that diffusion-only models fail to reproduce.

Abstract

We study generative modeling for time series using entropic optimal transport and the Schrödinger bridge (SB) framework, with a focus on applications in finance and energy modeling. Extending the diffusion-based approach of Hamdouche, Henry-Labordère, Pham, 2023, we introduce a jump-diffusion Schrödinger bridge model that allows for discontinuities in the generative dynamics. Starting from a Schrödinger bridge entropy minimization problem, we reformulate the task as a stochastic control problem whose solution characterizes the optimal controlled jump-diffusion process. When sampled on a fixed time grid, this process generates synthetic time series matching the joint distributions of the observed data. The model is fully data-driven, as both the drift and the jump intensity are learned directly from the data. We propose practical algorithms for training, sampling, and hyperparameter calibration. Numerical experiments on simulated and real datasets, including financial and energy time series, show that incorporating jumps substantially improves the realism of the generated data, in particular by capturing abrupt movements, heavy tails, and regime changes that diffusion-only models fail to reproduce. Comparisons with state-of-the-art generative models highlight the benefits and limitations of the proposed approach.
Paper Structure (31 sections, 2 theorems, 102 equations, 18 figures, 3 tables, 2 algorithms)

This paper contains 31 sections, 2 theorems, 102 equations, 18 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Generative modeling task.
  3. The Schrödinger bridge problem.
  4. Generative models based on Schrödinger bridges.
  5. Our contributions and organization of the paper.
  6. Setting and problem formulation
  7. Solution of the Schrödinger bridge problem with jumps for time series
  8. Approximation of the optimal drift and intensity
  9. Explicit formula for the drift and intensity of jumps
  10. Kernel regression estimators
  11. Simulation techniques
  12. Gaussian jump distribution
  13. Simulation of the jump term
  14. Euler scheme with Gaussian jumps
  15. Jump-adapted version of the Euler scheme
  16. ...and 16 more sections

Key Result

Theorem 3.1

The probability measure $\mathbb{P}^* \in \mathcal{P}(\Omega)$ defined by solves the SBJTS problem and it is the law of the optimal process with $X_0=0$, where $(W_t)_{t\in[0,T]}$ is a $\mathbb{P}^*$-Brownian motion, and $N$ is a Poisson random measure with intensity measure $\lambda^*_t(z)\nu^0(dz)dt$ under $\mathbb{P}^*$. For $i=0,\ldots,N-1$, $t\in (t_i,t_{i+1}]$, the drift $\alpha^*_t$ and t

Figures (18)

  • Figure 7.1: Simulated trajectories of Merton process \ref{['Merton_sde']} with parameters $a=0$, $b=2$, $\lambda_\eta =10$, $m_J=0$ and $v_J =0.8$.
  • Figure 7.2: Generation of synthetic time series by SBTS model: sample paths of real time series (left) and sample paths of synthetic time series with $h=0.1$ and $k=1$ (right).
  • Figure 7.3: Generation of synthetic time series by SBTS model: QQ-plot between the quantiles of the empirical distributions of $X_{t_{50}}$ on real and synthetic time series.
  • Figure 7.4: Discriminative scores tested on real time series and synthetic time series generated with different values of the hyperparameters $(\sigma, \gamma, \lambda^0)$.
  • Figure 7.5: Generation of synthetic time series by SBJTS model: sample paths of real time series (left), sample paths of synthetic time series with the choice $h=0.3$, $k=1$, $\sigma=2$, $\lambda^0=5$, $c=0$, and $\gamma =0.8$ (case (i) - middle), sample paths of synthetic time series with the choice $h=0.3$, $k=1$, $\sigma=1$, $\lambda^0=70$, $c=0$, and $\gamma =1$ (case (ii) - right).
  • ...and 13 more figures

Theorems & Definitions (8)

  • Theorem 3.1
  • proof
  • Proposition 4.1
  • proof
  • Remark 4.1
  • Remark 5.1
  • Remark 5.2
  • Remark 6.1