Paper: arXiv 2304.05093

Authors: Mohamed Hamdouche, Pierre Henry-Labordere, Huyên Pham

Abstract

We propose a novel generative model for time series based on Schr{ö}dinger bridge (SB) approach. This consists in the entropic interpolation via optimal transport between a reference probability measure on path space and a target measure consistent with the joint data distribution of the time series. The solution is characterized by a stochastic differential equation on finite horizon with a path-dependent drift function, hence respecting the temporal dynamics of the time series distribution. We can estimate the drift function from data samples either by kernel regression methods or with LSTM neural networks, and the simulation of the SB diffusion yields new synthetic data samples of the time series. The performance of our generative model is evaluated through a series of numerical experiments. First, we test with a toy autoregressive model, a GARCH Model, and the example of fractional Brownian motion, and measure the accuracy of our algorithm with marginal and temporal dependencies metrics. Next, we use our SB generated synthetic samples for the application to deep hedging on real-data sets. Finally, we illustrate the SB approach for generating sequence of images.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 7.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents a mathematically rigorous framework applying Schrödinger bridges and optimal transport to time series generation. It complements strong theoretical foundations with solid empirical experiments including synthetic benchmarks, deep hedging applications on real data, and high-dimensional image sequence generation.

Research Flowchart

  flowchart TD
    A[Goal: Time Series Generative Modeling via Schrödinger Bridge] --> B[Input: Data & Reference Path Measure]
    B --> C[Formulate Optimal Transport & Solve SDE with Path-Dependent Drift]
    C --> D[Estimate Drift via Kernel Regression or LSTMs]
    D --> E[Simulate SB Diffusion to Generate Synthetic Samples]
    E --> F[Evaluate: Toy Models, Deep Hedging & Image Sequences]
    F --> G[Outcome: Accurately Captures Temporal & Marginal Dependencies]