We propose a novel generative model for time series based on Schr{\"o}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.

翻译：我们提出了一种基于Schr{\"o}dinger桥（SB）方法的新型时间序列生成模型。这包括通过最优运输在路径空间上的参考概率测度和与时间序列的联合数据分布一致的目标测度之间的熵插值。该解决方案的特点是有限时间内的随机微分方程，具有路径相关的漂移函数，因此尊重时间序列分布的时间动态。我们可以使用内核回归方法或LSTM神经网络从数据样本中估计漂移函数，并通过模拟SB扩散生成时间序列的新合成数据样本。我们通过一系列数字实验评估了生成模型的性能。首先，我们测试玩具自回归模型，GARCH模型和分数布朗运动的例子，并使用边际和时间依赖关系指标测量算法的准确性。接下来，我们使用SB生成的合成样本应用于真实数据集上的深层对冲。最后，我们说明了SB方法用于生成图像序列的例子。