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薛定谔桥（Schrödinger bridge, SB）方法的时间序列生成新模型。该方法通过在路径空间上的参考概率测度与时间序列联合数据分布所对应的目标测度之间，利用最优传输实现熵插值。其解由有限时域上具有路径依赖漂移函数的随机微分方程刻画，因而能够尊重时间序列分布的时变动力学特性。我们可通过核回归方法或LSTM神经网络从数据样本中估计漂移函数，而SB扩散的模拟可生成新的时间序列合成数据样本。通过一系列数值实验评估了该生成模型的性能。首先，在玩具自回归模型、GARCH模型及分数布朗运动实例上进行测试，并利用边际依赖与时间依赖度量指标衡量算法准确性；其次，将SB生成的合成样本应用于真实数据集上的深度对冲；最后，展示了SB方法在图像序列生成中的应用。