Neural SDEs are continuous-time generative models for sequential data. State-of-the-art performance for irregular time series generation has been previously obtained by training these models adversarially as GANs. However, as typical for GAN architectures, training is notoriously unstable, often suffers from mode collapse, and requires specialised techniques such as weight clipping and gradient penalty to mitigate these issues. In this paper, we introduce a novel class of scoring rules on pathspace based on signature kernels and use them as objective for training Neural SDEs non-adversarially. By showing strict properness of such kernel scores and consistency of the corresponding estimators, we provide existence and uniqueness guarantees for the minimiser. With this formulation, evaluating the generator-discriminator pair amounts to solving a system of linear path-dependent PDEs which allows for memory-efficient adjoint-based backpropagation. Moreover, because the proposed kernel scores are well-defined for paths with values in infinite dimensional spaces of functions, our framework can be easily extended to generate spatiotemporal data. Our procedure permits conditioning on a rich variety of market conditions and significantly outperforms alternative ways of training Neural SDEs on a variety of tasks including the simulation of rough volatility models, the conditional probabilistic forecasts of real-world forex pairs where the conditioning variable is an observed past trajectory, and the mesh-free generation of limit order book dynamics.

翻译：神经随机微分方程是用于序列数据的连续时间生成模型。此前，不规则时间序列生成的最优性能是通过将这些模型作为生成对抗网络进行对抗训练获得的。然而，正如生成对抗架构的典型特征，训练过程极不稳定，常出现模式崩溃问题，且需要权重裁剪和梯度惩罚等专门技术来缓解这些问题。本文提出了一类基于特征核的路径空间新评分规则，并将其作为非对抗训练神经随机微分方程的目标函数。通过证明此类核评分的严格恰当性及相应估计量的一致性，我们给出了极小值存在性与唯一性的保证。在该框架下，评估生成器-判别器对等价于求解一组依赖路径的线性偏微分方程，从而能实现基于伴随方法的高效内存反向传播。此外，由于所提出的核评分对取值于无穷维函数空间的路径也具有良好定义，我们的框架可轻松扩展至时空数据生成。该方法允许对多种市场条件进行条件化处理，在粗糙波动率模型模拟、以观察到的过去轨迹为条件变量的真实外汇对条件概率预测以及限价订单簿动态的无网格生成等多项任务中，显著优于其他神经随机微分方程训练方式。