论文标题
学习条件定律：在过滤和预测扩散过程中的签名和条件剂量
Learning the conditional law: signatures and conditional GANs in filtering and prediction of diffusion processes
论文作者
论文摘要
我们考虑扩散过程的过滤和预测问题。信号和观察是由相关的维纳过程驱动的随机微分方程（SDE）建模的。在经典估计理论中，用于滤波和预测度量的测量值随机部分差分方程（SPDE）。这些方程可能很难在数值上求解。我们使用条件生成对抗网络（GAN）与签名（来自粗糙路径理论的对象）一起提供了近似算法。足够平滑路径的签名完全决定了路径。结果，在某些情况下，基于签名的gan被证明可以有效地近似随机过程的定律。对于我们的算法，我们将此方法扩展到从条件定律中采样的，鉴于嘈杂的部分观察结果。我们的发电机是使用神经微分方程（NDE）构建的，依赖于其通用近似属性。我们在提供严格的数学框架方面表现出良好的体现。数值结果显示了我们算法的效率。
We consider the filtering and prediction problem for a diffusion process. The signal and observation are modeled by stochastic differential equations (SDEs) driven by correlated Wiener processes. In classical estimation theory, measure-valued stochastic partial differential equations (SPDEs) are derived for the filtering and prediction measures. These equations can be hard to solve numerically. We provide an approximation algorithm using conditional generative adversarial networks (GANs) in combination with signatures, an object from rough path theory. The signature of a sufficiently smooth path determines the path completely. As a result, in some cases, GANs based on signatures have been shown to efficiently approximate the law of a stochastic process. For our algorithm we extend this method to sample from the conditional law, given noisy, partial observation. Our generator is constructed using neural differential equations (NDEs), relying on their universal approximator property. We show well-posedness in providing a rigorous mathematical framework. Numerical results show the efficiency of our algorithm.
