Deep Kalman filters (DKFs) are a class of neural network models that generate Gaussian probability measures from sequential data. Though DKFs are inspired by the Kalman filter, they lack concrete theoretical ties to the stochastic filtering problem, thus limiting their applicability to areas where traditional model-based filters have been used, e.g.\ model calibration for bond and option prices in mathematical finance. We address this issue in the mathematical foundations of deep learning by exhibiting a class of continuous-time DKFs which can approximately implement the conditional law of a broad class of non-Markovian and conditionally Gaussian signal processes given noisy continuous-times measurements. Our approximation results hold uniformly over sufficiently regular compact subsets of paths, where the approximation error is quantified by the worst-case 2-Wasserstein distance computed uniformly over the given compact set of paths.

翻译：深度卡尔曼滤波器（DKFs）是一类从序列数据生成高斯概率测度的神经网络模型。尽管DKFs受卡尔曼滤波器启发，但它们与随机滤波问题缺乏具体的理论联系，从而限制了其在传统基于模型的滤波器应用领域的适用性，例如数学金融中债券和期权价格的模型校准。我们在深度学习的数学基础中解决了这一问题，通过展示一类连续时间DKFs，它们能够近似实现一类广泛的非马尔可夫且条件高斯信号过程在给定噪声连续时间观测下的条件分布。我们的近似结果在路径的足够正则的紧致子集上一致成立，其中近似误差通过在该给定紧致路径集上一致计算的 worst-case 2-Wasserstein 距离进行量化。