  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.
