Deep Kalman Filters Can Filter

• URL: http://arxiv.org/abs/2310.19603v1
• Date: Mon, 30 Oct 2023 14:58:12 GMT
• Title: Deep Kalman Filters Can Filter
• Authors: Blanka Hovart, Anastasis Kratsios, Yannick Limmer, Xuwei Yang
• Abstract summary: Deep Kalman filters (DKFs) are a class of neural network models that generate Gaussian probability measures from sequential data. DKFs are inspired by the Kalman filter, but they lack concrete theoretical ties to the filtering problem. We show that continuous-time DKFs can implement the conditional law of a broad class of non-Markovian and conditionally Gaussian signal processes.
• Score: 9.131190818372474
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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