Problem
Research questions and friction points this paper is trying to address.
Depth Kalman Filters
Continuous Data Stream
Financial Prediction
Deep Kalman Filters Can Filter
🤖 AI Summary
Conventional deep Kalman filters (DKFs) lack theoretical guarantees for non-Markovian, conditionally Gaussian signal processes, limiting their applicability in mathematically rigorous domains such as mathematical finance. Method: We propose the continuous-time deep Kalman filter (CT-DKF), modeling signal dynamics via stochastic differential equations and quantifying distributional approximation error using the 2-Wasserstein distance. Our framework establishes uniform approximation of conditional distributions over regular compact path spaces. Contribution/Results: CT-DKF is the first DKF variant with a rigorous theoretical guarantee of consistent approximation to the optimal Bayesian filter for arbitrary non-Markovian, conditionally Gaussian processes. It overcomes the absence of convergence and generalization guarantees in existing DKFs; its approximation error is precisely bounded by the worst-case 2-Wasserstein distance. This provides a verifiable theoretical foundation for financial applications including bond and option pricing, and model calibration.
📝 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Continuous-Time Depth Kalman Filters
2-Wasserstein Distance
Stochastic Problem Handling
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