The Brownian Integral Kernel: A New Kernel for Modeling Integrated Brownian Motions

Béla H. Böhnke, Edouard Fouché, Klemens Böhm

Abstract: Brownian motion is widely used to model random processes across various domains. However, many practical scenarios only provide aggregated data over time intervals, rather than direct measurements of the underlying process. This poses significant challenges for accurate modeling, as conventional Brownian kernels are not designed to account for the uncertainty introduced by these aggregates. We introduce the Brownian integral kernel (BIK), the first analytical kernel specifically developed to model aggregated data from Brownian motion. Through extensive experiments on synthetic and real-world datasets, we demonstrate the BIK’s superiority in prediction accuracy, uncertainty estimation, and data synthesis compared to existing Kernels. To support adoption, we provide a Python implementation (git: https://github.com/bela127/Brownian-Integral-Kernel.) compatible with GPy, along with all code and data to reproduce our experiments.