Higher order kernel mean embeddings to capture filtrations of stochastic processes

Conference item

Stochastic processes are random variables with values in some space of paths. However, reducing a stochastic process to a path-valued random variable ignores its filtration, i.e. the flow of information carried by the process through time. By conditioning the process on its filtration, we introduce a family of higher order kernel mean embeddings (KMEs) that generalizes the notion of KME to capture additional information related to the filtration. We derive empirical estimators for the associated higher order maximum mean discrepancies (MMDs) and prove consistency. We then construct a filtration-sensitive kernel two-sample test able to capture information that gets missed by the standard MMD test. In addition, leveraging our higher order MMDs we construct a family of universal kernels on stochastic processes that allows to solve real-world calibration and optimal stopping problems in quantitative finance (such as the pricing of American options) via classical kernel-based regression methods. Finally, adapting existing tests for conditional independence to the case of stochastic processes, we design a causal-discovery algorithm to recover the causal graph of structural dependencies among interacting bodies solely from observations of their multidimensional trajectories.

Authors: Salvi, C; Lemercier, M; Liu, C; Horvath, B; Damoulas, T

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Curran Associates
• Host title: Advances in Neural Information Processing Systems 34 (NeurIPS 2021)
• Volume: 20
• Pages: 16635-16647
• Publication date: 2022-05-01
• Acceptance date: 2021-09-28
• Event title: 35th Conference on Neural Information Processing Systems (NeurIPS 2021)
• Event location: Virtual event
• Event website: https://nips.cc/Conferences/2021/
• Event start date: 2021-12-06
• Event end date: 2021-12-14
• ISSN: 1049-5258
• ISBN: 9781713845393
• Language: English