Open Access
June 2023 Adapted topologies and higher rank signatures
Patric Bonnier, Chong Liu, Harald Oberhauser
Author Affiliations +
Ann. Appl. Probab. 33(3): 2136-2175 (June 2023). DOI: 10.1214/22-AAP1862

Abstract

Two adapted stochastic processes can have similar laws but give different results in applications such as optimal stopping, queuing theory, or stochastic programming. The reason is that the topology of weak convergence does not account for the growth of information over time that is captured in the filtration of an adapted stochastic process. To address such discontinuities, Aldous introduced the extended weak topology, and subsequently, Hoover and Keisler showed that both, weak topology and extended weak topology, are just the first two topologies in a sequence of topologies that get increasingly finer. We introduce higher rank expected signatures to embed adapted processes into graded linear spaces and show that these embeddings induce the adapted topologies of Hoover–Keisler.

Funding Statement

PB is supported by the Engineering and Physical Sciences Research Council [EP/R513295/1].
CL is supported by the SNSF Grant [P2EZP2_188068].
HO is supported by the EPSRC grant “Datasig” [EP/S026347/1], the Alan Turing Institute, the Oxford-Man Institute, and the Centre for Intelligent Multidimensional Data Analysis (CIMDA).

Acknowledgments

HO would like to thank Manu Eder for helpful discussions. CL would like to thank Gudmund Pammer for pointing out the fact that the metric dr characterizes the convergence in τˆr when V is a locally compact space.

Citation

Download Citation

Patric Bonnier. Chong Liu. Harald Oberhauser. "Adapted topologies and higher rank signatures." Ann. Appl. Probab. 33 (3) 2136 - 2175, June 2023. https://doi.org/10.1214/22-AAP1862

Information

Received: 1 April 2021; Revised: 1 April 2022; Published: June 2023
First available in Project Euclid: 2 May 2023

MathSciNet: MR4583667
zbMATH: 07692314
Digital Object Identifier: 10.1214/22-AAP1862

Subjects:
Primary: 60G07
Secondary: 60L10

Keywords: Adapted topologies , extended weak convergence , signatures

Rights: This research was funded, in whole or in part, by [Swiss National Science Foundation, P2EZP2-188068]. A CC BY 4.0 license is applied to this article arising from this submission, in accordance with the grant's open access conditions.

Vol.33 • No. 3 • June 2023
Back to Top