Adapted Topologies and Higher Rank Signatures

TL;DR

This paper introduces higher rank expected signatures to embed adapted stochastic processes into graded linear spaces, providing a new way to induce the adapted topologies of Hoover--Keisler and better capture information growth over time.

Contribution

It develops a novel embedding of adapted processes using higher rank expected signatures, linking them to the Hoover--Keisler adapted topologies.

Findings

Higher rank expected signatures effectively embed adapted processes.

The embeddings induce the Hoover--Keisler adapted topologies.

The approach captures the growth of information over time.

Abstract

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. For example, two adapted stochastic processes can have very similar laws but give completely different results in applications such as optimal stopping, queuing theory, or stochastic programming. 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 use higher rank expected signatures to embed adapted processes into graded linear spaces and show that these embeddings induce the adapted topologies of Hoover--Keisler.