  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.
