Adapted topologies and higher rank signatures

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.