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 programming. The reason is that the topology of weak convergence does not account for growth information over time captured filtration an process. To address discontinuities, Aldous introduced extended topology, and subsequently, Hoover Keisler showed both, are just first two topologies a sequence get increasingly finer. We introduce higher rank expected signatures to embed into graded linear spaces show these embeddings induce Hoover–Keisler.