Stochastic processes are random variables with values in some space of paths.
However, reducing a stochastic process to a path-valued random variable ignores
its filtration, i.e. the flow of information carried by the process through time. By
conditioning the process on its filtration, we introduce a family of higher order
kernel mean embeddings (KMEs) that generalizes the notion of KME and captures
additional information related to the filtration. We derive empirical estimators
for the associated higher order maximum mean discrepancies (MMDs) and prove
consistency. We then construct a filtration-sensitive kernel two-sample test able
to pick up information that gets missed by the standard MMD test. In addition,
leveraging our higher order MMDs we construct a family of universal kernels
on stochastic processes that allows to solve real-world calibration and optimal
stopping problems in quantitative finance (such as the pricing of American options)
via classical kernel-based regression methods. Finally, adapting existing tests for
conditional independence to the case of stochastic processes, we design a causal-
discovery algorithm to recover the causal graph of structural dependencies among
interacting bodies solely from observations of their multidimensional trajectories.