Stochastic Process as a Functor

A stochastic process is a functor from time into a category of Markov kernels; the shape one gives time decides the kind of process.

A stochastic process is a functor into the Kleisli category of the Giry monad, . The order version takes time as a partially-ordered-set , and the functor sends each instant to a state space and each step to a transition kernel.

Functoriality is exactly the Chapman–Kolmogorov consistency, since composing the kernels along agrees with the direct kernel. The monoid version takes as a one-object category, so a functor out of it is a single endomorphism kernel iterated, a time-homogeneous chain whose law is invariant under time shifts.

Remark on observer versus state, by currying

A process is a joint map from outcomes and times to states, and its two curryings split the two readings of it. fixes an outcome and returns a whole trajectory, the observer’s family of random variables, one sample path per outcome. fixes a time and returns a random variable, the state, the execution behaviour of the process. The functor is the second reading made structural, and the Markov category is where its kernels live.

Example of a Gaussian process

A Gaussian process is a sub-process whose finite marginals are all Gaussian and mutually consistent under the kernels (a Kolmogorov-extension family). It is cut out of by requiring every transition to preserve Gaussianity, and is then presented compactly by a mean function and a covariance kernel rather than the full functor.

References

• Michèle Giry, “A categorical approach to probability theory”, Lecture Notes in Mathematics 915 (1982)