Representing general stochastic processes as martingale laws

Random variables Xi, i = 1, 2 are ‘probabilistically equivalent’ if they have the
same law. Moreover, in any class of equivalent random variables it is easy to select canonical
representatives.
The corresponding questions are more involved for processes Xi on filtered stochastic bases
(Ωi, Fi, Pi, (Fi
t )t∈[0,1]). Here equivalence in law does not capture relevant properties of pro-
cesses such as the solutions to stochastic control or multistage decision problems. This mo-
tivates Aldous to introduce the stronger notion of synonymity based on prediction processes.
Stronger still, Hoover–Keisler formalize what it means that Xi, i = 1, 2 have the same proba-
bilistic properties. We establish that canonical representatives of the Hoover–Keisler equiva-
lence classes are given precisely by the set of all Markov-martingale laws on a specific nested
path space M∞. As a consequence we obtain that, modulo Hoover–Keisler equivalence, the
class of stochastic processes forms a Polish space.
On this space, processes are topologically close iff they model similar probabilistic phe-
nomena. In particular this means that their laws as well as the information encoded in the
respective filtrations are similar. Importantly, compact sets of processes admit a Prohorov-
type characterization. We also obtain that for every stochastic process, defined on some
abstract basis, there exists a process with identical probabilistic properties which is defined
on a standard Borel space.