  In classical physics, the Kolmogorov extension theorem lays the foundation
for the theory of stochastic processes. It has been known for a long time that,
in its original form, this theorem does not hold in quantum mechanics. More
generally, it does not hold in any theory of stochastic processes -- classical,
quantum or beyond -- that does not just describe passive observations, but
allows for active interventions. Such processes form the basis of the study of
causal modelling across the sciences, including in the quantum domain. To date,
these frameworks have lacked a conceptual underpinning similar to that provided
by Kolmogorov's theorem for classical stochastic processes. We prove a
generalized extension theorem that applies to all theories of stochastic
processes, putting them on equally firm mathematical ground as their classical
counterpart. Additionally, we show that quantum causal modelling and quantum
stochastic processes are equivalent. This provides the correct framework for
the description of experiments involving continuous control, which play a
crucial role in the development of quantum technologies. Furthermore, we show
that the original extension theorem follows from the generalized one in the
correct limit, and elucidate how a comprehensive understanding of general
stochastic processes allows one to unambiguously define the distinction between
those that are classical and those that are quantum.
