  We study the Classical Probability analogue of the dilations of a quantum
dynamical semigroup defined in Quantum Probability via quantum stochastic
differential equations. Given a homogeneous Markov chain in continuous time in
a finite state space E, we introduce a second system, an environment, and a
deterministic invertible time-homogeneous global evolution of the system E with
this environment such that the original Markov evolution of E can be realized
by a proper choice of the initial random state of the environment. We also
compare this dilations with the dilations of a quantum dynamical semigroup in
Quantum Probability: given a classical Markov semigroup, we extend it to a
proper quantum dynamical semigroup for which we can find a Hudson-Parthasarathy
dilation which is itself an extension of our classical dilation.

Probability