  In this note, we propose two different approaches to rigorously justify a
pseudo-Markov property for controlled diffusion processes which is often
(explicitly or implicitly) used to prove the dynamic programming principle in
the stochastic control literature. The first approach develops a sketch of
proof proposed by Fleming and Souganidis~\cite{fleming-souganidis}. The second
approach is based on an enlargement of the original state space and a
controlled martingale problem. We clarify some measurability and topological
issues raised by these two approaches.
