From this Is Brownian bridge a Markov process , I can see that $X_{t}=B_t-tB_1$, Brownian Bridge, is a Markov Process.
Does exist another way to see it without using Ito processes but just the definition? What about the strong Markov property?
EDIT: I proved without using Ito processes that the Brownian Bridge is a Markov Process. I used the follow property:
$$X_t \mbox{ satisfies the Markov Property} \quad \mbox{ iff } \quad X_t \mbox{ is indipendent of } X_z \mbox{, if we know } X_s \quad \mbox{with } z<s<t$$
Now, I want to prove (or disprove) if $X_t$ satisfies the strong Markov Property. Could you help me?