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?

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    $\begingroup$ "What about the strong Markov property?" Yes, what about it? Sorry but what do you mean with this sentence? $\endgroup$ – Did Apr 7 '18 at 8:43
  • $\begingroup$ @Did. I proved the first my problem. Could you know how I can go on? $\endgroup$ – Skills Apr 7 '18 at 9:19
  • $\begingroup$ First, it is not true that the process $(X_t)$ satisfies the Markov property iff $X_t$ is independent of $X_z$ conditionally on $X_s$ for every $z<s<t$. $\endgroup$ – Did Apr 7 '18 at 9:43
  • $\begingroup$ @Did $X_t$ is a BM. Maybe is not a "iff" but only a sufficient condition? $\endgroup$ – Skills Apr 7 '18 at 12:31
  • $\begingroup$ Certainly not sufficient. And $X$ is not a Brownian motion. $\endgroup$ – Did Apr 7 '18 at 13:27

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