How to prove $W_t-tW_1$ is Markov?

Question

For a Brownian motion $W_t$, how do we prove the bridge process $W_t-tW_1$ is Markov? Essentially, we need to prove for $s< t$, \begin{align} \mathbb{P}(W_t-tW_1\in x\mid \mathcal{F}_s)=\mathbb{P}(W_t-tW_1\in x\mid W_s) \end{align}

I understand I need to use $W_t-W_s$ being independent of $\mathcal{F}_s$, but how to use that for proving the above statement?

Answer 1

The Brownian bridge is a Markov process because it is an Itô process. In particular, we may use Itô's formula to check that $Y_t = W_t - tW_1$ satisfies the following SDE: $$dY_t = - \frac{Y_t}{1-t}dt + dB_t$$