Hey guys I posted here a couple months ago and everyone was really helpful. I am reaching out tonight to make sure I am doing a problem correctly. I must determine if the following stochastic process is a martingale and if not how can I make it one.

Let W_t be a Wiener process and t denote time:

X_t = W_t -2t

I tried to follow another example to test if it is a martingale and came up with the following,

X_t = W_t -2t

E [X_{t+s} | X_t] = E[ W_{t+s} -2(t+s) | W_t -2t]

= E [ W_{t+s} | W_t - 2t ] -2 (t+s)

= W_t - 2(t+s)

\neq\ W_t - 2t

This is not a martingale and in order to make it a martingale I must drop the end term by adding 2t and then I am left with:

X_t = W_t

I am just looking for any help in regards if I did it correctly and if I made any mistakes or assumed something that may not be true.
Any help would be appreciated.