Girsanov Theorem - Corollary

Corollary

If X is a continuous process and W is Brownian Motion under measure P then

is Brownian motion under Q.

The fact that is continuous is trivial; by Girsanov's theorem it is a Q local martingale, and by computing the quadratic variation

it follows by Levy's characterization of Brownian Motion that this is a Q Brownian Motion.