I have a standard brownian motion under probability measure P, let Z(t) be the exponential martingale Z(t)=exp{-thetaB(t)-1/2(theta)^2t}. Define measure Q(A)=E[Z(t)1a] as a probability measure. I need to show that the process Xt=Bt+theta*t (Brownian motion with drift) is a standard brownian motion under measure Q WITHOUT using C-M or Girsanov theorems, but instead using Levy's characterisation of brownian motion to show that under Q, Xt is a cont. martingale.

I am pulling my hear out on this one!!