Change of Measure Using Levy Characterization

May 2010
2
0
I have a standard brownian motion under probability measure \(\displaystyle P\), let \(\displaystyle Z(t)\) be the exponential martingale \(\displaystyle Z(t)=exp{-thetaB(t)-1/2(theta)^2t}\). Define measure \(\displaystyle Q(A)=E[Z(t)1a]\) as a probability measure. I need to show that the process \(\displaystyle Xt=Bt+theta*t\) (Brownian motion with drift) is a standard brownian motion under measure \(\displaystyle Q\) WITHOUT using C-M or Girsanov theorems, but instead using Levy's characterisation of brownian motion to show that under \(\displaystyle Q\), \(\displaystyle Xt\) is a cont. martingale.

I am pulling my hear out on this one!!(Angry)