Let be a probability space, on which is filtration satisfying general conditions. is a standard Brownian motion. Let be a martingale given by
where is bounded measurable process.
for all constant .
To prove the above proposition, we will use following two facts. We define by
- By direct computation, one can have
- By time change argument, we have
Now we are ready to present the proof of Proposition~1.
Proof: Since is continuous process,
By Levy’s zero one law, we have
Therefore, it is enough to show that there exists such that
Note that, the martingale has the same distribution as a time-changed Brownian motion starting from state . Together with , we have for some standar Brownian motion ,
Since is independent to , and strictly less than , we can simply take .
To this end, one may ask if the condition of can be relaxed to without affecting the result? The answer is unfortunately NO. See counter-example here.