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.
Proposition 1 Assume 
for some positive constants 
and 
, then
for all constant 
.
To prove the above proposition, we will use following two facts. We define ![{f: \mathbb{R}^+ \times \mathbb{R}^+ \times \mathbb{R} \mapsto [0,1]}](https://s0.wp.com/latex.php?latex=%7Bf%3A+%5Cmathbb%7BR%7D%5E%2B+%5Ctimes+%5Cmathbb%7BR%7D%5E%2B+%5Ctimes+%5Cmathbb%7BR%7D+%5Cmapsto+%5B0%2C1%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002 "{f: \mathbb{R}^+ \times \mathbb{R}^+ \times \mathbb{R} \mapsto [0,1]}")
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.
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