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From: TefJlives on 10 Jul 2010 06:57
Hello all,

I wonder if anyone can help me with the question below. Suppose f=u
+iv
is holomorphic and B_s is a planar Brownian motion. We can find an
adapted process C_t such that


\int_0^t |f'(B_s)|ds = \int_0^{C_t} |f'(B_s)|^2ds


If we define V_t = u(B_{C_t}), we then have


<V>_t = |f'(B_t)|dt


But is V a local martingale? It is if C_t is a stopping time for each
t. To show that it is a stopping time we need to show


{C_t <= r} \in F_r


where F is the filtration. But


{C_t <= r} = {\int_0^t |f'(B_s)|ds <= \int_0^r |f'(B_s)|^2ds} \in
F_{max{r,t}}


So I don't see how we can conclude that C_t is a stopping time. And
yet, we should be able to adjust the speed to obtain a new process V
for which <V>_t = |f'(B_t)|dt, no? Can someone help me with this?
Thank you.


Greg


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