# time change

Discussion in 'Probability' started by TefJlives, Jul 9, 2010.

1. ### TefJlivesGuest

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

TefJlives, Jul 9, 2010

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