i*******t
发帖数: 16 | |
i*******t
发帖数: 16 | 2 不好意思，忘说了一点：t是固定的
【在 i*******t 的大作中提到】
: 麻烦见附件，谢谢大家：）
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i*******t
发帖数: 16 | 3 大牛们请支个招呀，谢谢啦：）
【在 i*******t 的大作中提到】
: 麻烦见附件，谢谢大家：）
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Q***5
发帖数: 994 | 4 I guess that probability should be 1 instead of 0. I think you can find the
answer in some text book, e.g. the one by Shreve and (what's the other
author's name?). |
i*******t
发帖数: 16 | 5 多谢指教：）
这就是那本书上的一道题（exercise 9.17）。从结果倒推的话，我觉得测度应该是0
the
【在 Q***5 的大作中提到】
: I guess that probability should be 1 instead of 0. I think you can find the
: answer in some text book, e.g. the one by Shreve and (what's the other
: author's name?).
|
Q***5
发帖数: 994 | 6 Here is one way to prove it, if we are allowed to use some other results
about Brownian motion:
Since B.M. is homogeneous, we only have to prove the conclusion at t = 0.
Proof by contradiction.
Let G(h) = (B(h)-B(0))/h
If the conclusion does not hold, then there a>0 and N>0, such that
prob(G(h)>-N)>0 for any 0
which means that with Prob>0 the Brownian motion B never goes below the line
x=-Nt
on the interval (0,a).
By Girsanov's Thm, W_t = B_t +Nt is a B.M. under an equivalent probability,
an |
Q***5
发帖数: 994 | 7 I mean: with prob>0 W_t never goes below 0 on (0,a) |
i*******t
发帖数: 16 | 8 Thank you for the detailed solution. You are really NIU!
I'll try to understand
【在 Q***5 的大作中提到】
: I mean: with prob>0 W_t never goes below 0 on (0,a)
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n******t
发帖数: 4406 | 9 You just need the simple fact that B(t+h)-B(t) has the same dist. as a Brown
ian Motion,
and the process Y(t) defined by tB(1/t) for t!=0 and 0 for t=0, is also a Br
ownian motion.
【在 i*******t 的大作中提到】
: Thank you for the detailed solution. You are really NIU!
: I'll try to understand
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i*******t
发帖数: 16 | 10 I got it!
Thank you very much!
Brown
Br
【在 n******t 的大作中提到】
: You just need the simple fact that B(t+h)-B(t) has the same dist. as a Brown
: ian Motion,
: and the process Y(t) defined by tB(1/t) for t!=0 and 0 for t=0, is also a Br
: ownian motion.
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