Brownian motion reflection principle result
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I'm studying about the reflection principle of the brownian motion, and I found that this result is a direct consequence of this principle:
Let $B_t$ a brownian motion, then for every $a in mathbb{R} $,
$$mathbb{P}(lim_{t to infty} sup_{sin [0,t]} B_s > a) = 1$$
I'm trying to prove this statement using the reflection principle but I'm totally lost. I can't see how are those results related.
real-analysis stochastic-processes brownian-motion
asked Dec 7 '18 at 3:03
frl93frl93
1169
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add a comment |
$begingroup$
I'm studying about the reflection principle of the brownian motion, and I found that this result is a direct consequence of this principle:
Let $B_t$ a brownian motion, then for every $a in mathbb{R} $,
$$mathbb{P}(lim_{t to infty} sup_{sin [0,t]} B_s > a) = 1$$
I'm trying to prove this statement using the reflection principle but I'm totally lost. I can't see how are those results related.
real-analysis stochastic-processes brownian-motion
asked Dec 7 '18 at 3:03
frl93frl93
1169
$endgroup$
$begingroup$
Which formulation of the reflection principle do you know/use? It is direct consequence of the reflection principle that $$M_t := sup_{s leq t} B_s$$ equals in distribution $|B_t$|. Knowing this, it shouldn't be difficult to prove the assertion.
$endgroup$
– saz
Dec 7 '18 at 7:51
add a comment |
$begingroup$
I'm studying about the reflection principle of the brownian motion, and I found that this result is a direct consequence of this principle:
Let $B_t$ a brownian motion, then for every $a in mathbb{R} $,
$$mathbb{P}(lim_{t to infty} sup_{sin [0,t]} B_s > a) = 1$$
I'm trying to prove this statement using the reflection principle but I'm totally lost. I can't see how are those results related.
real-analysis stochastic-processes brownian-motion
asked Dec 7 '18 at 3:03
frl93frl93
1169
$endgroup$
I'm studying about the reflection principle of the brownian motion, and I found that this result is a direct consequence of this principle:
Let $B_t$ a brownian motion, then for every $a in mathbb{R} $,
$$mathbb{P}(lim_{t to infty} sup_{sin [0,t]} B_s > a) = 1$$
I'm trying to prove this statement using the reflection principle but I'm totally lost. I can't see how are those results related.
real-analysis stochastic-processes brownian-motion
real-analysis stochastic-processes brownian-motion
asked Dec 7 '18 at 3:03
frl93frl93
1169
asked Dec 7 '18 at 3:03
frl93frl93
1169
asked Dec 7 '18 at 3:03
frl93frl93
1169
asked Dec 7 '18 at 3:03
frl93frl93
1169
asked Dec 7 '18 at 3:03
frl93frl93
1169
1169
$begingroup$
Which formulation of the reflection principle do you know/use? It is direct consequence of the reflection principle that $$M_t := sup_{s leq t} B_s$$ equals in distribution $|B_t$|. Knowing this, it shouldn't be difficult to prove the assertion.
$endgroup$
– saz
Dec 7 '18 at 7:51
add a comment |
$begingroup$
Which formulation of the reflection principle do you know/use? It is direct consequence of the reflection principle that $$M_t := sup_{s leq t} B_s$$ equals in distribution $|B_t$|. Knowing this, it shouldn't be difficult to prove the assertion.
$endgroup$
– saz
Dec 7 '18 at 7:51
$begingroup$
Which formulation of the reflection principle do you know/use? It is direct consequence of the reflection principle that $$M_t := sup_{s leq t} B_s$$ equals in distribution $|B_t$|. Knowing this, it shouldn't be difficult to prove the assertion.
$endgroup$
– saz
Dec 7 '18 at 7:51
$begingroup$
Which formulation of the reflection principle do you know/use? It is direct consequence of the reflection principle that $$M_t := sup_{s leq t} B_s$$ equals in distribution $|B_t$|. Knowing this, it shouldn't be difficult to prove the assertion.
$endgroup$
– saz
Dec 7 '18 at 7:51
add a comment |
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$begingroup$
Which formulation of the reflection principle do you know/use? It is direct consequence of the reflection principle that $$M_t := sup_{s leq t} B_s$$ equals in distribution $|B_t$|. Knowing this, it shouldn't be difficult to prove the assertion.
$endgroup$
– saz
Dec 7 '18 at 7:51