Mixing
Compute $mathbb{P}{ W_t < 0 , , text{for all} , , 1 < t < 2}$ for a Brownian motion $(W_t)_{t geq...
0
$begingroup$
Let $(W_t)_{t ge 0}$ be a standard one-dimensional Brownian motion.
Calculate $$mathbb{P}{ W_t < 0 , , text{for all} , , 1 < t < 2}.$$
I can only think that this will be conditional on $W_1$. Please tell how to proceed?
probability-theory stochastic-processes brownian-motion
edited Feb 3 at 10:07
saz
82.5k862131
asked Feb 3 at 7:35
SomSom
43
$endgroup$
closed as off-topic by Did, José Carlos Santos, mrtaurho, YiFan, Cesareo Feb 4 at 11:45
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Did, José Carlos Santos, mrtaurho, YiFan, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.
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show 4 more comments
0
$begingroup$
Let $(W_t)_{t ge 0}$ be a standard one-dimensional Brownian motion.
Calculate $$mathbb{P}{ W_t < 0 , , text{for all} , , 1 < t < 2}.$$
I can only think that this will be conditional on $W_1$. Please tell how to proceed?
probability-theory stochastic-processes brownian-motion
edited Feb 3 at 10:07
saz
82.5k862131
asked Feb 3 at 7:35
SomSom
43
$endgroup$
closed as off-topic by Did, José Carlos Santos, mrtaurho, YiFan, Cesareo Feb 4 at 11:45
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Did, José Carlos Santos, mrtaurho, YiFan, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.
$begingroup$
Condition on $W_1$.
$endgroup$
– Did
Feb 3 at 8:23
$begingroup$
@JoséCarlosSantos, Som: Why approve an absurd edit?
$endgroup$
– Did
Feb 3 at 8:24
$begingroup$
@pointguard0 More care with the edits you suggest, please.
$endgroup$
– Did
Feb 3 at 8:25
$begingroup$
P(Wt<0 for all 1<t<2)
$endgroup$
– Som
Feb 3 at 8:40
$begingroup$
Som: Very much so. Which is not P(Wt<0) for all 1<t<2, yes?
$endgroup$
– Did
Feb 3 at 8:43
|
show 4 more comments
0
0
0
2
$begingroup$
Let $(W_t)_{t ge 0}$ be a standard one-dimensional Brownian motion.
Calculate $$mathbb{P}{ W_t < 0 , , text{for all} , , 1 < t < 2}.$$
I can only think that this will be conditional on $W_1$. Please tell how to proceed?
probability-theory stochastic-processes brownian-motion
edited Feb 3 at 10:07
saz
82.5k862131
asked Feb 3 at 7:35
SomSom
43
$endgroup$
Let $(W_t)_{t ge 0}$ be a standard one-dimensional Brownian motion.
Calculate $$mathbb{P}{ W_t < 0 , , text{for all} , , 1 < t < 2}.$$
I can only think that this will be conditional on $W_1$. Please tell how to proceed?
probability-theory stochastic-processes brownian-motion
probability-theory stochastic-processes brownian-motion
edited Feb 3 at 10:07
saz
82.5k862131
asked Feb 3 at 7:35
SomSom
43
edited Feb 3 at 10:07
saz
82.5k862131
asked Feb 3 at 7:35
SomSom
43
edited Feb 3 at 10:07
saz
82.5k862131
edited Feb 3 at 10:07
saz
82.5k862131
edited Feb 3 at 10:07
saz
82.5k862131
82.5k862131
asked Feb 3 at 7:35
SomSom
43
asked Feb 3 at 7:35
SomSom
43
asked Feb 3 at 7:35
SomSom
43
43
closed as off-topic by Did, José Carlos Santos, mrtaurho, YiFan, Cesareo Feb 4 at 11:45
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Did, José Carlos Santos, mrtaurho, YiFan, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Did, José Carlos Santos, mrtaurho, YiFan, Cesareo Feb 4 at 11:45
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Did, José Carlos Santos, mrtaurho, YiFan, Cesareo
If this question can be reworded to fit the rules in the help center, please edit the question.
$begingroup$
Condition on $W_1$.
$endgroup$
– Did
Feb 3 at 8:23
$begingroup$
@JoséCarlosSantos, Som: Why approve an absurd edit?
$endgroup$
– Did
Feb 3 at 8:24
$begingroup$
@pointguard0 More care with the edits you suggest, please.
$endgroup$
– Did
Feb 3 at 8:25
$begingroup$
P(Wt<0 for all 1<t<2)
$endgroup$
– Som
Feb 3 at 8:40
$begingroup$
Som: Very much so. Which is not P(Wt<0) for all 1<t<2, yes?
$endgroup$
– Did
Feb 3 at 8:43
|
show 4 more comments
$begingroup$
Condition on $W_1$.
$endgroup$
– Did
Feb 3 at 8:23
$begingroup$
@JoséCarlosSantos, Som: Why approve an absurd edit?
$endgroup$
– Did
Feb 3 at 8:24
$begingroup$
@pointguard0 More care with the edits you suggest, please.
$endgroup$
– Did
Feb 3 at 8:25
$begingroup$
P(Wt<0 for all 1<t<2)
$endgroup$
– Som
Feb 3 at 8:40
$begingroup$
Som: Very much so. Which is not P(Wt<0) for all 1<t<2, yes?
$endgroup$
– Did
Feb 3 at 8:43
$begingroup$
Condition on $W_1$.
$endgroup$
– Did
Feb 3 at 8:23
$begingroup$
Condition on $W_1$.
$endgroup$
– Did
Feb 3 at 8:23
$begingroup$
@JoséCarlosSantos, Som: Why approve an absurd edit?
$endgroup$
– Did
Feb 3 at 8:24
$begingroup$
@JoséCarlosSantos, Som: Why approve an absurd edit?
$endgroup$
– Did
Feb 3 at 8:24
$begingroup$
@pointguard0 More care with the edits you suggest, please.
$endgroup$
– Did
Feb 3 at 8:25
$begingroup$
@pointguard0 More care with the edits you suggest, please.
$endgroup$
– Did
Feb 3 at 8:25
$begingroup$
P(Wt<0 for all 1<t<2)
$endgroup$
– Som
Feb 3 at 8:40
$begingroup$
P(Wt<0 for all 1<t<2)
$endgroup$
– Som
Feb 3 at 8:40
$begingroup$
Som: Very much so. Which is not P(Wt<0) for all 1<t<2, yes?
$endgroup$
– Did
Feb 3 at 8:43
$begingroup$
Som: Very much so. Which is not P(Wt<0) for all 1<t<2, yes?
$endgroup$
– Did
Feb 3 at 8:43
|
show 4 more comments
1 Answer
1
active
oldest
votes
3
$begingroup$
Hints: Set $$B_t := W_{1+t}-W_1.$$
- Check (or recall) that $(B_t)_{t geq 0}$ is a Brownian motion and that $(B_t)_{t geq 0}$ is independent from $mathcal{F}_1^W:=sigma(W_s; s leq 1)$.
- Use the independence of $(B_t)_{t geq 0}$ and $W_1$ to show that $$mathbb{P}(W_t < 0 , , text{for all} , , t in (1,2) mid W_1) = f(W_1)$$ where $$f(x) := mathbb{P}(x+B_t < 0, , text{for all $t in (0,1)$}). tag{1}$$
- Prove that $f(x)=0$ for all $x geq 0$. (Hint: What happens close to $t=0$?)
- Fix $x<0$. Show that $$f(x) = mathbb{P}(tau_{-x} geq 1)$$ for the stopping time $$tau_{-x} := inf{t>0; B_t geq -x}.$$ Conclude from the reflection principle that $$f(x) = mathbb{P}(|B_1|<-x),$$ and so $$f(x)=1-2Phi(x)$$ where $Phi$ is the cdf of the centered standard Gaussian distribution with density $varphi$.
- Combining the above steps gives begin{align*} mathbb{P}(W_t < 0 , , text{for all} , , t in (1,2)) &= mathbb{E} left[ left( 1- 2Phi(W_1)right) 1_{{W_1<0}} right] \ &= frac{1}{2} - 2 int_{-infty}^0 Phi(x) varphi(x) , dx. end{align*}
- Conclude that $$mathbb{P}(W_t < 0 , , text{for all} , , t in (1,2)) = frac{1}{4}; $$ see e.g. this answer for how to compute the integral in Step 5.
answered Feb 3 at 9:01
sazsaz
82.5k862131
$endgroup$
$begingroup$
Yep. (+1) $ $ $ $
$endgroup$
– Did
Feb 3 at 12:08
$begingroup$
@Did Thank you.
$endgroup$
– saz
Feb 3 at 12:13
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
3
$begingroup$
Hints: Set $$B_t := W_{1+t}-W_1.$$
- Check (or recall) that $(B_t)_{t geq 0}$ is a Brownian motion and that $(B_t)_{t geq 0}$ is independent from $mathcal{F}_1^W:=sigma(W_s; s leq 1)$.
- Use the independence of $(B_t)_{t geq 0}$ and $W_1$ to show that $$mathbb{P}(W_t < 0 , , text{for all} , , t in (1,2) mid W_1) = f(W_1)$$ where $$f(x) := mathbb{P}(x+B_t < 0, , text{for all $t in (0,1)$}). tag{1}$$
- Prove that $f(x)=0$ for all $x geq 0$. (Hint: What happens close to $t=0$?)
- Fix $x<0$. Show that $$f(x) = mathbb{P}(tau_{-x} geq 1)$$ for the stopping time $$tau_{-x} := inf{t>0; B_t geq -x}.$$ Conclude from the reflection principle that $$f(x) = mathbb{P}(|B_1|<-x),$$ and so $$f(x)=1-2Phi(x)$$ where $Phi$ is the cdf of the centered standard Gaussian distribution with density $varphi$.
- Combining the above steps gives begin{align*} mathbb{P}(W_t < 0 , , text{for all} , , t in (1,2)) &= mathbb{E} left[ left( 1- 2Phi(W_1)right) 1_{{W_1<0}} right] \ &= frac{1}{2} - 2 int_{-infty}^0 Phi(x) varphi(x) , dx. end{align*}
- Conclude that $$mathbb{P}(W_t < 0 , , text{for all} , , t in (1,2)) = frac{1}{4}; $$ see e.g. this answer for how to compute the integral in Step 5.
answered Feb 3 at 9:01
sazsaz
82.5k862131
$endgroup$
$begingroup$
Yep. (+1) $ $ $ $
$endgroup$
– Did
Feb 3 at 12:08
$begingroup$
@Did Thank you.
$endgroup$
– saz
Feb 3 at 12:13
add a comment |
3
$begingroup$
Hints: Set $$B_t := W_{1+t}-W_1.$$
- Check (or recall) that $(B_t)_{t geq 0}$ is a Brownian motion and that $(B_t)_{t geq 0}$ is independent from $mathcal{F}_1^W:=sigma(W_s; s leq 1)$.
- Use the independence of $(B_t)_{t geq 0}$ and $W_1$ to show that $$mathbb{P}(W_t < 0 , , text{for all} , , t in (1,2) mid W_1) = f(W_1)$$ where $$f(x) := mathbb{P}(x+B_t < 0, , text{for all $t in (0,1)$}). tag{1}$$
- Prove that $f(x)=0$ for all $x geq 0$. (Hint: What happens close to $t=0$?)
- Fix $x<0$. Show that $$f(x) = mathbb{P}(tau_{-x} geq 1)$$ for the stopping time $$tau_{-x} := inf{t>0; B_t geq -x}.$$ Conclude from the reflection principle that $$f(x) = mathbb{P}(|B_1|<-x),$$ and so $$f(x)=1-2Phi(x)$$ where $Phi$ is the cdf of the centered standard Gaussian distribution with density $varphi$.
- Combining the above steps gives begin{align*} mathbb{P}(W_t < 0 , , text{for all} , , t in (1,2)) &= mathbb{E} left[ left( 1- 2Phi(W_1)right) 1_{{W_1<0}} right] \ &= frac{1}{2} - 2 int_{-infty}^0 Phi(x) varphi(x) , dx. end{align*}
- Conclude that $$mathbb{P}(W_t < 0 , , text{for all} , , t in (1,2)) = frac{1}{4}; $$ see e.g. this answer for how to compute the integral in Step 5.
answered Feb 3 at 9:01
sazsaz
82.5k862131
$endgroup$
$begingroup$
Yep. (+1) $ $ $ $
$endgroup$
– Did
Feb 3 at 12:08
$begingroup$
@Did Thank you.
$endgroup$
– saz
Feb 3 at 12:13
add a comment |
3
3
3
$begingroup$
Hints: Set $$B_t := W_{1+t}-W_1.$$
- Check (or recall) that $(B_t)_{t geq 0}$ is a Brownian motion and that $(B_t)_{t geq 0}$ is independent from $mathcal{F}_1^W:=sigma(W_s; s leq 1)$.
- Use the independence of $(B_t)_{t geq 0}$ and $W_1$ to show that $$mathbb{P}(W_t < 0 , , text{for all} , , t in (1,2) mid W_1) = f(W_1)$$ where $$f(x) := mathbb{P}(x+B_t < 0, , text{for all $t in (0,1)$}). tag{1}$$
- Prove that $f(x)=0$ for all $x geq 0$. (Hint: What happens close to $t=0$?)
- Fix $x<0$. Show that $$f(x) = mathbb{P}(tau_{-x} geq 1)$$ for the stopping time $$tau_{-x} := inf{t>0; B_t geq -x}.$$ Conclude from the reflection principle that $$f(x) = mathbb{P}(|B_1|<-x),$$ and so $$f(x)=1-2Phi(x)$$ where $Phi$ is the cdf of the centered standard Gaussian distribution with density $varphi$.
- Combining the above steps gives begin{align*} mathbb{P}(W_t < 0 , , text{for all} , , t in (1,2)) &= mathbb{E} left[ left( 1- 2Phi(W_1)right) 1_{{W_1<0}} right] \ &= frac{1}{2} - 2 int_{-infty}^0 Phi(x) varphi(x) , dx. end{align*}
- Conclude that $$mathbb{P}(W_t < 0 , , text{for all} , , t in (1,2)) = frac{1}{4}; $$ see e.g. this answer for how to compute the integral in Step 5.
answered Feb 3 at 9:01
sazsaz
82.5k862131
$endgroup$
Hints: Set $$B_t := W_{1+t}-W_1.$$
- Check (or recall) that $(B_t)_{t geq 0}$ is a Brownian motion and that $(B_t)_{t geq 0}$ is independent from $mathcal{F}_1^W:=sigma(W_s; s leq 1)$.
- Use the independence of $(B_t)_{t geq 0}$ and $W_1$ to show that $$mathbb{P}(W_t < 0 , , text{for all} , , t in (1,2) mid W_1) = f(W_1)$$ where $$f(x) := mathbb{P}(x+B_t < 0, , text{for all $t in (0,1)$}). tag{1}$$
- Prove that $f(x)=0$ for all $x geq 0$. (Hint: What happens close to $t=0$?)
- Fix $x<0$. Show that $$f(x) = mathbb{P}(tau_{-x} geq 1)$$ for the stopping time $$tau_{-x} := inf{t>0; B_t geq -x}.$$ Conclude from the reflection principle that $$f(x) = mathbb{P}(|B_1|<-x),$$ and so $$f(x)=1-2Phi(x)$$ where $Phi$ is the cdf of the centered standard Gaussian distribution with density $varphi$.
- Combining the above steps gives begin{align*} mathbb{P}(W_t < 0 , , text{for all} , , t in (1,2)) &= mathbb{E} left[ left( 1- 2Phi(W_1)right) 1_{{W_1<0}} right] \ &= frac{1}{2} - 2 int_{-infty}^0 Phi(x) varphi(x) , dx. end{align*}
- Conclude that $$mathbb{P}(W_t < 0 , , text{for all} , , t in (1,2)) = frac{1}{4}; $$ see e.g. this answer for how to compute the integral in Step 5.
answered Feb 3 at 9:01
sazsaz
82.5k862131
edited Feb 3 at 18:54
answered Feb 3 at 9:01
sazsaz
82.5k862131
answered Feb 3 at 9:01
sazsaz
82.5k862131
answered Feb 3 at 9:01
sazsaz
82.5k862131
82.5k862131
$begingroup$
Yep. (+1) $ $ $ $
$endgroup$
– Did
Feb 3 at 12:08
$begingroup$
@Did Thank you.
$endgroup$
– saz
Feb 3 at 12:13
add a comment |
$begingroup$
Yep. (+1) $ $ $ $
$endgroup$
– Did
Feb 3 at 12:08
$begingroup$
@Did Thank you.
$endgroup$
– saz
Feb 3 at 12:13
$begingroup$
Yep. (+1) $ $ $ $
$endgroup$
– Did
Feb 3 at 12:08
$begingroup$
Yep. (+1) $ $ $ $
$endgroup$
– Did
Feb 3 at 12:08
$begingroup$
@Did Thank you.
$endgroup$
– saz
Feb 3 at 12:13
$begingroup$
@Did Thank you.
$endgroup$
– saz
Feb 3 at 12:13
add a comment |
$begingroup$
Condition on $W_1$.
$endgroup$
– Did
Feb 3 at 8:23
$begingroup$
@JoséCarlosSantos, Som: Why approve an absurd edit?
$endgroup$
– Did
Feb 3 at 8:24
$begingroup$
@pointguard0 More care with the edits you suggest, please.
$endgroup$
– Did
Feb 3 at 8:25
$begingroup$
P(Wt<0 for all 1<t<2)
$endgroup$
– Som
Feb 3 at 8:40
$begingroup$
Som: Very much so. Which is not P(Wt<0) for all 1<t<2, yes?
$endgroup$
– Did
Feb 3 at 8:43