Mixing
Collection: Results on stopping times for Brownian motion (with drift)
$begingroup$
The aim of this question is to collect results on stopping times of Brownian motion (possibly with drift), with a focus on distributional properties:
- distributions of stopping times (Laplace transform, moments,..)
- distributional properties of the stopped process (computation/finiteness of moments, ...)
Many of the results, which I have in mind, are typical homework problems.
What is the motivation for such a collection?
There is a number of "classical" stopping times for Brownian motion, but unfortunately these stopping times don't have a specific name (apart from "exit time", "hitting time", ... - which is also not very specific), and this makes it hard to find results here on StackExchange. Sometimes, when I'm looking for a result, I know that it is somewhere here on MSE but I'm simply not able to find it. For other questions, which are asked very frequently in MSE, it is often difficult to find a good "old" answer.
In any case, I believe that it would be a benefit to make the knowledge easier to access - both for students (who are trying to solve their homework problems) as for the "teachers" (who are answering questions on MSE).
To make this list a helpful tool (e.g. for answering questions) please make sure to give a short but concise description of each result which you list in your answer.
probability-theory brownian-motion big-list stopping-times
asked Jan 19 at 11:06
sazsaz
81.3k861127
$endgroup$
add a comment |
$begingroup$
The aim of this question is to collect results on stopping times of Brownian motion (possibly with drift), with a focus on distributional properties:
- distributions of stopping times (Laplace transform, moments,..)
- distributional properties of the stopped process (computation/finiteness of moments, ...)
Many of the results, which I have in mind, are typical homework problems.
What is the motivation for such a collection?
There is a number of "classical" stopping times for Brownian motion, but unfortunately these stopping times don't have a specific name (apart from "exit time", "hitting time", ... - which is also not very specific), and this makes it hard to find results here on StackExchange. Sometimes, when I'm looking for a result, I know that it is somewhere here on MSE but I'm simply not able to find it. For other questions, which are asked very frequently in MSE, it is often difficult to find a good "old" answer.
In any case, I believe that it would be a benefit to make the knowledge easier to access - both for students (who are trying to solve their homework problems) as for the "teachers" (who are answering questions on MSE).
To make this list a helpful tool (e.g. for answering questions) please make sure to give a short but concise description of each result which you list in your answer.
probability-theory brownian-motion big-list stopping-times
asked Jan 19 at 11:06
sazsaz
81.3k861127
$endgroup$
add a comment |
$begingroup$
The aim of this question is to collect results on stopping times of Brownian motion (possibly with drift), with a focus on distributional properties:
- distributions of stopping times (Laplace transform, moments,..)
- distributional properties of the stopped process (computation/finiteness of moments, ...)
Many of the results, which I have in mind, are typical homework problems.
What is the motivation for such a collection?
There is a number of "classical" stopping times for Brownian motion, but unfortunately these stopping times don't have a specific name (apart from "exit time", "hitting time", ... - which is also not very specific), and this makes it hard to find results here on StackExchange. Sometimes, when I'm looking for a result, I know that it is somewhere here on MSE but I'm simply not able to find it. For other questions, which are asked very frequently in MSE, it is often difficult to find a good "old" answer.
In any case, I believe that it would be a benefit to make the knowledge easier to access - both for students (who are trying to solve their homework problems) as for the "teachers" (who are answering questions on MSE).
To make this list a helpful tool (e.g. for answering questions) please make sure to give a short but concise description of each result which you list in your answer.
probability-theory brownian-motion big-list stopping-times
asked Jan 19 at 11:06
sazsaz
81.3k861127
$endgroup$
The aim of this question is to collect results on stopping times of Brownian motion (possibly with drift), with a focus on distributional properties:
- distributions of stopping times (Laplace transform, moments,..)
- distributional properties of the stopped process (computation/finiteness of moments, ...)
Many of the results, which I have in mind, are typical homework problems.
What is the motivation for such a collection?
There is a number of "classical" stopping times for Brownian motion, but unfortunately these stopping times don't have a specific name (apart from "exit time", "hitting time", ... - which is also not very specific), and this makes it hard to find results here on StackExchange. Sometimes, when I'm looking for a result, I know that it is somewhere here on MSE but I'm simply not able to find it. For other questions, which are asked very frequently in MSE, it is often difficult to find a good "old" answer.
In any case, I believe that it would be a benefit to make the knowledge easier to access - both for students (who are trying to solve their homework problems) as for the "teachers" (who are answering questions on MSE).
To make this list a helpful tool (e.g. for answering questions) please make sure to give a short but concise description of each result which you list in your answer.
probability-theory brownian-motion big-list stopping-times
probability-theory brownian-motion big-list stopping-times
asked Jan 19 at 11:06
sazsaz
81.3k861127
asked Jan 19 at 11:06
sazsaz
81.3k861127
asked Jan 19 at 11:06
sazsaz
81.3k861127
asked Jan 19 at 11:06
sazsaz
81.3k861127
asked Jan 19 at 11:06
sazsaz
81.3k861127
81.3k861127
add a comment |
add a comment |
1 Answer
1
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oldest
votes
$begingroup$
Below, $(X_t)_{t geq 0}$ is either a Brownian motion (BM, for short) or a Brownian motion with drift. For each of the items in my list I will indicate for which process the corresponding result was obtained.
$tau := inf{t geq 0; X_t = a}$ for $a>0$.
Note: We have $tau=inf{t geq 0; X_t geq a}$ a.s. if $(X_t)_{t geq 0}$ is a BM, see this answer.
- (BM) $mathbb{P}(tau<infty)=1$ (via martingale methods)
- (BM with non-negative drift) $mathbb{P}(tau<infty)=1$ (via law of iterated logarithm)
- (BM with negative drift) Computation of $mathbb{P}(tau<infty)$
- (BM) $mathbb{E}tau = infty$
- (BM with drift) density of the distribution of $tau$
- (BM with positive drift) Laplace transform of $tau$
$tau= inf{t geq 0; X_t notin [a,b]}$
- (BM) Distribution of $X_{tau}$
- (BM) Computation of $mathbb{E}(tau)$ (see this question for BM started at $X_0=x$)
- (BM) Computation of $mathbb{E}(tau^2)$
- (BM) $mathbb{E}(tau^p) < infty$ for all $p geq 1$
- (BM; $a=b$) Laplace transform of $tau$ (see also this question)
Hitting times for some curves
- (BM) $mathbb{E}tau=infty$ for $tau := inf{t; B_t^2 geq 1+t}$ (see also this question)
- (BM) $mathbb{E}(tau) =tfrac{1}{2}$ for $tau = inf{t; B_t^2 = 1-t}$
- (BM) $mathbb{E}tau<infty$ for $tau:=inf{t; |B_t| = tfrac{1}{2}(1+sqrt{1+t})}$
- (BM) $mathbb{E}(tau)=infty$ for $tau=inf{t; B_t geq e^{-lambda t}}$
Miscellaneous
- (BM) $tau=inf{t geq 0; X_t notin (a,b)}$ is not a stopping time with respect to the canocical filtration
- (BM) Proof of Wald's identity
answered Jan 19 at 11:09
sazsaz
81.3k861127
$endgroup$
add a comment |
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1 Answer
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1 Answer
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$begingroup$
Below, $(X_t)_{t geq 0}$ is either a Brownian motion (BM, for short) or a Brownian motion with drift. For each of the items in my list I will indicate for which process the corresponding result was obtained.
$tau := inf{t geq 0; X_t = a}$ for $a>0$.
Note: We have $tau=inf{t geq 0; X_t geq a}$ a.s. if $(X_t)_{t geq 0}$ is a BM, see this answer.
- (BM) $mathbb{P}(tau<infty)=1$ (via martingale methods)
- (BM with non-negative drift) $mathbb{P}(tau<infty)=1$ (via law of iterated logarithm)
- (BM with negative drift) Computation of $mathbb{P}(tau<infty)$
- (BM) $mathbb{E}tau = infty$
- (BM with drift) density of the distribution of $tau$
- (BM with positive drift) Laplace transform of $tau$
$tau= inf{t geq 0; X_t notin [a,b]}$
- (BM) Distribution of $X_{tau}$
- (BM) Computation of $mathbb{E}(tau)$ (see this question for BM started at $X_0=x$)
- (BM) Computation of $mathbb{E}(tau^2)$
- (BM) $mathbb{E}(tau^p) < infty$ for all $p geq 1$
- (BM; $a=b$) Laplace transform of $tau$ (see also this question)
Hitting times for some curves
- (BM) $mathbb{E}tau=infty$ for $tau := inf{t; B_t^2 geq 1+t}$ (see also this question)
- (BM) $mathbb{E}(tau) =tfrac{1}{2}$ for $tau = inf{t; B_t^2 = 1-t}$
- (BM) $mathbb{E}tau<infty$ for $tau:=inf{t; |B_t| = tfrac{1}{2}(1+sqrt{1+t})}$
- (BM) $mathbb{E}(tau)=infty$ for $tau=inf{t; B_t geq e^{-lambda t}}$
Miscellaneous
- (BM) $tau=inf{t geq 0; X_t notin (a,b)}$ is not a stopping time with respect to the canocical filtration
- (BM) Proof of Wald's identity
answered Jan 19 at 11:09
sazsaz
81.3k861127
$endgroup$
add a comment |
$begingroup$
Below, $(X_t)_{t geq 0}$ is either a Brownian motion (BM, for short) or a Brownian motion with drift. For each of the items in my list I will indicate for which process the corresponding result was obtained.
$tau := inf{t geq 0; X_t = a}$ for $a>0$.
Note: We have $tau=inf{t geq 0; X_t geq a}$ a.s. if $(X_t)_{t geq 0}$ is a BM, see this answer.
- (BM) $mathbb{P}(tau<infty)=1$ (via martingale methods)
- (BM with non-negative drift) $mathbb{P}(tau<infty)=1$ (via law of iterated logarithm)
- (BM with negative drift) Computation of $mathbb{P}(tau<infty)$
- (BM) $mathbb{E}tau = infty$
- (BM with drift) density of the distribution of $tau$
- (BM with positive drift) Laplace transform of $tau$
$tau= inf{t geq 0; X_t notin [a,b]}$
- (BM) Distribution of $X_{tau}$
- (BM) Computation of $mathbb{E}(tau)$ (see this question for BM started at $X_0=x$)
- (BM) Computation of $mathbb{E}(tau^2)$
- (BM) $mathbb{E}(tau^p) < infty$ for all $p geq 1$
- (BM; $a=b$) Laplace transform of $tau$ (see also this question)
Hitting times for some curves
- (BM) $mathbb{E}tau=infty$ for $tau := inf{t; B_t^2 geq 1+t}$ (see also this question)
- (BM) $mathbb{E}(tau) =tfrac{1}{2}$ for $tau = inf{t; B_t^2 = 1-t}$
- (BM) $mathbb{E}tau<infty$ for $tau:=inf{t; |B_t| = tfrac{1}{2}(1+sqrt{1+t})}$
- (BM) $mathbb{E}(tau)=infty$ for $tau=inf{t; B_t geq e^{-lambda t}}$
Miscellaneous
- (BM) $tau=inf{t geq 0; X_t notin (a,b)}$ is not a stopping time with respect to the canocical filtration
- (BM) Proof of Wald's identity
answered Jan 19 at 11:09
sazsaz
81.3k861127
$endgroup$
add a comment |
$begingroup$
Below, $(X_t)_{t geq 0}$ is either a Brownian motion (BM, for short) or a Brownian motion with drift. For each of the items in my list I will indicate for which process the corresponding result was obtained.
$tau := inf{t geq 0; X_t = a}$ for $a>0$.
Note: We have $tau=inf{t geq 0; X_t geq a}$ a.s. if $(X_t)_{t geq 0}$ is a BM, see this answer.
- (BM) $mathbb{P}(tau<infty)=1$ (via martingale methods)
- (BM with non-negative drift) $mathbb{P}(tau<infty)=1$ (via law of iterated logarithm)
- (BM with negative drift) Computation of $mathbb{P}(tau<infty)$
- (BM) $mathbb{E}tau = infty$
- (BM with drift) density of the distribution of $tau$
- (BM with positive drift) Laplace transform of $tau$
$tau= inf{t geq 0; X_t notin [a,b]}$
- (BM) Distribution of $X_{tau}$
- (BM) Computation of $mathbb{E}(tau)$ (see this question for BM started at $X_0=x$)
- (BM) Computation of $mathbb{E}(tau^2)$
- (BM) $mathbb{E}(tau^p) < infty$ for all $p geq 1$
- (BM; $a=b$) Laplace transform of $tau$ (see also this question)
Hitting times for some curves
- (BM) $mathbb{E}tau=infty$ for $tau := inf{t; B_t^2 geq 1+t}$ (see also this question)
- (BM) $mathbb{E}(tau) =tfrac{1}{2}$ for $tau = inf{t; B_t^2 = 1-t}$
- (BM) $mathbb{E}tau<infty$ for $tau:=inf{t; |B_t| = tfrac{1}{2}(1+sqrt{1+t})}$
- (BM) $mathbb{E}(tau)=infty$ for $tau=inf{t; B_t geq e^{-lambda t}}$
Miscellaneous
- (BM) $tau=inf{t geq 0; X_t notin (a,b)}$ is not a stopping time with respect to the canocical filtration
- (BM) Proof of Wald's identity
answered Jan 19 at 11:09
sazsaz
81.3k861127
$endgroup$
Below, $(X_t)_{t geq 0}$ is either a Brownian motion (BM, for short) or a Brownian motion with drift. For each of the items in my list I will indicate for which process the corresponding result was obtained.
$tau := inf{t geq 0; X_t = a}$ for $a>0$.
Note: We have $tau=inf{t geq 0; X_t geq a}$ a.s. if $(X_t)_{t geq 0}$ is a BM, see this answer.
- (BM) $mathbb{P}(tau<infty)=1$ (via martingale methods)
- (BM with non-negative drift) $mathbb{P}(tau<infty)=1$ (via law of iterated logarithm)
- (BM with negative drift) Computation of $mathbb{P}(tau<infty)$
- (BM) $mathbb{E}tau = infty$
- (BM with drift) density of the distribution of $tau$
- (BM with positive drift) Laplace transform of $tau$
$tau= inf{t geq 0; X_t notin [a,b]}$
- (BM) Distribution of $X_{tau}$
- (BM) Computation of $mathbb{E}(tau)$ (see this question for BM started at $X_0=x$)
- (BM) Computation of $mathbb{E}(tau^2)$
- (BM) $mathbb{E}(tau^p) < infty$ for all $p geq 1$
- (BM; $a=b$) Laplace transform of $tau$ (see also this question)
Hitting times for some curves
- (BM) $mathbb{E}tau=infty$ for $tau := inf{t; B_t^2 geq 1+t}$ (see also this question)
- (BM) $mathbb{E}(tau) =tfrac{1}{2}$ for $tau = inf{t; B_t^2 = 1-t}$
- (BM) $mathbb{E}tau<infty$ for $tau:=inf{t; |B_t| = tfrac{1}{2}(1+sqrt{1+t})}$
- (BM) $mathbb{E}(tau)=infty$ for $tau=inf{t; B_t geq e^{-lambda t}}$
Miscellaneous
- (BM) $tau=inf{t geq 0; X_t notin (a,b)}$ is not a stopping time with respect to the canocical filtration
- (BM) Proof of Wald's identity
answered Jan 19 at 11:09
sazsaz
81.3k861127
edited Feb 6 at 9:41
answered Jan 19 at 11:09
sazsaz
81.3k861127
answered Jan 19 at 11:09
sazsaz
81.3k861127
answered Jan 19 at 11:09
sazsaz
81.3k861127
81.3k861127
add a comment |
add a comment |
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