Example of an adapted and jointly measurable stochastic process which is NOT progressively measurable...
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I believe this question has been asked a few years ago, but nobody answered it. Hence the second take:
What is an example of an adapted and jointly measurable stochastic process which is NOT progressively measurable? I would appreciate any help.
First breakthrough: I found this example on p62 of Chung-Williams; however, I don't follow their argument due to the fact that they use analytic sets.
Chung-williams p62
probability-theory measure-theory stochastic-processes
asked Oct 10 '18 at 19:55
phantagarowphantagarow
24
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closed as off-topic by Xander Henderson, user21820, Chris Custer, amWhy, José Carlos Santos Jan 8 at 16:11
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." – Xander Henderson, user21820, Chris Custer, amWhy, José Carlos Santos
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
I believe this question has been asked a few years ago, but nobody answered it. Hence the second take:
What is an example of an adapted and jointly measurable stochastic process which is NOT progressively measurable? I would appreciate any help.
First breakthrough: I found this example on p62 of Chung-Williams; however, I don't follow their argument due to the fact that they use analytic sets.
Chung-williams p62
probability-theory measure-theory stochastic-processes
asked Oct 10 '18 at 19:55
phantagarowphantagarow
24
$endgroup$
closed as off-topic by Xander Henderson, user21820, Chris Custer, amWhy, José Carlos Santos Jan 8 at 16:11
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." – Xander Henderson, user21820, Chris Custer, amWhy, José Carlos Santos
If this question can be reworded to fit the rules in the help center, please edit the question.
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You are probably refering to this question.
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– saz
Oct 11 '18 at 6:31
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You're totally correct!
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– phantagarow
Oct 12 '18 at 20:59
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It appears to me that you have proved answers to this question which don't answer the question at all, but which instead seek to clarify what you are asking. Can you please delete those answers, then edit the question itself to include those comments as additional context?
$endgroup$
– Xander Henderson
Jan 7 at 20:34
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On the contrary, both of my posts provide an example of an adapted and jointly measurable process which is NOT progressively measurable.
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– phantagarow
Jan 7 at 22:11
add a comment |
$begingroup$
I believe this question has been asked a few years ago, but nobody answered it. Hence the second take:
What is an example of an adapted and jointly measurable stochastic process which is NOT progressively measurable? I would appreciate any help.
First breakthrough: I found this example on p62 of Chung-Williams; however, I don't follow their argument due to the fact that they use analytic sets.
Chung-williams p62
probability-theory measure-theory stochastic-processes
asked Oct 10 '18 at 19:55
phantagarowphantagarow
24
$endgroup$
I believe this question has been asked a few years ago, but nobody answered it. Hence the second take:
What is an example of an adapted and jointly measurable stochastic process which is NOT progressively measurable? I would appreciate any help.
First breakthrough: I found this example on p62 of Chung-Williams; however, I don't follow their argument due to the fact that they use analytic sets.
Chung-williams p62
probability-theory measure-theory stochastic-processes
probability-theory measure-theory stochastic-processes
asked Oct 10 '18 at 19:55
phantagarowphantagarow
24
asked Oct 10 '18 at 19:55
phantagarowphantagarow
24
edited Jan 8 at 2:01
phantagarow
asked Oct 10 '18 at 19:55
phantagarowphantagarow
24
asked Oct 10 '18 at 19:55
phantagarowphantagarow
24
asked Oct 10 '18 at 19:55
phantagarowphantagarow
24
24
closed as off-topic by Xander Henderson, user21820, Chris Custer, amWhy, José Carlos Santos Jan 8 at 16:11
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." – Xander Henderson, user21820, Chris Custer, amWhy, José Carlos Santos
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Xander Henderson, user21820, Chris Custer, amWhy, José Carlos Santos Jan 8 at 16:11
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." – Xander Henderson, user21820, Chris Custer, amWhy, José Carlos Santos
If this question can be reworded to fit the rules in the help center, please edit the question.
$begingroup$
You are probably refering to this question.
$endgroup$
– saz
Oct 11 '18 at 6:31
$begingroup$
You're totally correct!
$endgroup$
– phantagarow
Oct 12 '18 at 20:59
$begingroup$
It appears to me that you have proved answers to this question which don't answer the question at all, but which instead seek to clarify what you are asking. Can you please delete those answers, then edit the question itself to include those comments as additional context?
$endgroup$
– Xander Henderson
Jan 7 at 20:34
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On the contrary, both of my posts provide an example of an adapted and jointly measurable process which is NOT progressively measurable.
$endgroup$
– phantagarow
Jan 7 at 22:11
add a comment |
$begingroup$
You are probably refering to this question.
$endgroup$
– saz
Oct 11 '18 at 6:31
$begingroup$
You're totally correct!
$endgroup$
– phantagarow
Oct 12 '18 at 20:59
$begingroup$
It appears to me that you have proved answers to this question which don't answer the question at all, but which instead seek to clarify what you are asking. Can you please delete those answers, then edit the question itself to include those comments as additional context?
$endgroup$
– Xander Henderson
Jan 7 at 20:34
$begingroup$
On the contrary, both of my posts provide an example of an adapted and jointly measurable process which is NOT progressively measurable.
$endgroup$
– phantagarow
Jan 7 at 22:11
$begingroup$
You are probably refering to this question.
$endgroup$
– saz
Oct 11 '18 at 6:31
$begingroup$
You are probably refering to this question.
$endgroup$
– saz
Oct 11 '18 at 6:31
$begingroup$
You're totally correct!
$endgroup$
– phantagarow
Oct 12 '18 at 20:59
$begingroup$
You're totally correct!
$endgroup$
– phantagarow
Oct 12 '18 at 20:59
$begingroup$
It appears to me that you have proved answers to this question which don't answer the question at all, but which instead seek to clarify what you are asking. Can you please delete those answers, then edit the question itself to include those comments as additional context?
$endgroup$
– Xander Henderson
Jan 7 at 20:34
$begingroup$
It appears to me that you have proved answers to this question which don't answer the question at all, but which instead seek to clarify what you are asking. Can you please delete those answers, then edit the question itself to include those comments as additional context?
$endgroup$
– Xander Henderson
Jan 7 at 20:34
$begingroup$
On the contrary, both of my posts provide an example of an adapted and jointly measurable process which is NOT progressively measurable.
$endgroup$
– phantagarow
Jan 7 at 22:11
$begingroup$
On the contrary, both of my posts provide an example of an adapted and jointly measurable process which is NOT progressively measurable.
$endgroup$
– phantagarow
Jan 7 at 22:11
add a comment |
1 Answer
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Second breakthrough: There is a simpler example: $text{Let } Omega = [0,1], mathcal{F}=mathcal{B}[0,1], mathbb{P}=text{Lebesgue measure}, mathcal{F}_{t}=text{completion of }{emptyset, Omega} text{wrt Lebesgue measure }forall t in [0,1], Delta=text{diagonal in } Omega times [0,1]. text{Finally set} $
$$
Y =mathbb{1}_{Delta}:Omega times[0,1]rightarrowmathbb{R}
$$
Then $Y$ is adapted and jointly measurable, but not progressively measurable.
answered Jan 7 at 20:08
phantagarowphantagarow
24
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add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Second breakthrough: There is a simpler example: $text{Let } Omega = [0,1], mathcal{F}=mathcal{B}[0,1], mathbb{P}=text{Lebesgue measure}, mathcal{F}_{t}=text{completion of }{emptyset, Omega} text{wrt Lebesgue measure }forall t in [0,1], Delta=text{diagonal in } Omega times [0,1]. text{Finally set} $
$$
Y =mathbb{1}_{Delta}:Omega times[0,1]rightarrowmathbb{R}
$$
Then $Y$ is adapted and jointly measurable, but not progressively measurable.
answered Jan 7 at 20:08
phantagarowphantagarow
24
$endgroup$
add a comment |
$begingroup$
Second breakthrough: There is a simpler example: $text{Let } Omega = [0,1], mathcal{F}=mathcal{B}[0,1], mathbb{P}=text{Lebesgue measure}, mathcal{F}_{t}=text{completion of }{emptyset, Omega} text{wrt Lebesgue measure }forall t in [0,1], Delta=text{diagonal in } Omega times [0,1]. text{Finally set} $
$$
Y =mathbb{1}_{Delta}:Omega times[0,1]rightarrowmathbb{R}
$$
Then $Y$ is adapted and jointly measurable, but not progressively measurable.
answered Jan 7 at 20:08
phantagarowphantagarow
24
$endgroup$
add a comment |
$begingroup$
Second breakthrough: There is a simpler example: $text{Let } Omega = [0,1], mathcal{F}=mathcal{B}[0,1], mathbb{P}=text{Lebesgue measure}, mathcal{F}_{t}=text{completion of }{emptyset, Omega} text{wrt Lebesgue measure }forall t in [0,1], Delta=text{diagonal in } Omega times [0,1]. text{Finally set} $
$$
Y =mathbb{1}_{Delta}:Omega times[0,1]rightarrowmathbb{R}
$$
Then $Y$ is adapted and jointly measurable, but not progressively measurable.
answered Jan 7 at 20:08
phantagarowphantagarow
24
$endgroup$
Second breakthrough: There is a simpler example: $text{Let } Omega = [0,1], mathcal{F}=mathcal{B}[0,1], mathbb{P}=text{Lebesgue measure}, mathcal{F}_{t}=text{completion of }{emptyset, Omega} text{wrt Lebesgue measure }forall t in [0,1], Delta=text{diagonal in } Omega times [0,1]. text{Finally set} $
$$
Y =mathbb{1}_{Delta}:Omega times[0,1]rightarrowmathbb{R}
$$
Then $Y$ is adapted and jointly measurable, but not progressively measurable.
answered Jan 7 at 20:08
phantagarowphantagarow
24
edited Jan 8 at 2:03
answered Jan 7 at 20:08
phantagarowphantagarow
24
answered Jan 7 at 20:08
phantagarowphantagarow
24
answered Jan 7 at 20:08
phantagarowphantagarow
24
24
add a comment |
add a comment |
$begingroup$
You are probably refering to this question.
$endgroup$
– saz
Oct 11 '18 at 6:31
$begingroup$
You're totally correct!
$endgroup$
– phantagarow
Oct 12 '18 at 20:59
$begingroup$
It appears to me that you have proved answers to this question which don't answer the question at all, but which instead seek to clarify what you are asking. Can you please delete those answers, then edit the question itself to include those comments as additional context?
$endgroup$
– Xander Henderson
Jan 7 at 20:34
$begingroup$
On the contrary, both of my posts provide an example of an adapted and jointly measurable process which is NOT progressively measurable.
$endgroup$
– phantagarow
Jan 7 at 22:11