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Why do we need to include filtrations in the definition of probability spaces when talking about stochastic processes.
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In the first line of these notes the author defines his stochastic process. Using a filtration. What is the importance of filtrations?
https://warwick.ac.uk/fac/sci/maths/people/staff/stefan_grosskinsky/ma4h3/ma4h3-0809/notes_ips_book.pdf
probability-theory stochastic-processes filtrations
asked Jul 24 at 16:48
Monty
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In the first line of these notes the author defines his stochastic process. Using a filtration. What is the importance of filtrations?
https://warwick.ac.uk/fac/sci/maths/people/staff/stefan_grosskinsky/ma4h3/ma4h3-0809/notes_ips_book.pdf
probability-theory stochastic-processes filtrations
asked Jul 24 at 16:48
Monty
15212
1
We want the stochastic process to be measurable for every time $t$. "Measurable with respect to what?" The filtration.
– user223391
Jul 24 at 16:50
I can't access that link.
– saulspatz
Jul 24 at 16:51
Cant you say measurable with respect to F instead of ${F_t$ where F is the smallest sigma-algebra which makes the mapping measurable for any t>0
– Monty
Jul 24 at 16:53
Only if $F_t=F$, which is rarely the case.
– user223391
Jul 24 at 17:06
1
Which mapping? Details and carefully written propositions already answer the question. To have a quick idea, imagine instead of a filtration $F=(F_t)_tin T=0,1,2,3,dots$ and and adapted process to it $X$ the following situation. The filtration is a newspaper providing information and $X$ is a TV sender using only the information from the newspaper. So at day $t$, the program $X_t$ can only tell stuff in the past journals. (To have a random variable, consider only the stock prices from the journal.)
– dan_fulea
Jul 24 at 17:11
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up vote
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down vote
favorite
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1
In the first line of these notes the author defines his stochastic process. Using a filtration. What is the importance of filtrations?
https://warwick.ac.uk/fac/sci/maths/people/staff/stefan_grosskinsky/ma4h3/ma4h3-0809/notes_ips_book.pdf
probability-theory stochastic-processes filtrations
asked Jul 24 at 16:48
Monty
15212
In the first line of these notes the author defines his stochastic process. Using a filtration. What is the importance of filtrations?
https://warwick.ac.uk/fac/sci/maths/people/staff/stefan_grosskinsky/ma4h3/ma4h3-0809/notes_ips_book.pdf
probability-theory stochastic-processes filtrations
asked Jul 24 at 16:48
Monty
15212
asked Jul 24 at 16:48
Monty
15212
asked Jul 24 at 16:48
Monty
15212
asked Jul 24 at 16:48
Monty
15212
15212
1
We want the stochastic process to be measurable for every time $t$. "Measurable with respect to what?" The filtration.
– user223391
Jul 24 at 16:50
I can't access that link.
– saulspatz
Jul 24 at 16:51
Cant you say measurable with respect to F instead of ${F_t$ where F is the smallest sigma-algebra which makes the mapping measurable for any t>0
– Monty
Jul 24 at 16:53
Only if $F_t=F$, which is rarely the case.
– user223391
Jul 24 at 17:06
1
Which mapping? Details and carefully written propositions already answer the question. To have a quick idea, imagine instead of a filtration $F=(F_t)_tin T=0,1,2,3,dots$ and and adapted process to it $X$ the following situation. The filtration is a newspaper providing information and $X$ is a TV sender using only the information from the newspaper. So at day $t$, the program $X_t$ can only tell stuff in the past journals. (To have a random variable, consider only the stock prices from the journal.)
– dan_fulea
Jul 24 at 17:11
add a comment |Â
1
We want the stochastic process to be measurable for every time $t$. "Measurable with respect to what?" The filtration.
– user223391
Jul 24 at 16:50
I can't access that link.
– saulspatz
Jul 24 at 16:51
Cant you say measurable with respect to F instead of ${F_t$ where F is the smallest sigma-algebra which makes the mapping measurable for any t>0
– Monty
Jul 24 at 16:53
Only if $F_t=F$, which is rarely the case.
– user223391
Jul 24 at 17:06
1
Which mapping? Details and carefully written propositions already answer the question. To have a quick idea, imagine instead of a filtration $F=(F_t)_tin T=0,1,2,3,dots$ and and adapted process to it $X$ the following situation. The filtration is a newspaper providing information and $X$ is a TV sender using only the information from the newspaper. So at day $t$, the program $X_t$ can only tell stuff in the past journals. (To have a random variable, consider only the stock prices from the journal.)
– dan_fulea
Jul 24 at 17:11
1
1
We want the stochastic process to be measurable for every time $t$. "Measurable with respect to what?" The filtration.
– user223391
Jul 24 at 16:50
We want the stochastic process to be measurable for every time $t$. "Measurable with respect to what?" The filtration.
– user223391
Jul 24 at 16:50
I can't access that link.
– saulspatz
Jul 24 at 16:51
I can't access that link.
– saulspatz
Jul 24 at 16:51
Cant you say measurable with respect to F instead of ${F_t$ where F is the smallest sigma-algebra which makes the mapping measurable for any t>0
– Monty
Jul 24 at 16:53
Cant you say measurable with respect to F instead of ${F_t$ where F is the smallest sigma-algebra which makes the mapping measurable for any t>0
– Monty
Jul 24 at 16:53
Only if $F_t=F$, which is rarely the case.
– user223391
Jul 24 at 17:06
Only if $F_t=F$, which is rarely the case.
– user223391
Jul 24 at 17:06
1
1
Which mapping? Details and carefully written propositions already answer the question. To have a quick idea, imagine instead of a filtration $F=(F_t)_tin T=0,1,2,3,dots$ and and adapted process to it $X$ the following situation. The filtration is a newspaper providing information and $X$ is a TV sender using only the information from the newspaper. So at day $t$, the program $X_t$ can only tell stuff in the past journals. (To have a random variable, consider only the stock prices from the journal.)
– dan_fulea
Jul 24 at 17:11
Which mapping? Details and carefully written propositions already answer the question. To have a quick idea, imagine instead of a filtration $F=(F_t)_tin T=0,1,2,3,dots$ and and adapted process to it $X$ the following situation. The filtration is a newspaper providing information and $X$ is a TV sender using only the information from the newspaper. So at day $t$, the program $X_t$ can only tell stuff in the past journals. (To have a random variable, consider only the stock prices from the journal.)
– dan_fulea
Jul 24 at 17:11
add a comment |Â
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1
We want the stochastic process to be measurable for every time $t$. "Measurable with respect to what?" The filtration.
– user223391
Jul 24 at 16:50
I can't access that link.
– saulspatz
Jul 24 at 16:51
Cant you say measurable with respect to F instead of ${F_t$ where F is the smallest sigma-algebra which makes the mapping measurable for any t>0
– Monty
Jul 24 at 16:53
Only if $F_t=F$, which is rarely the case.
– user223391
Jul 24 at 17:06
1
Which mapping? Details and carefully written propositions already answer the question. To have a quick idea, imagine instead of a filtration $F=(F_t)_tin T=0,1,2,3,dots$ and and adapted process to it $X$ the following situation. The filtration is a newspaper providing information and $X$ is a TV sender using only the information from the newspaper. So at day $t$, the program $X_t$ can only tell stuff in the past journals. (To have a random variable, consider only the stock prices from the journal.)
– dan_fulea
Jul 24 at 17:11