Mixing
adapted process, translation between measurable and information?
Clash Royale CLAN TAG#URR8PPP
up vote
1
down vote
favorite
Although there are plenty of questions and answers on understanding the intuition for adapted process like this post and this post I am still unclear on how an adapted filtration 'captures the information of the process up to time $t$.'
First some notation, let
$(Omega, mathcalF, P)$ be a probability space.
$X:Ttimes Omega to (S, Sigma)$ be a stochastic process. That is $X_t:= X(t,cdot)$ is an $(mathcalF, Sigma)$-measurable function for all $tin T.$
$mathcalF_t$ such that $mathcalF_ssubset mathcalF_t$ for $sleq t$ be a filtration of $mathcalF.$
Definition: We say $X_t$ is adapted to the filtration $mathcalF_t$ if $X_t$ is $mathcalF_t$-measurable for all $t.$
The definition is clear as day but I am confused about the interpretation/intuition/motivation. A classic example is the price of a stock, $S_t$ which is adapted to the natural filtration of the Brownian motion on which it is modeled $S_t = exp(ut + sigma B_t).$ The interpretation on the adapted filtration condition is that we only know the current price of the stock (and its history) but we don't know the future price of the stock.
Question: (stock version): Why does the stock being adapted filtration have anything to do with knowing its current and historical prices?
I don't see the connection mathematically. In real life we have a stock and we know its price and its history. I can't see the connection between this and that condition that $S_t$ be $mathcalF_t$-measurable. A discrete example might help illustrate my confusion.
Discrete Example
Consider the example of flipping a coin 3 times. With $X_i$ being 1 if the $i$th flip is heads and $X_i$ is 0 if the $i$th flip is tails. Let
$$Omega = hhh, hht, hth, htt,ttt, tth, tht, thh$$ be the state space of the experiment and $mathcalF$ all subsets of $Omega.$ Then we define the stochastic process $X$ by $$X:1,2,3times Omega to (S,Sigma)$$ with $S=0,1$ and $Sigma$ all subsets of $S.$
The smallest possible filtration to which $X$ is adapted is the natural filtration. We compute
$$beginalign
mathcalF_1 &= sigma(X_1^-1(A))\
&= sigma(X_1^-1(emptyset), X_1^-1(0), X_1^-1(1), X_1^-1(0,1)\
&= emptyset, ttt,tth,tht,thh, hhh,hht,hth,htt, Omega
endalign$$
In this case adaptability translates to forcing $X_1$ to constant on the sets $ttt,tth,tht,thh$ and $hhh,hht,hth,htt.$ But how does this help us? It certainly doesn't tell us which set happened and which didn't. I can see that $X_2$ is not $mathcalF_1$-measurable, but so what? Knowing $mathcalF_1$ doesn't give us any knowledge about the result of the first toss, other that it was either a heads or a tails, which we already knew. So now I am ready to state my more general question
Question Why does the $sigma$-algebra $mathcalF_t$ 'contain information' about the process up to time $t$ and what does this mean? How can we use it to say something concrete about what has actually happened up to time $t$ and how can we justify this mathematically?
measure-theory stochastic-processes finance filtrations
asked Jul 23 at 20:07
moquant
82
add a comment |Â
up vote
1
down vote
favorite
Although there are plenty of questions and answers on understanding the intuition for adapted process like this post and this post I am still unclear on how an adapted filtration 'captures the information of the process up to time $t$.'
First some notation, let
$(Omega, mathcalF, P)$ be a probability space.
$X:Ttimes Omega to (S, Sigma)$ be a stochastic process. That is $X_t:= X(t,cdot)$ is an $(mathcalF, Sigma)$-measurable function for all $tin T.$
$mathcalF_t$ such that $mathcalF_ssubset mathcalF_t$ for $sleq t$ be a filtration of $mathcalF.$
Definition: We say $X_t$ is adapted to the filtration $mathcalF_t$ if $X_t$ is $mathcalF_t$-measurable for all $t.$
The definition is clear as day but I am confused about the interpretation/intuition/motivation. A classic example is the price of a stock, $S_t$ which is adapted to the natural filtration of the Brownian motion on which it is modeled $S_t = exp(ut + sigma B_t).$ The interpretation on the adapted filtration condition is that we only know the current price of the stock (and its history) but we don't know the future price of the stock.
Question: (stock version): Why does the stock being adapted filtration have anything to do with knowing its current and historical prices?
I don't see the connection mathematically. In real life we have a stock and we know its price and its history. I can't see the connection between this and that condition that $S_t$ be $mathcalF_t$-measurable. A discrete example might help illustrate my confusion.
Discrete Example
Consider the example of flipping a coin 3 times. With $X_i$ being 1 if the $i$th flip is heads and $X_i$ is 0 if the $i$th flip is tails. Let
$$Omega = hhh, hht, hth, htt,ttt, tth, tht, thh$$ be the state space of the experiment and $mathcalF$ all subsets of $Omega.$ Then we define the stochastic process $X$ by $$X:1,2,3times Omega to (S,Sigma)$$ with $S=0,1$ and $Sigma$ all subsets of $S.$
The smallest possible filtration to which $X$ is adapted is the natural filtration. We compute
$$beginalign
mathcalF_1 &= sigma(X_1^-1(A))\
&= sigma(X_1^-1(emptyset), X_1^-1(0), X_1^-1(1), X_1^-1(0,1)\
&= emptyset, ttt,tth,tht,thh, hhh,hht,hth,htt, Omega
endalign$$
In this case adaptability translates to forcing $X_1$ to constant on the sets $ttt,tth,tht,thh$ and $hhh,hht,hth,htt.$ But how does this help us? It certainly doesn't tell us which set happened and which didn't. I can see that $X_2$ is not $mathcalF_1$-measurable, but so what? Knowing $mathcalF_1$ doesn't give us any knowledge about the result of the first toss, other that it was either a heads or a tails, which we already knew. So now I am ready to state my more general question
Question Why does the $sigma$-algebra $mathcalF_t$ 'contain information' about the process up to time $t$ and what does this mean? How can we use it to say something concrete about what has actually happened up to time $t$ and how can we justify this mathematically?
measure-theory stochastic-processes finance filtrations
asked Jul 23 at 20:07
moquant
82
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Although there are plenty of questions and answers on understanding the intuition for adapted process like this post and this post I am still unclear on how an adapted filtration 'captures the information of the process up to time $t$.'
First some notation, let
$(Omega, mathcalF, P)$ be a probability space.
$X:Ttimes Omega to (S, Sigma)$ be a stochastic process. That is $X_t:= X(t,cdot)$ is an $(mathcalF, Sigma)$-measurable function for all $tin T.$
$mathcalF_t$ such that $mathcalF_ssubset mathcalF_t$ for $sleq t$ be a filtration of $mathcalF.$
Definition: We say $X_t$ is adapted to the filtration $mathcalF_t$ if $X_t$ is $mathcalF_t$-measurable for all $t.$
The definition is clear as day but I am confused about the interpretation/intuition/motivation. A classic example is the price of a stock, $S_t$ which is adapted to the natural filtration of the Brownian motion on which it is modeled $S_t = exp(ut + sigma B_t).$ The interpretation on the adapted filtration condition is that we only know the current price of the stock (and its history) but we don't know the future price of the stock.
Question: (stock version): Why does the stock being adapted filtration have anything to do with knowing its current and historical prices?
I don't see the connection mathematically. In real life we have a stock and we know its price and its history. I can't see the connection between this and that condition that $S_t$ be $mathcalF_t$-measurable. A discrete example might help illustrate my confusion.
Discrete Example
Consider the example of flipping a coin 3 times. With $X_i$ being 1 if the $i$th flip is heads and $X_i$ is 0 if the $i$th flip is tails. Let
$$Omega = hhh, hht, hth, htt,ttt, tth, tht, thh$$ be the state space of the experiment and $mathcalF$ all subsets of $Omega.$ Then we define the stochastic process $X$ by $$X:1,2,3times Omega to (S,Sigma)$$ with $S=0,1$ and $Sigma$ all subsets of $S.$
The smallest possible filtration to which $X$ is adapted is the natural filtration. We compute
$$beginalign
mathcalF_1 &= sigma(X_1^-1(A))\
&= sigma(X_1^-1(emptyset), X_1^-1(0), X_1^-1(1), X_1^-1(0,1)\
&= emptyset, ttt,tth,tht,thh, hhh,hht,hth,htt, Omega
endalign$$
In this case adaptability translates to forcing $X_1$ to constant on the sets $ttt,tth,tht,thh$ and $hhh,hht,hth,htt.$ But how does this help us? It certainly doesn't tell us which set happened and which didn't. I can see that $X_2$ is not $mathcalF_1$-measurable, but so what? Knowing $mathcalF_1$ doesn't give us any knowledge about the result of the first toss, other that it was either a heads or a tails, which we already knew. So now I am ready to state my more general question
Question Why does the $sigma$-algebra $mathcalF_t$ 'contain information' about the process up to time $t$ and what does this mean? How can we use it to say something concrete about what has actually happened up to time $t$ and how can we justify this mathematically?
measure-theory stochastic-processes finance filtrations
asked Jul 23 at 20:07
moquant
82
Although there are plenty of questions and answers on understanding the intuition for adapted process like this post and this post I am still unclear on how an adapted filtration 'captures the information of the process up to time $t$.'
First some notation, let
$(Omega, mathcalF, P)$ be a probability space.
$X:Ttimes Omega to (S, Sigma)$ be a stochastic process. That is $X_t:= X(t,cdot)$ is an $(mathcalF, Sigma)$-measurable function for all $tin T.$
$mathcalF_t$ such that $mathcalF_ssubset mathcalF_t$ for $sleq t$ be a filtration of $mathcalF.$
Definition: We say $X_t$ is adapted to the filtration $mathcalF_t$ if $X_t$ is $mathcalF_t$-measurable for all $t.$
The definition is clear as day but I am confused about the interpretation/intuition/motivation. A classic example is the price of a stock, $S_t$ which is adapted to the natural filtration of the Brownian motion on which it is modeled $S_t = exp(ut + sigma B_t).$ The interpretation on the adapted filtration condition is that we only know the current price of the stock (and its history) but we don't know the future price of the stock.
Question: (stock version): Why does the stock being adapted filtration have anything to do with knowing its current and historical prices?
I don't see the connection mathematically. In real life we have a stock and we know its price and its history. I can't see the connection between this and that condition that $S_t$ be $mathcalF_t$-measurable. A discrete example might help illustrate my confusion.
Discrete Example
Consider the example of flipping a coin 3 times. With $X_i$ being 1 if the $i$th flip is heads and $X_i$ is 0 if the $i$th flip is tails. Let
$$Omega = hhh, hht, hth, htt,ttt, tth, tht, thh$$ be the state space of the experiment and $mathcalF$ all subsets of $Omega.$ Then we define the stochastic process $X$ by $$X:1,2,3times Omega to (S,Sigma)$$ with $S=0,1$ and $Sigma$ all subsets of $S.$
The smallest possible filtration to which $X$ is adapted is the natural filtration. We compute
$$beginalign
mathcalF_1 &= sigma(X_1^-1(A))\
&= sigma(X_1^-1(emptyset), X_1^-1(0), X_1^-1(1), X_1^-1(0,1)\
&= emptyset, ttt,tth,tht,thh, hhh,hht,hth,htt, Omega
endalign$$
In this case adaptability translates to forcing $X_1$ to constant on the sets $ttt,tth,tht,thh$ and $hhh,hht,hth,htt.$ But how does this help us? It certainly doesn't tell us which set happened and which didn't. I can see that $X_2$ is not $mathcalF_1$-measurable, but so what? Knowing $mathcalF_1$ doesn't give us any knowledge about the result of the first toss, other that it was either a heads or a tails, which we already knew. So now I am ready to state my more general question
Question Why does the $sigma$-algebra $mathcalF_t$ 'contain information' about the process up to time $t$ and what does this mean? How can we use it to say something concrete about what has actually happened up to time $t$ and how can we justify this mathematically?
measure-theory stochastic-processes finance filtrations
asked Jul 23 at 20:07
moquant
82
asked Jul 23 at 20:07
moquant
82
asked Jul 23 at 20:07
moquant
82
asked Jul 23 at 20:07
moquant
82
82
add a comment |Â
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
4
down vote
accepted
It's not that $mathcalF_t$ itself "contains information about the process up to time $t$" in the way you seem to be assuming. Rather, elements of $mathcalF_t$ are allowed to depend on information about the process up to time $t$. For instance, in your example, the set $ttt,tth,tht,thh$ has some very useful information about the first toss: it tells us that the first toss was tails!
To put it another way, $mathcalF_t$ should be thought of as the set of all events which depend only on what has happened up to time $t$. So in your example, one of the events in $mathcalF_1$ is that the first coin toss was tails, represented by the set $ttt,tth,tht,thh$. Another event in $mathcalF_1$ is that the first coin toss was heads, represented by the set $hhh,hht,hth,htt$.
Now, if $X$ is a random variable, to say that $X$ is $mathcalF_t$ measurable means that every event we can define in terms of $X$ is in $mathcalF_t$ (more precisely, for any Borel set $A$, $omega:X(omega)in AinmathcalF_t$). That is, the event of $X$ having any particular value (or any particular range of values) is in $mathcalF_t$, meaning that this event depends only on what has happened up to time $t$. So, to say that $S_t$ is adapted to the filtration means exactly that any event regarding the value of $S_t$ only depends on what happens up to time $t$. In other words, the current price (at some time $t$) can't depend on what happens in the future.
I feel you might find this unsatisfying, and if you do, my response would be that you are simply misunderstanding the purpose of the notion of a process adapted to a filtration. This is just a technical definition for describing stochastic processes as a mathematical structure. It doesn't have any special powers and isn't going to magically answer any questions about stock prices. Treat it as just a definition and nothing more, and be patient until you see applications where the definition is useful.
answered Jul 23 at 20:28
Eric Wofsey
162k12189300
Thanks for the answer. I think maybe I need to go back and look at exactly how the assumption of adaptability is used in the Ito integral. I understand your point about the events in $mathcalF_1$ representing the outcome of the first toss, but why not just take $mathcalF_t = mathcalF$ in which case those events are also present. What advantage do we get from the definition of adaptability, other than being more general than requiring $S_t$ to $mathcalF$-measurable?
– moquant
Jul 23 at 21:06
Well, that's where you'll have to be patient and wait for the applications, like I said. One really useful thing you can do with a filtration is define conditional expectation. That is, given a random variable $X$, you can define another random variable $E(X|mathcalF_t)$ which roughly represents "the expected value of $X$ knowing only what happens up to time $t$". For instance, if $X$ is the value of some stock at time $2$, then $E(X|mathcalF_1)$ is the random variable whose value is the expectation of what the value will be at time $2$, given what you know at time $1$.
– Eric Wofsey
Jul 23 at 21:18
Another point which you may be missing here is that you should really think of the filtration as coming first. It's not that we start with a stochastic process, and then pick some filtration which it is adapted to. Rather, we start with a filtration, which represents the passage of time in the way I described. Then, given a stochastic process, we can express the idea that this stochastic process takes place over time by stating that it is adapted to our filtration.
– Eric Wofsey
Jul 23 at 21:20
Note also that we might use the same filtration for many different stochastic processes. The filtration gives us a fixed notion of "time" which we can use with many different stochastic processes at once, as long as they are all adapted to it.
– Eric Wofsey
Jul 23 at 21:25
Ok, thanks for pointing me in the right direction. I think like you said really understanding the applications will help build the intuition for the definition and the common interpretation/meaning in real terms. I have considered the point of view of starting with the filtration first but it essentially led me to the same place of confusion. I think what you've said combined with some self-study will clear up any remaining confusions.
– moquant
Jul 23 at 21:36
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
accepted
It's not that $mathcalF_t$ itself "contains information about the process up to time $t$" in the way you seem to be assuming. Rather, elements of $mathcalF_t$ are allowed to depend on information about the process up to time $t$. For instance, in your example, the set $ttt,tth,tht,thh$ has some very useful information about the first toss: it tells us that the first toss was tails!
To put it another way, $mathcalF_t$ should be thought of as the set of all events which depend only on what has happened up to time $t$. So in your example, one of the events in $mathcalF_1$ is that the first coin toss was tails, represented by the set $ttt,tth,tht,thh$. Another event in $mathcalF_1$ is that the first coin toss was heads, represented by the set $hhh,hht,hth,htt$.
Now, if $X$ is a random variable, to say that $X$ is $mathcalF_t$ measurable means that every event we can define in terms of $X$ is in $mathcalF_t$ (more precisely, for any Borel set $A$, $omega:X(omega)in AinmathcalF_t$). That is, the event of $X$ having any particular value (or any particular range of values) is in $mathcalF_t$, meaning that this event depends only on what has happened up to time $t$. So, to say that $S_t$ is adapted to the filtration means exactly that any event regarding the value of $S_t$ only depends on what happens up to time $t$. In other words, the current price (at some time $t$) can't depend on what happens in the future.
I feel you might find this unsatisfying, and if you do, my response would be that you are simply misunderstanding the purpose of the notion of a process adapted to a filtration. This is just a technical definition for describing stochastic processes as a mathematical structure. It doesn't have any special powers and isn't going to magically answer any questions about stock prices. Treat it as just a definition and nothing more, and be patient until you see applications where the definition is useful.
answered Jul 23 at 20:28
Eric Wofsey
162k12189300
Thanks for the answer. I think maybe I need to go back and look at exactly how the assumption of adaptability is used in the Ito integral. I understand your point about the events in $mathcalF_1$ representing the outcome of the first toss, but why not just take $mathcalF_t = mathcalF$ in which case those events are also present. What advantage do we get from the definition of adaptability, other than being more general than requiring $S_t$ to $mathcalF$-measurable?
– moquant
Jul 23 at 21:06
Well, that's where you'll have to be patient and wait for the applications, like I said. One really useful thing you can do with a filtration is define conditional expectation. That is, given a random variable $X$, you can define another random variable $E(X|mathcalF_t)$ which roughly represents "the expected value of $X$ knowing only what happens up to time $t$". For instance, if $X$ is the value of some stock at time $2$, then $E(X|mathcalF_1)$ is the random variable whose value is the expectation of what the value will be at time $2$, given what you know at time $1$.
– Eric Wofsey
Jul 23 at 21:18
Another point which you may be missing here is that you should really think of the filtration as coming first. It's not that we start with a stochastic process, and then pick some filtration which it is adapted to. Rather, we start with a filtration, which represents the passage of time in the way I described. Then, given a stochastic process, we can express the idea that this stochastic process takes place over time by stating that it is adapted to our filtration.
– Eric Wofsey
Jul 23 at 21:20
Note also that we might use the same filtration for many different stochastic processes. The filtration gives us a fixed notion of "time" which we can use with many different stochastic processes at once, as long as they are all adapted to it.
– Eric Wofsey
Jul 23 at 21:25
Ok, thanks for pointing me in the right direction. I think like you said really understanding the applications will help build the intuition for the definition and the common interpretation/meaning in real terms. I have considered the point of view of starting with the filtration first but it essentially led me to the same place of confusion. I think what you've said combined with some self-study will clear up any remaining confusions.
– moquant
Jul 23 at 21:36
add a comment |Â
up vote
4
down vote
accepted
It's not that $mathcalF_t$ itself "contains information about the process up to time $t$" in the way you seem to be assuming. Rather, elements of $mathcalF_t$ are allowed to depend on information about the process up to time $t$. For instance, in your example, the set $ttt,tth,tht,thh$ has some very useful information about the first toss: it tells us that the first toss was tails!
To put it another way, $mathcalF_t$ should be thought of as the set of all events which depend only on what has happened up to time $t$. So in your example, one of the events in $mathcalF_1$ is that the first coin toss was tails, represented by the set $ttt,tth,tht,thh$. Another event in $mathcalF_1$ is that the first coin toss was heads, represented by the set $hhh,hht,hth,htt$.
Now, if $X$ is a random variable, to say that $X$ is $mathcalF_t$ measurable means that every event we can define in terms of $X$ is in $mathcalF_t$ (more precisely, for any Borel set $A$, $omega:X(omega)in AinmathcalF_t$). That is, the event of $X$ having any particular value (or any particular range of values) is in $mathcalF_t$, meaning that this event depends only on what has happened up to time $t$. So, to say that $S_t$ is adapted to the filtration means exactly that any event regarding the value of $S_t$ only depends on what happens up to time $t$. In other words, the current price (at some time $t$) can't depend on what happens in the future.
I feel you might find this unsatisfying, and if you do, my response would be that you are simply misunderstanding the purpose of the notion of a process adapted to a filtration. This is just a technical definition for describing stochastic processes as a mathematical structure. It doesn't have any special powers and isn't going to magically answer any questions about stock prices. Treat it as just a definition and nothing more, and be patient until you see applications where the definition is useful.
answered Jul 23 at 20:28
Eric Wofsey
162k12189300
Thanks for the answer. I think maybe I need to go back and look at exactly how the assumption of adaptability is used in the Ito integral. I understand your point about the events in $mathcalF_1$ representing the outcome of the first toss, but why not just take $mathcalF_t = mathcalF$ in which case those events are also present. What advantage do we get from the definition of adaptability, other than being more general than requiring $S_t$ to $mathcalF$-measurable?
– moquant
Jul 23 at 21:06
Well, that's where you'll have to be patient and wait for the applications, like I said. One really useful thing you can do with a filtration is define conditional expectation. That is, given a random variable $X$, you can define another random variable $E(X|mathcalF_t)$ which roughly represents "the expected value of $X$ knowing only what happens up to time $t$". For instance, if $X$ is the value of some stock at time $2$, then $E(X|mathcalF_1)$ is the random variable whose value is the expectation of what the value will be at time $2$, given what you know at time $1$.
– Eric Wofsey
Jul 23 at 21:18
Another point which you may be missing here is that you should really think of the filtration as coming first. It's not that we start with a stochastic process, and then pick some filtration which it is adapted to. Rather, we start with a filtration, which represents the passage of time in the way I described. Then, given a stochastic process, we can express the idea that this stochastic process takes place over time by stating that it is adapted to our filtration.
– Eric Wofsey
Jul 23 at 21:20
Note also that we might use the same filtration for many different stochastic processes. The filtration gives us a fixed notion of "time" which we can use with many different stochastic processes at once, as long as they are all adapted to it.
– Eric Wofsey
Jul 23 at 21:25
Ok, thanks for pointing me in the right direction. I think like you said really understanding the applications will help build the intuition for the definition and the common interpretation/meaning in real terms. I have considered the point of view of starting with the filtration first but it essentially led me to the same place of confusion. I think what you've said combined with some self-study will clear up any remaining confusions.
– moquant
Jul 23 at 21:36
add a comment |Â
up vote
4
down vote
accepted
up vote
4
down vote
accepted
It's not that $mathcalF_t$ itself "contains information about the process up to time $t$" in the way you seem to be assuming. Rather, elements of $mathcalF_t$ are allowed to depend on information about the process up to time $t$. For instance, in your example, the set $ttt,tth,tht,thh$ has some very useful information about the first toss: it tells us that the first toss was tails!
To put it another way, $mathcalF_t$ should be thought of as the set of all events which depend only on what has happened up to time $t$. So in your example, one of the events in $mathcalF_1$ is that the first coin toss was tails, represented by the set $ttt,tth,tht,thh$. Another event in $mathcalF_1$ is that the first coin toss was heads, represented by the set $hhh,hht,hth,htt$.
Now, if $X$ is a random variable, to say that $X$ is $mathcalF_t$ measurable means that every event we can define in terms of $X$ is in $mathcalF_t$ (more precisely, for any Borel set $A$, $omega:X(omega)in AinmathcalF_t$). That is, the event of $X$ having any particular value (or any particular range of values) is in $mathcalF_t$, meaning that this event depends only on what has happened up to time $t$. So, to say that $S_t$ is adapted to the filtration means exactly that any event regarding the value of $S_t$ only depends on what happens up to time $t$. In other words, the current price (at some time $t$) can't depend on what happens in the future.
I feel you might find this unsatisfying, and if you do, my response would be that you are simply misunderstanding the purpose of the notion of a process adapted to a filtration. This is just a technical definition for describing stochastic processes as a mathematical structure. It doesn't have any special powers and isn't going to magically answer any questions about stock prices. Treat it as just a definition and nothing more, and be patient until you see applications where the definition is useful.
answered Jul 23 at 20:28
Eric Wofsey
162k12189300
It's not that $mathcalF_t$ itself "contains information about the process up to time $t$" in the way you seem to be assuming. Rather, elements of $mathcalF_t$ are allowed to depend on information about the process up to time $t$. For instance, in your example, the set $ttt,tth,tht,thh$ has some very useful information about the first toss: it tells us that the first toss was tails!
To put it another way, $mathcalF_t$ should be thought of as the set of all events which depend only on what has happened up to time $t$. So in your example, one of the events in $mathcalF_1$ is that the first coin toss was tails, represented by the set $ttt,tth,tht,thh$. Another event in $mathcalF_1$ is that the first coin toss was heads, represented by the set $hhh,hht,hth,htt$.
Now, if $X$ is a random variable, to say that $X$ is $mathcalF_t$ measurable means that every event we can define in terms of $X$ is in $mathcalF_t$ (more precisely, for any Borel set $A$, $omega:X(omega)in AinmathcalF_t$). That is, the event of $X$ having any particular value (or any particular range of values) is in $mathcalF_t$, meaning that this event depends only on what has happened up to time $t$. So, to say that $S_t$ is adapted to the filtration means exactly that any event regarding the value of $S_t$ only depends on what happens up to time $t$. In other words, the current price (at some time $t$) can't depend on what happens in the future.
I feel you might find this unsatisfying, and if you do, my response would be that you are simply misunderstanding the purpose of the notion of a process adapted to a filtration. This is just a technical definition for describing stochastic processes as a mathematical structure. It doesn't have any special powers and isn't going to magically answer any questions about stock prices. Treat it as just a definition and nothing more, and be patient until you see applications where the definition is useful.
answered Jul 23 at 20:28
Eric Wofsey
162k12189300
answered Jul 23 at 20:28
Eric Wofsey
162k12189300
answered Jul 23 at 20:28
Eric Wofsey
162k12189300
answered Jul 23 at 20:28
Eric Wofsey
162k12189300
162k12189300
Thanks for the answer. I think maybe I need to go back and look at exactly how the assumption of adaptability is used in the Ito integral. I understand your point about the events in $mathcalF_1$ representing the outcome of the first toss, but why not just take $mathcalF_t = mathcalF$ in which case those events are also present. What advantage do we get from the definition of adaptability, other than being more general than requiring $S_t$ to $mathcalF$-measurable?
– moquant
Jul 23 at 21:06
Well, that's where you'll have to be patient and wait for the applications, like I said. One really useful thing you can do with a filtration is define conditional expectation. That is, given a random variable $X$, you can define another random variable $E(X|mathcalF_t)$ which roughly represents "the expected value of $X$ knowing only what happens up to time $t$". For instance, if $X$ is the value of some stock at time $2$, then $E(X|mathcalF_1)$ is the random variable whose value is the expectation of what the value will be at time $2$, given what you know at time $1$.
– Eric Wofsey
Jul 23 at 21:18
Another point which you may be missing here is that you should really think of the filtration as coming first. It's not that we start with a stochastic process, and then pick some filtration which it is adapted to. Rather, we start with a filtration, which represents the passage of time in the way I described. Then, given a stochastic process, we can express the idea that this stochastic process takes place over time by stating that it is adapted to our filtration.
– Eric Wofsey
Jul 23 at 21:20
Note also that we might use the same filtration for many different stochastic processes. The filtration gives us a fixed notion of "time" which we can use with many different stochastic processes at once, as long as they are all adapted to it.
– Eric Wofsey
Jul 23 at 21:25
Ok, thanks for pointing me in the right direction. I think like you said really understanding the applications will help build the intuition for the definition and the common interpretation/meaning in real terms. I have considered the point of view of starting with the filtration first but it essentially led me to the same place of confusion. I think what you've said combined with some self-study will clear up any remaining confusions.
– moquant
Jul 23 at 21:36
add a comment |Â
Thanks for the answer. I think maybe I need to go back and look at exactly how the assumption of adaptability is used in the Ito integral. I understand your point about the events in $mathcalF_1$ representing the outcome of the first toss, but why not just take $mathcalF_t = mathcalF$ in which case those events are also present. What advantage do we get from the definition of adaptability, other than being more general than requiring $S_t$ to $mathcalF$-measurable?
– moquant
Jul 23 at 21:06
Well, that's where you'll have to be patient and wait for the applications, like I said. One really useful thing you can do with a filtration is define conditional expectation. That is, given a random variable $X$, you can define another random variable $E(X|mathcalF_t)$ which roughly represents "the expected value of $X$ knowing only what happens up to time $t$". For instance, if $X$ is the value of some stock at time $2$, then $E(X|mathcalF_1)$ is the random variable whose value is the expectation of what the value will be at time $2$, given what you know at time $1$.
– Eric Wofsey
Jul 23 at 21:18
Another point which you may be missing here is that you should really think of the filtration as coming first. It's not that we start with a stochastic process, and then pick some filtration which it is adapted to. Rather, we start with a filtration, which represents the passage of time in the way I described. Then, given a stochastic process, we can express the idea that this stochastic process takes place over time by stating that it is adapted to our filtration.
– Eric Wofsey
Jul 23 at 21:20
Note also that we might use the same filtration for many different stochastic processes. The filtration gives us a fixed notion of "time" which we can use with many different stochastic processes at once, as long as they are all adapted to it.
– Eric Wofsey
Jul 23 at 21:25
Ok, thanks for pointing me in the right direction. I think like you said really understanding the applications will help build the intuition for the definition and the common interpretation/meaning in real terms. I have considered the point of view of starting with the filtration first but it essentially led me to the same place of confusion. I think what you've said combined with some self-study will clear up any remaining confusions.
– moquant
Jul 23 at 21:36
Thanks for the answer. I think maybe I need to go back and look at exactly how the assumption of adaptability is used in the Ito integral. I understand your point about the events in $mathcalF_1$ representing the outcome of the first toss, but why not just take $mathcalF_t = mathcalF$ in which case those events are also present. What advantage do we get from the definition of adaptability, other than being more general than requiring $S_t$ to $mathcalF$-measurable?
– moquant
Jul 23 at 21:06
Thanks for the answer. I think maybe I need to go back and look at exactly how the assumption of adaptability is used in the Ito integral. I understand your point about the events in $mathcalF_1$ representing the outcome of the first toss, but why not just take $mathcalF_t = mathcalF$ in which case those events are also present. What advantage do we get from the definition of adaptability, other than being more general than requiring $S_t$ to $mathcalF$-measurable?
– moquant
Jul 23 at 21:06
Well, that's where you'll have to be patient and wait for the applications, like I said. One really useful thing you can do with a filtration is define conditional expectation. That is, given a random variable $X$, you can define another random variable $E(X|mathcalF_t)$ which roughly represents "the expected value of $X$ knowing only what happens up to time $t$". For instance, if $X$ is the value of some stock at time $2$, then $E(X|mathcalF_1)$ is the random variable whose value is the expectation of what the value will be at time $2$, given what you know at time $1$.
– Eric Wofsey
Jul 23 at 21:18
Well, that's where you'll have to be patient and wait for the applications, like I said. One really useful thing you can do with a filtration is define conditional expectation. That is, given a random variable $X$, you can define another random variable $E(X|mathcalF_t)$ which roughly represents "the expected value of $X$ knowing only what happens up to time $t$". For instance, if $X$ is the value of some stock at time $2$, then $E(X|mathcalF_1)$ is the random variable whose value is the expectation of what the value will be at time $2$, given what you know at time $1$.
– Eric Wofsey
Jul 23 at 21:18
Another point which you may be missing here is that you should really think of the filtration as coming first. It's not that we start with a stochastic process, and then pick some filtration which it is adapted to. Rather, we start with a filtration, which represents the passage of time in the way I described. Then, given a stochastic process, we can express the idea that this stochastic process takes place over time by stating that it is adapted to our filtration.
– Eric Wofsey
Jul 23 at 21:20
Another point which you may be missing here is that you should really think of the filtration as coming first. It's not that we start with a stochastic process, and then pick some filtration which it is adapted to. Rather, we start with a filtration, which represents the passage of time in the way I described. Then, given a stochastic process, we can express the idea that this stochastic process takes place over time by stating that it is adapted to our filtration.
– Eric Wofsey
Jul 23 at 21:20
Note also that we might use the same filtration for many different stochastic processes. The filtration gives us a fixed notion of "time" which we can use with many different stochastic processes at once, as long as they are all adapted to it.
– Eric Wofsey
Jul 23 at 21:25
Note also that we might use the same filtration for many different stochastic processes. The filtration gives us a fixed notion of "time" which we can use with many different stochastic processes at once, as long as they are all adapted to it.
– Eric Wofsey
Jul 23 at 21:25
Ok, thanks for pointing me in the right direction. I think like you said really understanding the applications will help build the intuition for the definition and the common interpretation/meaning in real terms. I have considered the point of view of starting with the filtration first but it essentially led me to the same place of confusion. I think what you've said combined with some self-study will clear up any remaining confusions.
– moquant
Jul 23 at 21:36
Ok, thanks for pointing me in the right direction. I think like you said really understanding the applications will help build the intuition for the definition and the common interpretation/meaning in real terms. I have considered the point of view of starting with the filtration first but it essentially led me to the same place of confusion. I think what you've said combined with some self-study will clear up any remaining confusions.
– moquant
Jul 23 at 21:36
add a comment |Â
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2860741%2fadapted-process-translation-between-measurable-and-information%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password