Conditional expectation of martingale and two bounded stopping times
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I am trying to prove the following:
Let $(X_n)$ be a martingale with respect to $(mathcal{F}_n)$ and suppose $tau_1$
and $tau_2$ are bounded stopping times such that $tau_1le tau_2 < B$, where $B < infty$ is the bound. Then $E(X_{tau_2}|mathcal{F}_{tau_1}) = X_{tau_1}.$
What I've learned so far are definition and some properties of conditional expectation and martingales and I think no advanced knowledge is necessary to solve this.
But I could not combine that definition and properties to produce good solution to this.
If anyone has any tips on how to do, I'd appreciate it a lot.
conditional-expectation martingales stopping-times
asked Dec 3 '18 at 12:00
EuduardoEuduardo
1288
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add a comment |
$begingroup$
I am trying to prove the following:
Let $(X_n)$ be a martingale with respect to $(mathcal{F}_n)$ and suppose $tau_1$
and $tau_2$ are bounded stopping times such that $tau_1le tau_2 < B$, where $B < infty$ is the bound. Then $E(X_{tau_2}|mathcal{F}_{tau_1}) = X_{tau_1}.$
What I've learned so far are definition and some properties of conditional expectation and martingales and I think no advanced knowledge is necessary to solve this.
But I could not combine that definition and properties to produce good solution to this.
If anyone has any tips on how to do, I'd appreciate it a lot.
conditional-expectation martingales stopping-times
asked Dec 3 '18 at 12:00
EuduardoEuduardo
1288
$endgroup$
$begingroup$
This is a (special case of a ) basic theorem called the Optional Sampling Theorem.
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– Kavi Rama Murthy
Dec 3 '18 at 12:08
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@KaviRamaMurthy But I found this problem located before the Optional Sampling Theorem (and martingale transform ...)
$endgroup$
– Euduardo
Dec 3 '18 at 12:14
2
$begingroup$
Take a look at this question: math.stackexchange.com/q/3017226/36150
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– saz
Dec 3 '18 at 15:15
add a comment |
$begingroup$
I am trying to prove the following:
Let $(X_n)$ be a martingale with respect to $(mathcal{F}_n)$ and suppose $tau_1$
and $tau_2$ are bounded stopping times such that $tau_1le tau_2 < B$, where $B < infty$ is the bound. Then $E(X_{tau_2}|mathcal{F}_{tau_1}) = X_{tau_1}.$
What I've learned so far are definition and some properties of conditional expectation and martingales and I think no advanced knowledge is necessary to solve this.
But I could not combine that definition and properties to produce good solution to this.
If anyone has any tips on how to do, I'd appreciate it a lot.
conditional-expectation martingales stopping-times
asked Dec 3 '18 at 12:00
EuduardoEuduardo
1288
$endgroup$
I am trying to prove the following:
Let $(X_n)$ be a martingale with respect to $(mathcal{F}_n)$ and suppose $tau_1$
and $tau_2$ are bounded stopping times such that $tau_1le tau_2 < B$, where $B < infty$ is the bound. Then $E(X_{tau_2}|mathcal{F}_{tau_1}) = X_{tau_1}.$
What I've learned so far are definition and some properties of conditional expectation and martingales and I think no advanced knowledge is necessary to solve this.
But I could not combine that definition and properties to produce good solution to this.
If anyone has any tips on how to do, I'd appreciate it a lot.
conditional-expectation martingales stopping-times
conditional-expectation martingales stopping-times
asked Dec 3 '18 at 12:00
EuduardoEuduardo
1288
asked Dec 3 '18 at 12:00
EuduardoEuduardo
1288
asked Dec 3 '18 at 12:00
EuduardoEuduardo
1288
asked Dec 3 '18 at 12:00
EuduardoEuduardo
1288
asked Dec 3 '18 at 12:00
EuduardoEuduardo
1288
1288
$begingroup$
This is a (special case of a ) basic theorem called the Optional Sampling Theorem.
$endgroup$
– Kavi Rama Murthy
Dec 3 '18 at 12:08
$begingroup$
@KaviRamaMurthy But I found this problem located before the Optional Sampling Theorem (and martingale transform ...)
$endgroup$
– Euduardo
Dec 3 '18 at 12:14
2
$begingroup$
Take a look at this question: math.stackexchange.com/q/3017226/36150
$endgroup$
– saz
Dec 3 '18 at 15:15
add a comment |
$begingroup$
This is a (special case of a ) basic theorem called the Optional Sampling Theorem.
$endgroup$
– Kavi Rama Murthy
Dec 3 '18 at 12:08
$begingroup$
@KaviRamaMurthy But I found this problem located before the Optional Sampling Theorem (and martingale transform ...)
$endgroup$
– Euduardo
Dec 3 '18 at 12:14
2
$begingroup$
Take a look at this question: math.stackexchange.com/q/3017226/36150
$endgroup$
– saz
Dec 3 '18 at 15:15
$begingroup$
This is a (special case of a ) basic theorem called the Optional Sampling Theorem.
$endgroup$
– Kavi Rama Murthy
Dec 3 '18 at 12:08
$begingroup$
This is a (special case of a ) basic theorem called the Optional Sampling Theorem.
$endgroup$
– Kavi Rama Murthy
Dec 3 '18 at 12:08
$begingroup$
@KaviRamaMurthy But I found this problem located before the Optional Sampling Theorem (and martingale transform ...)
$endgroup$
– Euduardo
Dec 3 '18 at 12:14
$begingroup$
@KaviRamaMurthy But I found this problem located before the Optional Sampling Theorem (and martingale transform ...)
$endgroup$
– Euduardo
Dec 3 '18 at 12:14
2
2
$begingroup$
Take a look at this question: math.stackexchange.com/q/3017226/36150
$endgroup$
– saz
Dec 3 '18 at 15:15
$begingroup$
Take a look at this question: math.stackexchange.com/q/3017226/36150
$endgroup$
– saz
Dec 3 '18 at 15:15
add a comment |
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$begingroup$
This is a (special case of a ) basic theorem called the Optional Sampling Theorem.
$endgroup$
– Kavi Rama Murthy
Dec 3 '18 at 12:08
$begingroup$
@KaviRamaMurthy But I found this problem located before the Optional Sampling Theorem (and martingale transform ...)
$endgroup$
– Euduardo
Dec 3 '18 at 12:14
2
$begingroup$
Take a look at this question: math.stackexchange.com/q/3017226/36150
$endgroup$
– saz
Dec 3 '18 at 15:15