Show that τ is a stopping time with respect to FX
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https://i.stack.imgur.com/gvLDp.png
Given the probability space in the attached image I would like to Show that τ is a stopping time with respect to FX.
I have calculated the FX to be:
F0X=σ({X0})=(ØΩ)
F1X= σ({X0,X1)}= σ({a,b},{c,d})
F2X= σ(X0,X1,X2})= σ({a},{b},{c},{d})=2Ω
and I know from here my next step would be to calculate τ of each event. This is where I come unstuck as it appears that τ is actually 0,0,0,0 because at Time 0 the value is already above the stopping time of 10 therefore we never reach event 1 or 2. But if τ = 0 my understanding is it does not fit the criteria to be a stopping time. Excuse the beginner level question but can anyone help me work through this problem?
Many thanks
probability stopping-times
asked Nov 18 at 1:51
Sam.U
112
add a comment |
up vote
1
down vote
favorite
https://i.stack.imgur.com/gvLDp.png
Given the probability space in the attached image I would like to Show that τ is a stopping time with respect to FX.
I have calculated the FX to be:
F0X=σ({X0})=(ØΩ)
F1X= σ({X0,X1)}= σ({a,b},{c,d})
F2X= σ(X0,X1,X2})= σ({a},{b},{c},{d})=2Ω
and I know from here my next step would be to calculate τ of each event. This is where I come unstuck as it appears that τ is actually 0,0,0,0 because at Time 0 the value is already above the stopping time of 10 therefore we never reach event 1 or 2. But if τ = 0 my understanding is it does not fit the criteria to be a stopping time. Excuse the beginner level question but can anyone help me work through this problem?
Many thanks
probability stopping-times
asked Nov 18 at 1:51
Sam.U
112
2
Please use MathJax in future, @Sam.U
– Shaun
Nov 18 at 1:54
Why do you think that $tau=0$ is not a stopping time?
– saz
Nov 18 at 6:29
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
https://i.stack.imgur.com/gvLDp.png
Given the probability space in the attached image I would like to Show that τ is a stopping time with respect to FX.
I have calculated the FX to be:
F0X=σ({X0})=(ØΩ)
F1X= σ({X0,X1)}= σ({a,b},{c,d})
F2X= σ(X0,X1,X2})= σ({a},{b},{c},{d})=2Ω
and I know from here my next step would be to calculate τ of each event. This is where I come unstuck as it appears that τ is actually 0,0,0,0 because at Time 0 the value is already above the stopping time of 10 therefore we never reach event 1 or 2. But if τ = 0 my understanding is it does not fit the criteria to be a stopping time. Excuse the beginner level question but can anyone help me work through this problem?
Many thanks
probability stopping-times
asked Nov 18 at 1:51
Sam.U
112
https://i.stack.imgur.com/gvLDp.png
Given the probability space in the attached image I would like to Show that τ is a stopping time with respect to FX.
I have calculated the FX to be:
F0X=σ({X0})=(ØΩ)
F1X= σ({X0,X1)}= σ({a,b},{c,d})
F2X= σ(X0,X1,X2})= σ({a},{b},{c},{d})=2Ω
and I know from here my next step would be to calculate τ of each event. This is where I come unstuck as it appears that τ is actually 0,0,0,0 because at Time 0 the value is already above the stopping time of 10 therefore we never reach event 1 or 2. But if τ = 0 my understanding is it does not fit the criteria to be a stopping time. Excuse the beginner level question but can anyone help me work through this problem?
Many thanks
probability stopping-times
probability stopping-times
asked Nov 18 at 1:51
Sam.U
112
asked Nov 18 at 1:51
Sam.U
112
asked Nov 18 at 1:51
Sam.U
112
asked Nov 18 at 1:51
Sam.U
112
asked Nov 18 at 1:51
Sam.U
112
112
2
Please use MathJax in future, @Sam.U
– Shaun
Nov 18 at 1:54
Why do you think that $tau=0$ is not a stopping time?
– saz
Nov 18 at 6:29
add a comment |
2
Please use MathJax in future, @Sam.U
– Shaun
Nov 18 at 1:54
Why do you think that $tau=0$ is not a stopping time?
– saz
Nov 18 at 6:29
2
2
Please use MathJax in future, @Sam.U
– Shaun
Nov 18 at 1:54
Please use MathJax in future, @Sam.U
– Shaun
Nov 18 at 1:54
Why do you think that $tau=0$ is not a stopping time?
– saz
Nov 18 at 6:29
Why do you think that $tau=0$ is not a stopping time?
– saz
Nov 18 at 6:29
add a comment |
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2
Please use MathJax in future, @Sam.U
– Shaun
Nov 18 at 1:54
Why do you think that $tau=0$ is not a stopping time?
– saz
Nov 18 at 6:29