Variance of the stochastic process
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I am asked to find $Var(aW_t+bint_0^tf(s)W_sds)$. This is what I did until now:
begin{align*} &mathbb{E}left(a^2W_t^2+2abW_tint_0^tf(s)W_sds+b^2left[int_0^tf(s)W_s , dsright]^2 right) \&=a^2t+2abint_0^tf(s)mathbb{E}(W_t,W_s)ds+b^2mathbb{E}left(left[int_0^tf(s)W_s , ds right]^2 right) \ &=a^2t+2abint_0^tf(s)sds+b^2mathbb{E}left(left[int_0^tf(s)W_s , ds right]^2 right) end{align*}
but I do not know if I am doing it correctly and how to calculate the last expectation.
stochastic-processes stochastic-calculus variance
edited Dec 13 '18 at 18:31
saz
80.5k860125
asked Dec 13 '18 at 18:01
T.SokhT.Sokh
6
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add a comment |
$begingroup$
I am asked to find $Var(aW_t+bint_0^tf(s)W_sds)$. This is what I did until now:
begin{align*} &mathbb{E}left(a^2W_t^2+2abW_tint_0^tf(s)W_sds+b^2left[int_0^tf(s)W_s , dsright]^2 right) \&=a^2t+2abint_0^tf(s)mathbb{E}(W_t,W_s)ds+b^2mathbb{E}left(left[int_0^tf(s)W_s , ds right]^2 right) \ &=a^2t+2abint_0^tf(s)sds+b^2mathbb{E}left(left[int_0^tf(s)W_s , ds right]^2 right) end{align*}
but I do not know if I am doing it correctly and how to calculate the last expectation.
stochastic-processes stochastic-calculus variance
edited Dec 13 '18 at 18:31
saz
80.5k860125
asked Dec 13 '18 at 18:01
T.SokhT.Sokh
6
$endgroup$
1
$begingroup$
Write $$left( int_0^t f(s) W_s , ds right)^2 = int_0^t int_0^t f(s) W_s f(u) W_u , ds , du$$ and apply Fubini's theorem
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– saz
Dec 13 '18 at 18:33
$begingroup$
Why not? You just need to compute $mathbb{E}(W_s W_u)$
$endgroup$
– saz
Dec 13 '18 at 20:22
$begingroup$
I got it thank you!
$endgroup$
– T.Sokh
Dec 13 '18 at 20:25
add a comment |
$begingroup$
I am asked to find $Var(aW_t+bint_0^tf(s)W_sds)$. This is what I did until now:
begin{align*} &mathbb{E}left(a^2W_t^2+2abW_tint_0^tf(s)W_sds+b^2left[int_0^tf(s)W_s , dsright]^2 right) \&=a^2t+2abint_0^tf(s)mathbb{E}(W_t,W_s)ds+b^2mathbb{E}left(left[int_0^tf(s)W_s , ds right]^2 right) \ &=a^2t+2abint_0^tf(s)sds+b^2mathbb{E}left(left[int_0^tf(s)W_s , ds right]^2 right) end{align*}
but I do not know if I am doing it correctly and how to calculate the last expectation.
stochastic-processes stochastic-calculus variance
edited Dec 13 '18 at 18:31
saz
80.5k860125
asked Dec 13 '18 at 18:01
T.SokhT.Sokh
6
$endgroup$
I am asked to find $Var(aW_t+bint_0^tf(s)W_sds)$. This is what I did until now:
begin{align*} &mathbb{E}left(a^2W_t^2+2abW_tint_0^tf(s)W_sds+b^2left[int_0^tf(s)W_s , dsright]^2 right) \&=a^2t+2abint_0^tf(s)mathbb{E}(W_t,W_s)ds+b^2mathbb{E}left(left[int_0^tf(s)W_s , ds right]^2 right) \ &=a^2t+2abint_0^tf(s)sds+b^2mathbb{E}left(left[int_0^tf(s)W_s , ds right]^2 right) end{align*}
but I do not know if I am doing it correctly and how to calculate the last expectation.
stochastic-processes stochastic-calculus variance
stochastic-processes stochastic-calculus variance
edited Dec 13 '18 at 18:31
saz
80.5k860125
asked Dec 13 '18 at 18:01
T.SokhT.Sokh
6
edited Dec 13 '18 at 18:31
saz
80.5k860125
asked Dec 13 '18 at 18:01
T.SokhT.Sokh
6
edited Dec 13 '18 at 18:31
saz
80.5k860125
edited Dec 13 '18 at 18:31
saz
80.5k860125
edited Dec 13 '18 at 18:31
saz
80.5k860125
80.5k860125
asked Dec 13 '18 at 18:01
T.SokhT.Sokh
6
asked Dec 13 '18 at 18:01
T.SokhT.Sokh
6
asked Dec 13 '18 at 18:01
T.SokhT.Sokh
6
6
1
$begingroup$
Write $$left( int_0^t f(s) W_s , ds right)^2 = int_0^t int_0^t f(s) W_s f(u) W_u , ds , du$$ and apply Fubini's theorem
$endgroup$
– saz
Dec 13 '18 at 18:33
$begingroup$
Why not? You just need to compute $mathbb{E}(W_s W_u)$
$endgroup$
– saz
Dec 13 '18 at 20:22
$begingroup$
I got it thank you!
$endgroup$
– T.Sokh
Dec 13 '18 at 20:25
add a comment |
1
$begingroup$
Write $$left( int_0^t f(s) W_s , ds right)^2 = int_0^t int_0^t f(s) W_s f(u) W_u , ds , du$$ and apply Fubini's theorem
$endgroup$
– saz
Dec 13 '18 at 18:33
$begingroup$
Why not? You just need to compute $mathbb{E}(W_s W_u)$
$endgroup$
– saz
Dec 13 '18 at 20:22
$begingroup$
I got it thank you!
$endgroup$
– T.Sokh
Dec 13 '18 at 20:25
1
1
$begingroup$
Write $$left( int_0^t f(s) W_s , ds right)^2 = int_0^t int_0^t f(s) W_s f(u) W_u , ds , du$$ and apply Fubini's theorem
$endgroup$
– saz
Dec 13 '18 at 18:33
$begingroup$
Write $$left( int_0^t f(s) W_s , ds right)^2 = int_0^t int_0^t f(s) W_s f(u) W_u , ds , du$$ and apply Fubini's theorem
$endgroup$
– saz
Dec 13 '18 at 18:33
$begingroup$
Why not? You just need to compute $mathbb{E}(W_s W_u)$
$endgroup$
– saz
Dec 13 '18 at 20:22
$begingroup$
Why not? You just need to compute $mathbb{E}(W_s W_u)$
$endgroup$
– saz
Dec 13 '18 at 20:22
$begingroup$
I got it thank you!
$endgroup$
– T.Sokh
Dec 13 '18 at 20:25
$begingroup$
I got it thank you!
$endgroup$
– T.Sokh
Dec 13 '18 at 20:25
add a comment |
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$begingroup$
Write $$left( int_0^t f(s) W_s , ds right)^2 = int_0^t int_0^t f(s) W_s f(u) W_u , ds , du$$ and apply Fubini's theorem
$endgroup$
– saz
Dec 13 '18 at 18:33
$begingroup$
Why not? You just need to compute $mathbb{E}(W_s W_u)$
$endgroup$
– saz
Dec 13 '18 at 20:22
$begingroup$
I got it thank you!
$endgroup$
– T.Sokh
Dec 13 '18 at 20:25