For the Brownian motion integrate
I want to calculate
$$operatorname{E} left[ int_0^1{W(t)dt cdot int_0^1{t^2W(t)dt}} right].$$
I discovered that the first integral is $operatorname{N}(0, frac{1}{3})$ but I don't know how to get the other one and the full answer of their multiplied expectation.
stochastic-calculus
edited Dec 9 at 12:56
skoestlmeier
9661425
asked Dec 9 at 12:27
Hobong
101
add a comment |
I want to calculate
$$operatorname{E} left[ int_0^1{W(t)dt cdot int_0^1{t^2W(t)dt}} right].$$
I discovered that the first integral is $operatorname{N}(0, frac{1}{3})$ but I don't know how to get the other one and the full answer of their multiplied expectation.
stochastic-calculus
edited Dec 9 at 12:56
skoestlmeier
9661425
asked Dec 9 at 12:27
Hobong
101
add a comment |
I want to calculate
$$operatorname{E} left[ int_0^1{W(t)dt cdot int_0^1{t^2W(t)dt}} right].$$
I discovered that the first integral is $operatorname{N}(0, frac{1}{3})$ but I don't know how to get the other one and the full answer of their multiplied expectation.
stochastic-calculus
edited Dec 9 at 12:56
skoestlmeier
9661425
asked Dec 9 at 12:27
Hobong
101
I want to calculate
$$operatorname{E} left[ int_0^1{W(t)dt cdot int_0^1{t^2W(t)dt}} right].$$
I discovered that the first integral is $operatorname{N}(0, frac{1}{3})$ but I don't know how to get the other one and the full answer of their multiplied expectation.
stochastic-calculus
stochastic-calculus
edited Dec 9 at 12:56
skoestlmeier
9661425
asked Dec 9 at 12:27
Hobong
101
edited Dec 9 at 12:56
skoestlmeier
9661425
asked Dec 9 at 12:27
Hobong
101
edited Dec 9 at 12:56
skoestlmeier
9661425
edited Dec 9 at 12:56
skoestlmeier
9661425
edited Dec 9 at 12:56
skoestlmeier
9661425
9661425
asked Dec 9 at 12:27
Hobong
101
asked Dec 9 at 12:27
Hobong
101
asked Dec 9 at 12:27
Hobong
101
101
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
Note that
begin{align*}
Eleft(int_0^1 W_t, dt int_0^1 t^2W_t, dt right) &= Eleft(int_0^1!!!int_0^1 s^2 W_s W_t, dsdt right)\
&=int_0^1!!!int_0^1 s^2 E(W_s W_t), dsdt\
&=int_0^1!!!int_0^1 s^2 (swedge t), dsdt\
&=int_0^1 s^2,ds int_0^1 swedge t, dt\
&=int_0^1 s^2,ds left(int_0^s t,dt + int_s^1 s,dtright).
end{align*}
The remaining is now straightforward.
answered Dec 9 at 14:42
Gordon
14.4k11658
Thank you very much. But the rest can be solved after that? Or just calculate 1/3(s^2/2+s-s^2)
– Hobong
Dec 9 at 15:23
1
The rest is just calculus.
– Gordon
Dec 9 at 15:27
add a comment |
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1 Answer
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active
oldest
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1 Answer
1
active
oldest
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active
oldest
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active
oldest
votes
Note that
begin{align*}
Eleft(int_0^1 W_t, dt int_0^1 t^2W_t, dt right) &= Eleft(int_0^1!!!int_0^1 s^2 W_s W_t, dsdt right)\
&=int_0^1!!!int_0^1 s^2 E(W_s W_t), dsdt\
&=int_0^1!!!int_0^1 s^2 (swedge t), dsdt\
&=int_0^1 s^2,ds int_0^1 swedge t, dt\
&=int_0^1 s^2,ds left(int_0^s t,dt + int_s^1 s,dtright).
end{align*}
The remaining is now straightforward.
answered Dec 9 at 14:42
Gordon
14.4k11658
Thank you very much. But the rest can be solved after that? Or just calculate 1/3(s^2/2+s-s^2)
– Hobong
Dec 9 at 15:23
1
The rest is just calculus.
– Gordon
Dec 9 at 15:27
add a comment |
Note that
begin{align*}
Eleft(int_0^1 W_t, dt int_0^1 t^2W_t, dt right) &= Eleft(int_0^1!!!int_0^1 s^2 W_s W_t, dsdt right)\
&=int_0^1!!!int_0^1 s^2 E(W_s W_t), dsdt\
&=int_0^1!!!int_0^1 s^2 (swedge t), dsdt\
&=int_0^1 s^2,ds int_0^1 swedge t, dt\
&=int_0^1 s^2,ds left(int_0^s t,dt + int_s^1 s,dtright).
end{align*}
The remaining is now straightforward.
answered Dec 9 at 14:42
Gordon
14.4k11658
Thank you very much. But the rest can be solved after that? Or just calculate 1/3(s^2/2+s-s^2)
– Hobong
Dec 9 at 15:23
1
The rest is just calculus.
– Gordon
Dec 9 at 15:27
add a comment |
Note that
begin{align*}
Eleft(int_0^1 W_t, dt int_0^1 t^2W_t, dt right) &= Eleft(int_0^1!!!int_0^1 s^2 W_s W_t, dsdt right)\
&=int_0^1!!!int_0^1 s^2 E(W_s W_t), dsdt\
&=int_0^1!!!int_0^1 s^2 (swedge t), dsdt\
&=int_0^1 s^2,ds int_0^1 swedge t, dt\
&=int_0^1 s^2,ds left(int_0^s t,dt + int_s^1 s,dtright).
end{align*}
The remaining is now straightforward.
answered Dec 9 at 14:42
Gordon
14.4k11658
Note that
begin{align*}
Eleft(int_0^1 W_t, dt int_0^1 t^2W_t, dt right) &= Eleft(int_0^1!!!int_0^1 s^2 W_s W_t, dsdt right)\
&=int_0^1!!!int_0^1 s^2 E(W_s W_t), dsdt\
&=int_0^1!!!int_0^1 s^2 (swedge t), dsdt\
&=int_0^1 s^2,ds int_0^1 swedge t, dt\
&=int_0^1 s^2,ds left(int_0^s t,dt + int_s^1 s,dtright).
end{align*}
The remaining is now straightforward.
answered Dec 9 at 14:42
Gordon
14.4k11658
answered Dec 9 at 14:42
Gordon
14.4k11658
answered Dec 9 at 14:42
Gordon
14.4k11658
answered Dec 9 at 14:42
Gordon
14.4k11658
14.4k11658
Thank you very much. But the rest can be solved after that? Or just calculate 1/3(s^2/2+s-s^2)
– Hobong
Dec 9 at 15:23
1
The rest is just calculus.
– Gordon
Dec 9 at 15:27
add a comment |
Thank you very much. But the rest can be solved after that? Or just calculate 1/3(s^2/2+s-s^2)
– Hobong
Dec 9 at 15:23
1
The rest is just calculus.
– Gordon
Dec 9 at 15:27
Thank you very much. But the rest can be solved after that? Or just calculate 1/3(s^2/2+s-s^2)
– Hobong
Dec 9 at 15:23
Thank you very much. But the rest can be solved after that? Or just calculate 1/3(s^2/2+s-s^2)
– Hobong
Dec 9 at 15:23
1
1
The rest is just calculus.
– Gordon
Dec 9 at 15:27
The rest is just calculus.
– Gordon
Dec 9 at 15:27
add a comment |
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