Variance of Stochastic Integral $int_0^1t^2 dWt$Expected value of the stochastic integral $int_0^t e^as dW_s$Expectation and variance of this stochastic processVariance of Ito IntegralFind a process $f=f(t,W_t)$ such that another process is a martingaleStochastic Calculus - Ito decompositionIto formula - some doubtsDistribution of stochastic integralWhat are the components in the Ito Process?Representation of Ito integralCalculating Variance of the stochastic process
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Variance of Stochastic Integral $int_0^1t^2 dWt$
Expected value of the stochastic integral $int_0^t e^as dW_s$Expectation and variance of this stochastic processVariance of Ito IntegralFind a process $f=f(t,W_t)$ such that another process is a martingaleStochastic Calculus - Ito decompositionIto formula - some doubtsDistribution of stochastic integralWhat are the components in the Ito Process?Representation of Ito integralCalculating Variance of the stochastic process
$begingroup$
I want to find Variance of integral $int_0^1t^2 dWt$
$W_t$ is Brownian motion
What I did:
I used Ito formula and got:
$int_0^1t^2 dWt = W_1 - int_0^1 2tW_t dt$ (correct me if I'm wrong)
I do not know how to compute $int_0^1 2tW_t dt$ or compute variance for whole answer.
stochastic-processes stochastic-calculus stochastic-integrals
asked yesterday
mahdimahdi
7412
$endgroup$
add a comment |
$begingroup$
I want to find Variance of integral $int_0^1t^2 dWt$
$W_t$ is Brownian motion
What I did:
I used Ito formula and got:
$int_0^1t^2 dWt = W_1 - int_0^1 2tW_t dt$ (correct me if I'm wrong)
I do not know how to compute $int_0^1 2tW_t dt$ or compute variance for whole answer.
stochastic-processes stochastic-calculus stochastic-integrals
asked yesterday
mahdimahdi
7412
$endgroup$
2
$begingroup$
by Ito the variance is $int_0^1 t^4 dt$
$endgroup$
– Math-fun
yesterday
add a comment |
$begingroup$
I want to find Variance of integral $int_0^1t^2 dWt$
$W_t$ is Brownian motion
What I did:
I used Ito formula and got:
$int_0^1t^2 dWt = W_1 - int_0^1 2tW_t dt$ (correct me if I'm wrong)
I do not know how to compute $int_0^1 2tW_t dt$ or compute variance for whole answer.
stochastic-processes stochastic-calculus stochastic-integrals
asked yesterday
mahdimahdi
7412
$endgroup$
I want to find Variance of integral $int_0^1t^2 dWt$
$W_t$ is Brownian motion
What I did:
I used Ito formula and got:
$int_0^1t^2 dWt = W_1 - int_0^1 2tW_t dt$ (correct me if I'm wrong)
I do not know how to compute $int_0^1 2tW_t dt$ or compute variance for whole answer.
stochastic-processes stochastic-calculus stochastic-integrals
stochastic-processes stochastic-calculus stochastic-integrals
asked yesterday
mahdimahdi
7412
asked yesterday
mahdimahdi
7412
edited 10 hours ago
mahdi
asked yesterday
mahdimahdi
7412
asked yesterday
mahdimahdi
7412
asked yesterday
mahdimahdi
7412
7412
2
$begingroup$
by Ito the variance is $int_0^1 t^4 dt$
$endgroup$
– Math-fun
yesterday
add a comment |
2
$begingroup$
by Ito the variance is $int_0^1 t^4 dt$
$endgroup$
– Math-fun
yesterday
2
2
$begingroup$
by Ito the variance is $int_0^1 t^4 dt$
$endgroup$
– Math-fun
yesterday
$begingroup$
by Ito the variance is $int_0^1 t^4 dt$
$endgroup$
– Math-fun
yesterday
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Expectation of Ito integral of a deterministic function wrt to a Brownian motion is $0$. Thus,
$$ mathbbVarleft[int_0^1 t^2 d W_t right]= mathbbEleft[left(int_0^1 t^2 d W_tright)^2right]$$
Applying Ito Isometry, we obtain
$$ mathbbEleft[left(int_0^1 t^2 d W_tright)^2right] = mathbbEleft[int_0^1 t^4 dtright] = int_0^1 t^4 dt = frac15 $$
answered 15 hours ago
MdocMdoc
556515
$endgroup$
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Expectation of Ito integral of a deterministic function wrt to a Brownian motion is $0$. Thus,
$$ mathbbVarleft[int_0^1 t^2 d W_t right]= mathbbEleft[left(int_0^1 t^2 d W_tright)^2right]$$
Applying Ito Isometry, we obtain
$$ mathbbEleft[left(int_0^1 t^2 d W_tright)^2right] = mathbbEleft[int_0^1 t^4 dtright] = int_0^1 t^4 dt = frac15 $$
answered 15 hours ago
MdocMdoc
556515
$endgroup$
add a comment |
$begingroup$
Expectation of Ito integral of a deterministic function wrt to a Brownian motion is $0$. Thus,
$$ mathbbVarleft[int_0^1 t^2 d W_t right]= mathbbEleft[left(int_0^1 t^2 d W_tright)^2right]$$
Applying Ito Isometry, we obtain
$$ mathbbEleft[left(int_0^1 t^2 d W_tright)^2right] = mathbbEleft[int_0^1 t^4 dtright] = int_0^1 t^4 dt = frac15 $$
answered 15 hours ago
MdocMdoc
556515
$endgroup$
add a comment |
$begingroup$
Expectation of Ito integral of a deterministic function wrt to a Brownian motion is $0$. Thus,
$$ mathbbVarleft[int_0^1 t^2 d W_t right]= mathbbEleft[left(int_0^1 t^2 d W_tright)^2right]$$
Applying Ito Isometry, we obtain
$$ mathbbEleft[left(int_0^1 t^2 d W_tright)^2right] = mathbbEleft[int_0^1 t^4 dtright] = int_0^1 t^4 dt = frac15 $$
answered 15 hours ago
MdocMdoc
556515
$endgroup$
Expectation of Ito integral of a deterministic function wrt to a Brownian motion is $0$. Thus,
$$ mathbbVarleft[int_0^1 t^2 d W_t right]= mathbbEleft[left(int_0^1 t^2 d W_tright)^2right]$$
Applying Ito Isometry, we obtain
$$ mathbbEleft[left(int_0^1 t^2 d W_tright)^2right] = mathbbEleft[int_0^1 t^4 dtright] = int_0^1 t^4 dt = frac15 $$
answered 15 hours ago
MdocMdoc
556515
answered 15 hours ago
MdocMdoc
556515
answered 15 hours ago
MdocMdoc
556515
answered 15 hours ago
MdocMdoc
556515
556515
add a comment |
add a comment |
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$begingroup$
by Ito the variance is $int_0^1 t^4 dt$
$endgroup$
– Math-fun
yesterday