Show that a bivariate Ito process has normal distribution
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I have to show that, if $W_t$ is a 1-d Brownian motion then
$biggl(W_t, int_0^t W_s dsbiggr)$ has normal distribution.
Hint: apply Ito formula to this bivariate process.
Any idea or suggestion on how to solve it?
I tried to show that with characteristic function approach, since the marginal distributions have both normal distribution
If I want to show that the couple is bivariate gaussian I have to prove that , $forall lambda_1 , lambda_2 in mathbb{R}$ $Z_t=lambda_1 W_t + lambda_2 int_0^t W_s ds$ is normal, so $dZ_t=lambda_1dW_t+lambda_2W_tdt$ and if I compute $phi_{Z_t}(eta)$ and $d(exp{ieta Z_t})$ in the end I get $phi_{Z_t}(eta)=int_0^t mathbb{E}(exp{(ieta Z_s)} cdot ietalambda_2 W_s)ds-int_0^t mathbb{E}(phi_{Z_s}) cdot eta^2 cdot 1/2 cdot ds$ so I don't know how to solve the first integral.
probability-theory stochastic-processes stochastic-calculus
asked Jan 15 at 10:10
Eva FacchiniEva Facchini
12
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add a comment |
$begingroup$
I have to show that, if $W_t$ is a 1-d Brownian motion then
$biggl(W_t, int_0^t W_s dsbiggr)$ has normal distribution.
Hint: apply Ito formula to this bivariate process.
Any idea or suggestion on how to solve it?
I tried to show that with characteristic function approach, since the marginal distributions have both normal distribution
If I want to show that the couple is bivariate gaussian I have to prove that , $forall lambda_1 , lambda_2 in mathbb{R}$ $Z_t=lambda_1 W_t + lambda_2 int_0^t W_s ds$ is normal, so $dZ_t=lambda_1dW_t+lambda_2W_tdt$ and if I compute $phi_{Z_t}(eta)$ and $d(exp{ieta Z_t})$ in the end I get $phi_{Z_t}(eta)=int_0^t mathbb{E}(exp{(ieta Z_s)} cdot ietalambda_2 W_s)ds-int_0^t mathbb{E}(phi_{Z_s}) cdot eta^2 cdot 1/2 cdot ds$ so I don't know how to solve the first integral.
probability-theory stochastic-processes stochastic-calculus
asked Jan 15 at 10:10
Eva FacchiniEva Facchini
12
$endgroup$
$begingroup$
So where exactly did you faiil/get stuck when you tried the characteristic function approach?
$endgroup$
– saz
Jan 15 at 11:22
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Hi @saz : I edited the text to show where I got stuck in the algebra.
$endgroup$
– Eva Facchini
Jan 15 at 11:56
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@EvaFacchini I don't see how the hint, which you were given, is particularly helpful. From my point og view, it is easier to reason using Riemann sums, i.e. to note that $$left( W_t, frac{1}{n} sum_{j=1}^n W_{tj/n} right)$$ is Gaussian (because $W$ is a Gaussian process) and hence its pointwise limit $(W_t, int_0^t W_s , ds)$ is Gaussian.
$endgroup$
– saz
Jan 16 at 13:59
add a comment |
$begingroup$
I have to show that, if $W_t$ is a 1-d Brownian motion then
$biggl(W_t, int_0^t W_s dsbiggr)$ has normal distribution.
Hint: apply Ito formula to this bivariate process.
Any idea or suggestion on how to solve it?
I tried to show that with characteristic function approach, since the marginal distributions have both normal distribution
If I want to show that the couple is bivariate gaussian I have to prove that , $forall lambda_1 , lambda_2 in mathbb{R}$ $Z_t=lambda_1 W_t + lambda_2 int_0^t W_s ds$ is normal, so $dZ_t=lambda_1dW_t+lambda_2W_tdt$ and if I compute $phi_{Z_t}(eta)$ and $d(exp{ieta Z_t})$ in the end I get $phi_{Z_t}(eta)=int_0^t mathbb{E}(exp{(ieta Z_s)} cdot ietalambda_2 W_s)ds-int_0^t mathbb{E}(phi_{Z_s}) cdot eta^2 cdot 1/2 cdot ds$ so I don't know how to solve the first integral.
probability-theory stochastic-processes stochastic-calculus
asked Jan 15 at 10:10
Eva FacchiniEva Facchini
12
$endgroup$
I have to show that, if $W_t$ is a 1-d Brownian motion then
$biggl(W_t, int_0^t W_s dsbiggr)$ has normal distribution.
Hint: apply Ito formula to this bivariate process.
Any idea or suggestion on how to solve it?
I tried to show that with characteristic function approach, since the marginal distributions have both normal distribution
If I want to show that the couple is bivariate gaussian I have to prove that , $forall lambda_1 , lambda_2 in mathbb{R}$ $Z_t=lambda_1 W_t + lambda_2 int_0^t W_s ds$ is normal, so $dZ_t=lambda_1dW_t+lambda_2W_tdt$ and if I compute $phi_{Z_t}(eta)$ and $d(exp{ieta Z_t})$ in the end I get $phi_{Z_t}(eta)=int_0^t mathbb{E}(exp{(ieta Z_s)} cdot ietalambda_2 W_s)ds-int_0^t mathbb{E}(phi_{Z_s}) cdot eta^2 cdot 1/2 cdot ds$ so I don't know how to solve the first integral.
probability-theory stochastic-processes stochastic-calculus
probability-theory stochastic-processes stochastic-calculus
asked Jan 15 at 10:10
Eva FacchiniEva Facchini
12
asked Jan 15 at 10:10
Eva FacchiniEva Facchini
12
edited Jan 15 at 13:05
Eva Facchini
asked Jan 15 at 10:10
Eva FacchiniEva Facchini
12
asked Jan 15 at 10:10
Eva FacchiniEva Facchini
12
asked Jan 15 at 10:10
Eva FacchiniEva Facchini
12
12
$begingroup$
So where exactly did you faiil/get stuck when you tried the characteristic function approach?
$endgroup$
– saz
Jan 15 at 11:22
$begingroup$
Hi @saz : I edited the text to show where I got stuck in the algebra.
$endgroup$
– Eva Facchini
Jan 15 at 11:56
$begingroup$
@EvaFacchini I don't see how the hint, which you were given, is particularly helpful. From my point og view, it is easier to reason using Riemann sums, i.e. to note that $$left( W_t, frac{1}{n} sum_{j=1}^n W_{tj/n} right)$$ is Gaussian (because $W$ is a Gaussian process) and hence its pointwise limit $(W_t, int_0^t W_s , ds)$ is Gaussian.
$endgroup$
– saz
Jan 16 at 13:59
add a comment |
$begingroup$
So where exactly did you faiil/get stuck when you tried the characteristic function approach?
$endgroup$
– saz
Jan 15 at 11:22
$begingroup$
Hi @saz : I edited the text to show where I got stuck in the algebra.
$endgroup$
– Eva Facchini
Jan 15 at 11:56
$begingroup$
@EvaFacchini I don't see how the hint, which you were given, is particularly helpful. From my point og view, it is easier to reason using Riemann sums, i.e. to note that $$left( W_t, frac{1}{n} sum_{j=1}^n W_{tj/n} right)$$ is Gaussian (because $W$ is a Gaussian process) and hence its pointwise limit $(W_t, int_0^t W_s , ds)$ is Gaussian.
$endgroup$
– saz
Jan 16 at 13:59
$begingroup$
So where exactly did you faiil/get stuck when you tried the characteristic function approach?
$endgroup$
– saz
Jan 15 at 11:22
$begingroup$
So where exactly did you faiil/get stuck when you tried the characteristic function approach?
$endgroup$
– saz
Jan 15 at 11:22
$begingroup$
Hi @saz : I edited the text to show where I got stuck in the algebra.
$endgroup$
– Eva Facchini
Jan 15 at 11:56
$begingroup$
Hi @saz : I edited the text to show where I got stuck in the algebra.
$endgroup$
– Eva Facchini
Jan 15 at 11:56
$begingroup$
@EvaFacchini I don't see how the hint, which you were given, is particularly helpful. From my point og view, it is easier to reason using Riemann sums, i.e. to note that $$left( W_t, frac{1}{n} sum_{j=1}^n W_{tj/n} right)$$ is Gaussian (because $W$ is a Gaussian process) and hence its pointwise limit $(W_t, int_0^t W_s , ds)$ is Gaussian.
$endgroup$
– saz
Jan 16 at 13:59
$begingroup$
@EvaFacchini I don't see how the hint, which you were given, is particularly helpful. From my point og view, it is easier to reason using Riemann sums, i.e. to note that $$left( W_t, frac{1}{n} sum_{j=1}^n W_{tj/n} right)$$ is Gaussian (because $W$ is a Gaussian process) and hence its pointwise limit $(W_t, int_0^t W_s , ds)$ is Gaussian.
$endgroup$
– saz
Jan 16 at 13:59
add a comment |
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$begingroup$
So where exactly did you faiil/get stuck when you tried the characteristic function approach?
$endgroup$
– saz
Jan 15 at 11:22
$begingroup$
Hi @saz : I edited the text to show where I got stuck in the algebra.
$endgroup$
– Eva Facchini
Jan 15 at 11:56
$begingroup$
@EvaFacchini I don't see how the hint, which you were given, is particularly helpful. From my point og view, it is easier to reason using Riemann sums, i.e. to note that $$left( W_t, frac{1}{n} sum_{j=1}^n W_{tj/n} right)$$ is Gaussian (because $W$ is a Gaussian process) and hence its pointwise limit $(W_t, int_0^t W_s , ds)$ is Gaussian.
$endgroup$
– saz
Jan 16 at 13:59