Law of Ito Integral












0












$begingroup$


Let $sigma$ be an $mathcal{F}_t$-adapted caglad stochastic process. Let $W$ be a Brownian motion independent of $sigma$. Let $r>0$ be a strictly positive real number. How can I prove that
$$
left|int_{0}^1sigma_s,dW_sright|^r stackrel{d}{=} left|Uright|^r,left(int_0^1sigma_s^2,dsright)^{r/2},
$$
where $Ustackrel{d}{=}$ is a $text{N}(0,1)$ gaussian variable?






probability-distributions stochastic-integrals







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    Have a look at this
    $endgroup$
    – saz
    Nov 26 '18 at 18:37










  • $begingroup$
    @saz: Do you have any reference for proving that $$ sum_{j=0}^{n-1} sigma_{j/n} (W_{(j+1)/n}-W_{j/n}) xrightarrow{n to infty} int_0^1 sigma_s , dW_s quadtag{$star$} $$ holds in $L^{r}$, for any $r>0$ ? The proof for $r=2$ is easily obtained with the Ito-isometry and implies convergence in $L^r$ for any $1leq r<2$, but what about a generic $r>0$ not integer?
    $endgroup$
    – AlmostSureUser
    Nov 27 '18 at 11:13








  • 1




    $begingroup$
    You can use the Burkholder-Davis-Gundy inequality to prove convergence in $L^r$ for any $r>0$.
    $endgroup$
    – saz
    Nov 27 '18 at 11:36










  • $begingroup$
    @saz I think I have proved the $L^r$ convergence, nevertheless I am still confused. Since $r$ is not an integer, even if the $L^r$ convergence can be used to conclude that $$ mathbb{E} left( left( int_0^1 sigma_s , dW_s right)^r mid mathcal{F}_{sigma} right) = lim_{n to infty} mathbb{E}left( left( sum_{j=0}^{n-1} sigma_{j/n} (W_{(j+1)/n}-W_{j/n}) right)^r mid mathcal{F}_{sigma} right) $$ how can we conclude that $$ left|int_{0}^1sigma_s,dW_sright|^r stackrel{d}{=} left|Uright|^r,left(int_0^1sigma_s^2,dsright)^{r/2}quad ? $$
    $endgroup$
    – AlmostSureUser
    Nov 28 '18 at 9:00








  • 1




    $begingroup$
    Ah, sorry, that's not what I meant; I should have been more precise from the beginning. Using the linked answer, you can show that the random variables $$X := int_0^1 sigma(s) , dW_s$$ and $$Y:= U sqrt{int_0^1 sigma(s)^2 , ds}$$ are equal in distribution for $U sim N(0,1)$ independent from $sigma$. This entails $$f(X) stackrel{d}{=} f(Y)$$ for any function $f$; in particular, we can choose $f(x) = |x|^r$.
    $endgroup$
    – saz
    Nov 28 '18 at 10:19


















0












$begingroup$


Let $sigma$ be an $mathcal{F}_t$-adapted caglad stochastic process. Let $W$ be a Brownian motion independent of $sigma$. Let $r>0$ be a strictly positive real number. How can I prove that
$$
left|int_{0}^1sigma_s,dW_sright|^r stackrel{d}{=} left|Uright|^r,left(int_0^1sigma_s^2,dsright)^{r/2},
$$
where $Ustackrel{d}{=}$ is a $text{N}(0,1)$ gaussian variable?






probability-distributions stochastic-integrals







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    Have a look at this
    $endgroup$
    – saz
    Nov 26 '18 at 18:37










  • $begingroup$
    @saz: Do you have any reference for proving that $$ sum_{j=0}^{n-1} sigma_{j/n} (W_{(j+1)/n}-W_{j/n}) xrightarrow{n to infty} int_0^1 sigma_s , dW_s quadtag{$star$} $$ holds in $L^{r}$, for any $r>0$ ? The proof for $r=2$ is easily obtained with the Ito-isometry and implies convergence in $L^r$ for any $1leq r<2$, but what about a generic $r>0$ not integer?
    $endgroup$
    – AlmostSureUser
    Nov 27 '18 at 11:13








  • 1




    $begingroup$
    You can use the Burkholder-Davis-Gundy inequality to prove convergence in $L^r$ for any $r>0$.
    $endgroup$
    – saz
    Nov 27 '18 at 11:36










  • $begingroup$
    @saz I think I have proved the $L^r$ convergence, nevertheless I am still confused. Since $r$ is not an integer, even if the $L^r$ convergence can be used to conclude that $$ mathbb{E} left( left( int_0^1 sigma_s , dW_s right)^r mid mathcal{F}_{sigma} right) = lim_{n to infty} mathbb{E}left( left( sum_{j=0}^{n-1} sigma_{j/n} (W_{(j+1)/n}-W_{j/n}) right)^r mid mathcal{F}_{sigma} right) $$ how can we conclude that $$ left|int_{0}^1sigma_s,dW_sright|^r stackrel{d}{=} left|Uright|^r,left(int_0^1sigma_s^2,dsright)^{r/2}quad ? $$
    $endgroup$
    – AlmostSureUser
    Nov 28 '18 at 9:00








  • 1




    $begingroup$
    Ah, sorry, that's not what I meant; I should have been more precise from the beginning. Using the linked answer, you can show that the random variables $$X := int_0^1 sigma(s) , dW_s$$ and $$Y:= U sqrt{int_0^1 sigma(s)^2 , ds}$$ are equal in distribution for $U sim N(0,1)$ independent from $sigma$. This entails $$f(X) stackrel{d}{=} f(Y)$$ for any function $f$; in particular, we can choose $f(x) = |x|^r$.
    $endgroup$
    – saz
    Nov 28 '18 at 10:19
















0












0








0





$begingroup$


Let $sigma$ be an $mathcal{F}_t$-adapted caglad stochastic process. Let $W$ be a Brownian motion independent of $sigma$. Let $r>0$ be a strictly positive real number. How can I prove that
$$
left|int_{0}^1sigma_s,dW_sright|^r stackrel{d}{=} left|Uright|^r,left(int_0^1sigma_s^2,dsright)^{r/2},
$$
where $Ustackrel{d}{=}$ is a $text{N}(0,1)$ gaussian variable?






probability-distributions stochastic-integrals







share|cite|improve this question









$endgroup$




Let $sigma$ be an $mathcal{F}_t$-adapted caglad stochastic process. Let $W$ be a Brownian motion independent of $sigma$. Let $r>0$ be a strictly positive real number. How can I prove that
$$
left|int_{0}^1sigma_s,dW_sright|^r stackrel{d}{=} left|Uright|^r,left(int_0^1sigma_s^2,dsright)^{r/2},
$$
where $Ustackrel{d}{=}$ is a $text{N}(0,1)$ gaussian variable?






probability-distributions stochastic-integrals




probability-distributions stochastic-integrals






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Nov 26 '18 at 16:39









AlmostSureUserAlmostSureUser

321417




321417








  • 1




    $begingroup$
    Have a look at this
    $endgroup$
    – saz
    Nov 26 '18 at 18:37










  • $begingroup$
    @saz: Do you have any reference for proving that $$ sum_{j=0}^{n-1} sigma_{j/n} (W_{(j+1)/n}-W_{j/n}) xrightarrow{n to infty} int_0^1 sigma_s , dW_s quadtag{$star$} $$ holds in $L^{r}$, for any $r>0$ ? The proof for $r=2$ is easily obtained with the Ito-isometry and implies convergence in $L^r$ for any $1leq r<2$, but what about a generic $r>0$ not integer?
    $endgroup$
    – AlmostSureUser
    Nov 27 '18 at 11:13








  • 1




    $begingroup$
    You can use the Burkholder-Davis-Gundy inequality to prove convergence in $L^r$ for any $r>0$.
    $endgroup$
    – saz
    Nov 27 '18 at 11:36










  • $begingroup$
    @saz I think I have proved the $L^r$ convergence, nevertheless I am still confused. Since $r$ is not an integer, even if the $L^r$ convergence can be used to conclude that $$ mathbb{E} left( left( int_0^1 sigma_s , dW_s right)^r mid mathcal{F}_{sigma} right) = lim_{n to infty} mathbb{E}left( left( sum_{j=0}^{n-1} sigma_{j/n} (W_{(j+1)/n}-W_{j/n}) right)^r mid mathcal{F}_{sigma} right) $$ how can we conclude that $$ left|int_{0}^1sigma_s,dW_sright|^r stackrel{d}{=} left|Uright|^r,left(int_0^1sigma_s^2,dsright)^{r/2}quad ? $$
    $endgroup$
    – AlmostSureUser
    Nov 28 '18 at 9:00








  • 1




    $begingroup$
    Ah, sorry, that's not what I meant; I should have been more precise from the beginning. Using the linked answer, you can show that the random variables $$X := int_0^1 sigma(s) , dW_s$$ and $$Y:= U sqrt{int_0^1 sigma(s)^2 , ds}$$ are equal in distribution for $U sim N(0,1)$ independent from $sigma$. This entails $$f(X) stackrel{d}{=} f(Y)$$ for any function $f$; in particular, we can choose $f(x) = |x|^r$.
    $endgroup$
    – saz
    Nov 28 '18 at 10:19
















  • 1




    $begingroup$
    Have a look at this
    $endgroup$
    – saz
    Nov 26 '18 at 18:37










  • $begingroup$
    @saz: Do you have any reference for proving that $$ sum_{j=0}^{n-1} sigma_{j/n} (W_{(j+1)/n}-W_{j/n}) xrightarrow{n to infty} int_0^1 sigma_s , dW_s quadtag{$star$} $$ holds in $L^{r}$, for any $r>0$ ? The proof for $r=2$ is easily obtained with the Ito-isometry and implies convergence in $L^r$ for any $1leq r<2$, but what about a generic $r>0$ not integer?
    $endgroup$
    – AlmostSureUser
    Nov 27 '18 at 11:13








  • 1




    $begingroup$
    You can use the Burkholder-Davis-Gundy inequality to prove convergence in $L^r$ for any $r>0$.
    $endgroup$
    – saz
    Nov 27 '18 at 11:36










  • $begingroup$
    @saz I think I have proved the $L^r$ convergence, nevertheless I am still confused. Since $r$ is not an integer, even if the $L^r$ convergence can be used to conclude that $$ mathbb{E} left( left( int_0^1 sigma_s , dW_s right)^r mid mathcal{F}_{sigma} right) = lim_{n to infty} mathbb{E}left( left( sum_{j=0}^{n-1} sigma_{j/n} (W_{(j+1)/n}-W_{j/n}) right)^r mid mathcal{F}_{sigma} right) $$ how can we conclude that $$ left|int_{0}^1sigma_s,dW_sright|^r stackrel{d}{=} left|Uright|^r,left(int_0^1sigma_s^2,dsright)^{r/2}quad ? $$
    $endgroup$
    – AlmostSureUser
    Nov 28 '18 at 9:00








  • 1




    $begingroup$
    Ah, sorry, that's not what I meant; I should have been more precise from the beginning. Using the linked answer, you can show that the random variables $$X := int_0^1 sigma(s) , dW_s$$ and $$Y:= U sqrt{int_0^1 sigma(s)^2 , ds}$$ are equal in distribution for $U sim N(0,1)$ independent from $sigma$. This entails $$f(X) stackrel{d}{=} f(Y)$$ for any function $f$; in particular, we can choose $f(x) = |x|^r$.
    $endgroup$
    – saz
    Nov 28 '18 at 10:19










1




1




$begingroup$
Have a look at this
$endgroup$
– saz
Nov 26 '18 at 18:37




$begingroup$
Have a look at this
$endgroup$
– saz
Nov 26 '18 at 18:37












$begingroup$
@saz: Do you have any reference for proving that $$ sum_{j=0}^{n-1} sigma_{j/n} (W_{(j+1)/n}-W_{j/n}) xrightarrow{n to infty} int_0^1 sigma_s , dW_s quadtag{$star$} $$ holds in $L^{r}$, for any $r>0$ ? The proof for $r=2$ is easily obtained with the Ito-isometry and implies convergence in $L^r$ for any $1leq r<2$, but what about a generic $r>0$ not integer?
$endgroup$
– AlmostSureUser
Nov 27 '18 at 11:13






$begingroup$
@saz: Do you have any reference for proving that $$ sum_{j=0}^{n-1} sigma_{j/n} (W_{(j+1)/n}-W_{j/n}) xrightarrow{n to infty} int_0^1 sigma_s , dW_s quadtag{$star$} $$ holds in $L^{r}$, for any $r>0$ ? The proof for $r=2$ is easily obtained with the Ito-isometry and implies convergence in $L^r$ for any $1leq r<2$, but what about a generic $r>0$ not integer?
$endgroup$
– AlmostSureUser
Nov 27 '18 at 11:13






1




1




$begingroup$
You can use the Burkholder-Davis-Gundy inequality to prove convergence in $L^r$ for any $r>0$.
$endgroup$
– saz
Nov 27 '18 at 11:36




$begingroup$
You can use the Burkholder-Davis-Gundy inequality to prove convergence in $L^r$ for any $r>0$.
$endgroup$
– saz
Nov 27 '18 at 11:36












$begingroup$
@saz I think I have proved the $L^r$ convergence, nevertheless I am still confused. Since $r$ is not an integer, even if the $L^r$ convergence can be used to conclude that $$ mathbb{E} left( left( int_0^1 sigma_s , dW_s right)^r mid mathcal{F}_{sigma} right) = lim_{n to infty} mathbb{E}left( left( sum_{j=0}^{n-1} sigma_{j/n} (W_{(j+1)/n}-W_{j/n}) right)^r mid mathcal{F}_{sigma} right) $$ how can we conclude that $$ left|int_{0}^1sigma_s,dW_sright|^r stackrel{d}{=} left|Uright|^r,left(int_0^1sigma_s^2,dsright)^{r/2}quad ? $$
$endgroup$
– AlmostSureUser
Nov 28 '18 at 9:00






$begingroup$
@saz I think I have proved the $L^r$ convergence, nevertheless I am still confused. Since $r$ is not an integer, even if the $L^r$ convergence can be used to conclude that $$ mathbb{E} left( left( int_0^1 sigma_s , dW_s right)^r mid mathcal{F}_{sigma} right) = lim_{n to infty} mathbb{E}left( left( sum_{j=0}^{n-1} sigma_{j/n} (W_{(j+1)/n}-W_{j/n}) right)^r mid mathcal{F}_{sigma} right) $$ how can we conclude that $$ left|int_{0}^1sigma_s,dW_sright|^r stackrel{d}{=} left|Uright|^r,left(int_0^1sigma_s^2,dsright)^{r/2}quad ? $$
$endgroup$
– AlmostSureUser
Nov 28 '18 at 9:00






1




1




$begingroup$
Ah, sorry, that's not what I meant; I should have been more precise from the beginning. Using the linked answer, you can show that the random variables $$X := int_0^1 sigma(s) , dW_s$$ and $$Y:= U sqrt{int_0^1 sigma(s)^2 , ds}$$ are equal in distribution for $U sim N(0,1)$ independent from $sigma$. This entails $$f(X) stackrel{d}{=} f(Y)$$ for any function $f$; in particular, we can choose $f(x) = |x|^r$.
$endgroup$
– saz
Nov 28 '18 at 10:19






$begingroup$
Ah, sorry, that's not what I meant; I should have been more precise from the beginning. Using the linked answer, you can show that the random variables $$X := int_0^1 sigma(s) , dW_s$$ and $$Y:= U sqrt{int_0^1 sigma(s)^2 , ds}$$ are equal in distribution for $U sim N(0,1)$ independent from $sigma$. This entails $$f(X) stackrel{d}{=} f(Y)$$ for any function $f$; in particular, we can choose $f(x) = |x|^r$.
$endgroup$
– saz
Nov 28 '18 at 10:19












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