extension of discrete time process to the unit interval
$begingroup$
I am reading a proof and i don't understand some lines of cumputations.
We have a Euler Scheme SDE given by
$X_{n}(frac{k+1}{n}) =X_{n}(frac{k}{n}) +b(X_{n}(frac{k}{n}))/n + sigma(X_{n}(frac{k}{n}))mu_{k+1}/ sqrt{n}$
with $mu_{k}$ are iid zero mean and unit variance.
They define the extension of the discret process $X_{n}(frac{k}{n})$ to the unit interval by : $X_{n}(t)=X_{n}([(t/n])$ with [x]is the integral part of x.
They said that this extension satisfy over [0,1] the following equation :
$X_{n}(t)=x_{0}+ int_{0}^{t}b(X_{n}(frac{[sn]}{n}-)drho_{n}(s) + int_{0}^{t}mu(X_{n}(frac{[sn]}{n}-)dB_{n}(s)$
where $X_{n}(s-)$ denotes the left limit when t goes to s of $X_{n}(t)$;$X_{n}(0-)=x_{0}$ ; $rho_{n}(s)=frac{[ns]}{n}$; $B_{n}(s)= sum_{k=1}^{[sn]}mu_{k}/ sqrt{n} $
Can someone please give me an idea about how they constructed the different integrals ?
Thank you for any help !
integration schemes discrete-calculus
asked Jan 11 at 15:24
nournour
316
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add a comment |
$begingroup$
I am reading a proof and i don't understand some lines of cumputations.
We have a Euler Scheme SDE given by
$X_{n}(frac{k+1}{n}) =X_{n}(frac{k}{n}) +b(X_{n}(frac{k}{n}))/n + sigma(X_{n}(frac{k}{n}))mu_{k+1}/ sqrt{n}$
with $mu_{k}$ are iid zero mean and unit variance.
They define the extension of the discret process $X_{n}(frac{k}{n})$ to the unit interval by : $X_{n}(t)=X_{n}([(t/n])$ with [x]is the integral part of x.
They said that this extension satisfy over [0,1] the following equation :
$X_{n}(t)=x_{0}+ int_{0}^{t}b(X_{n}(frac{[sn]}{n}-)drho_{n}(s) + int_{0}^{t}mu(X_{n}(frac{[sn]}{n}-)dB_{n}(s)$
where $X_{n}(s-)$ denotes the left limit when t goes to s of $X_{n}(t)$;$X_{n}(0-)=x_{0}$ ; $rho_{n}(s)=frac{[ns]}{n}$; $B_{n}(s)= sum_{k=1}^{[sn]}mu_{k}/ sqrt{n} $
Can someone please give me an idea about how they constructed the different integrals ?
Thank you for any help !
integration schemes discrete-calculus
asked Jan 11 at 15:24
nournour
316
$endgroup$
add a comment |
$begingroup$
I am reading a proof and i don't understand some lines of cumputations.
We have a Euler Scheme SDE given by
$X_{n}(frac{k+1}{n}) =X_{n}(frac{k}{n}) +b(X_{n}(frac{k}{n}))/n + sigma(X_{n}(frac{k}{n}))mu_{k+1}/ sqrt{n}$
with $mu_{k}$ are iid zero mean and unit variance.
They define the extension of the discret process $X_{n}(frac{k}{n})$ to the unit interval by : $X_{n}(t)=X_{n}([(t/n])$ with [x]is the integral part of x.
They said that this extension satisfy over [0,1] the following equation :
$X_{n}(t)=x_{0}+ int_{0}^{t}b(X_{n}(frac{[sn]}{n}-)drho_{n}(s) + int_{0}^{t}mu(X_{n}(frac{[sn]}{n}-)dB_{n}(s)$
where $X_{n}(s-)$ denotes the left limit when t goes to s of $X_{n}(t)$;$X_{n}(0-)=x_{0}$ ; $rho_{n}(s)=frac{[ns]}{n}$; $B_{n}(s)= sum_{k=1}^{[sn]}mu_{k}/ sqrt{n} $
Can someone please give me an idea about how they constructed the different integrals ?
Thank you for any help !
integration schemes discrete-calculus
asked Jan 11 at 15:24
nournour
316
$endgroup$
I am reading a proof and i don't understand some lines of cumputations.
We have a Euler Scheme SDE given by
$X_{n}(frac{k+1}{n}) =X_{n}(frac{k}{n}) +b(X_{n}(frac{k}{n}))/n + sigma(X_{n}(frac{k}{n}))mu_{k+1}/ sqrt{n}$
with $mu_{k}$ are iid zero mean and unit variance.
They define the extension of the discret process $X_{n}(frac{k}{n})$ to the unit interval by : $X_{n}(t)=X_{n}([(t/n])$ with [x]is the integral part of x.
They said that this extension satisfy over [0,1] the following equation :
$X_{n}(t)=x_{0}+ int_{0}^{t}b(X_{n}(frac{[sn]}{n}-)drho_{n}(s) + int_{0}^{t}mu(X_{n}(frac{[sn]}{n}-)dB_{n}(s)$
where $X_{n}(s-)$ denotes the left limit when t goes to s of $X_{n}(t)$;$X_{n}(0-)=x_{0}$ ; $rho_{n}(s)=frac{[ns]}{n}$; $B_{n}(s)= sum_{k=1}^{[sn]}mu_{k}/ sqrt{n} $
Can someone please give me an idea about how they constructed the different integrals ?
Thank you for any help !
integration schemes discrete-calculus
integration schemes discrete-calculus
asked Jan 11 at 15:24
nournour
316
asked Jan 11 at 15:24
nournour
316
asked Jan 11 at 15:24
nournour
316
asked Jan 11 at 15:24
nournour
316
asked Jan 11 at 15:24
nournour
316
316
add a comment |
add a comment |
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