The problem is about the expection of the exitpoint distance for the symmetric random walk.
$begingroup$
Let $nu(x)$ be a symmetric probability measure with respect to the origin on $xin[-1,1]$ such that $nu({0})neq 1$.
Consider a random walk started at $S_0=0$, denoted $S_n=X_1+cdots+X_n$, where $X_1,X_2, cdots$ are the i.i.d sequences such $X_isim nu(x)$. For some $1leq L<infty$, denote $tau=inf{ngeq0: S_n>L}$.
Let $hbar_{nu,L}=mathbb{E}(S_tau)-L$, in other words, $hbar_{nu,L}$ is the mean value of exitpoint distance from $L$.
$textbf{My question is how to derive the explicit formula for}$ $bf{hbar_{nu,L}}$$textbf{?}$
Mey be one can start by some simple $nu(x)$ and fix $L=1$. Let $mu(x)$ be the probability density function of $nu(x)$, for example,
$textbf{(i)} $ $mu(x)=1/2,~ xin[-1,1];$
$textbf{(ii)} $$mu(x)=frac{2}{pi}sqrt{1-x^2},~ xin[-1,1];$
If possible，could you recommend some relevant papers or books for me？ Anyway, any hints or help would be appreciated. Thank you very much.
probability probability-theory martingales random-walk stopping-times
asked Dec 8 '18 at 5:41
lang zoulang zou
252
$endgroup$
add a comment |
$begingroup$
Let $nu(x)$ be a symmetric probability measure with respect to the origin on $xin[-1,1]$ such that $nu({0})neq 1$.
Consider a random walk started at $S_0=0$, denoted $S_n=X_1+cdots+X_n$, where $X_1,X_2, cdots$ are the i.i.d sequences such $X_isim nu(x)$. For some $1leq L<infty$, denote $tau=inf{ngeq0: S_n>L}$.
Let $hbar_{nu,L}=mathbb{E}(S_tau)-L$, in other words, $hbar_{nu,L}$ is the mean value of exitpoint distance from $L$.
$textbf{My question is how to derive the explicit formula for}$ $bf{hbar_{nu,L}}$$textbf{?}$
Mey be one can start by some simple $nu(x)$ and fix $L=1$. Let $mu(x)$ be the probability density function of $nu(x)$, for example,
$textbf{(i)} $ $mu(x)=1/2,~ xin[-1,1];$
$textbf{(ii)} $$mu(x)=frac{2}{pi}sqrt{1-x^2},~ xin[-1,1];$
If possible，could you recommend some relevant papers or books for me？ Anyway, any hints or help would be appreciated. Thank you very much.
probability probability-theory martingales random-walk stopping-times
asked Dec 8 '18 at 5:41
lang zoulang zou
252
$endgroup$
add a comment |
$begingroup$
Let $nu(x)$ be a symmetric probability measure with respect to the origin on $xin[-1,1]$ such that $nu({0})neq 1$.
Consider a random walk started at $S_0=0$, denoted $S_n=X_1+cdots+X_n$, where $X_1,X_2, cdots$ are the i.i.d sequences such $X_isim nu(x)$. For some $1leq L<infty$, denote $tau=inf{ngeq0: S_n>L}$.
Let $hbar_{nu,L}=mathbb{E}(S_tau)-L$, in other words, $hbar_{nu,L}$ is the mean value of exitpoint distance from $L$.
$textbf{My question is how to derive the explicit formula for}$ $bf{hbar_{nu,L}}$$textbf{?}$
Mey be one can start by some simple $nu(x)$ and fix $L=1$. Let $mu(x)$ be the probability density function of $nu(x)$, for example,
$textbf{(i)} $ $mu(x)=1/2,~ xin[-1,1];$
$textbf{(ii)} $$mu(x)=frac{2}{pi}sqrt{1-x^2},~ xin[-1,1];$
If possible，could you recommend some relevant papers or books for me？ Anyway, any hints or help would be appreciated. Thank you very much.
probability probability-theory martingales random-walk stopping-times
asked Dec 8 '18 at 5:41
lang zoulang zou
252
$endgroup$
Let $nu(x)$ be a symmetric probability measure with respect to the origin on $xin[-1,1]$ such that $nu({0})neq 1$.
Consider a random walk started at $S_0=0$, denoted $S_n=X_1+cdots+X_n$, where $X_1,X_2, cdots$ are the i.i.d sequences such $X_isim nu(x)$. For some $1leq L<infty$, denote $tau=inf{ngeq0: S_n>L}$.
Let $hbar_{nu,L}=mathbb{E}(S_tau)-L$, in other words, $hbar_{nu,L}$ is the mean value of exitpoint distance from $L$.
$textbf{My question is how to derive the explicit formula for}$ $bf{hbar_{nu,L}}$$textbf{?}$
Mey be one can start by some simple $nu(x)$ and fix $L=1$. Let $mu(x)$ be the probability density function of $nu(x)$, for example,
$textbf{(i)} $ $mu(x)=1/2,~ xin[-1,1];$
$textbf{(ii)} $$mu(x)=frac{2}{pi}sqrt{1-x^2},~ xin[-1,1];$
If possible，could you recommend some relevant papers or books for me？ Anyway, any hints or help would be appreciated. Thank you very much.
probability probability-theory martingales random-walk stopping-times
probability probability-theory martingales random-walk stopping-times
asked Dec 8 '18 at 5:41
lang zoulang zou
252
asked Dec 8 '18 at 5:41
lang zoulang zou
252
asked Dec 8 '18 at 5:41
lang zoulang zou
252
asked Dec 8 '18 at 5:41
lang zoulang zou
252
asked Dec 8 '18 at 5:41
lang zoulang zou
252
252
add a comment |
add a comment |
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