Let $\nu(x)$ be a symmetric probability measure with respect to the origin on $x\in[-1,1]$ 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 $X_i\sim \nu(x)$. For some $1\leq L<\infty$, denote $\tau=\inf\{n\geq0: S_n>L\}$.

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)}\ \ \ $ $\mu(x)=1/2,~ x\in[-1,1];$

$\textbf{(ii)}\ \ \ $$\mu(x)=\frac{2}{\pi}\sqrt{1-x^2},~ x\in[-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.


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