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 $S_n=X_1+\dotsb+X_n$, where $X_1,X_2, \dotsc$ are i.i.d. random variables such that $X_i\sim \nu(x)$.

Fix $1\leq L<\infty$, and put $\tau=\inf\{n\geq0: S_n>L\}$ and $\hbar_{\nu,L}=\mathbb{E}(S_\tau)-L$. In other words, $\hbar_{\nu,L}$ is the mean value of exitpoint distance from $L$.

**My question is how to derive the explicit formula for $\hbar_{\nu,L}$.**

Maybe one can start by fixing $L = 1$ and choosing some simple $\nu(x)$, say with probability density function $\mu(x)$ given by

$\mu(x)=1/2$, $x\in[-1,1]$ or

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

Could you recommend some relevant papers or books for me? Anyway, any hints or help would be appreciated.

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forces bold in Markdown, and Markdown natively supports enumerated lists; neither should be faked with an equation environment. $\endgroup$ – LSpice Dec 9 '18 at 2:43