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A simple random walk $S_n = X_1 +\cdots +X_n$, where $P(X_i = 1) = p \not = 0.5$ and $P(X_i=-1)= q \triangleq 1-p$, admits the following probability $$P(S_n \textrm{ reaches } a \textrm{ before} -b) = \left(\frac{1-\left(\frac{q}{p}\right)^{b}}{1-\left(\frac{q}{p}\right)^{a+b}}\right),$$ for $a$ and $b$ positive integers. The expected number of steps to hit either of these boundaries is $$\frac{b}{q - p} - \left(\frac{a+b}{q-p}\right)\left(\frac{1-\left(\frac{q}{p}\right)^{b}}{1-\left(\frac{q}{p}\right)^{a+b}}\right).$$

My question is whether there are analogues to these simple expressions for positive $a$ and $b$ (not necessarily integer) when the $X_i$ are continuous i.i.d. random variables on $[-1,1]$ with a given density $f$? I'm particularly interested in a case where the $X_i$ have a finite positive mean $\mu_X > 0$.

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    $\begingroup$ I don't think you'll find exact expressions like this, but the asymptotic behaviour of the random walk should be similar. You might check out Spitzer's book: principles of random walk. $\endgroup$ – Anthony Quas Dec 17 '15 at 23:35
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    $\begingroup$ The condition $\sigma_X^2 < \infty$ is redundant, being implied by the hypothesis that $|X_i| \leq 1$ with probability $1$. $\endgroup$ – Noam D. Elkies Dec 18 '15 at 1:05
  • $\begingroup$ Thank you both for your comments (problem statement edited to remove redundancy). $\endgroup$ – mikew Dec 18 '15 at 19:10
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As already pointed out by Anthony in a comment, you can't really expect explicit formulae. Let's write $p(x)$ for the probability that the RW starting at $x\in (a,b)$ hits $a$ before it hits $b$. Then, by the same argument as in the discrete case (condition on what happens on the very next step), $p(x) = \int p(x+t)f(t)\, dt$; here, we must interpret $p(s)=1$ for $s\le a$ and $p(s)=0$ for $s\ge b$ on the right-hand side. So by splitting off the parts where $x+t\notin (a,b)$, we can rewrite the integral equation as $$ p(x) = F(a-x) + \int_a^b f(t-x)p(t)\, dt ,\quad a<x<b . $$ This is a Fredholm equation of the second type. It won't be easy to come up with an $f$ for which this has an explicit solution. You can obtain a series representation of the solution by iterating, see here for example (notice that the $\phi_n$ increase pointwise and stay $<1$, so convergence is guaranteed).

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