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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    $\begingroup$ checkout pg. 10 of these notes: columbia.edu/~ks20/FE-Notes/4700-07-Notes-BM.pdf Hitting times for brownian motion with drift. In some sense, the continuous time analogue of a random walk is brownian motion. $\endgroup$ – Matthew Gunn Dec 18 '15 at 1:14
  • $\begingroup$ Thanks for the pointer, though ideally I'd like to preserve the integer nature of the number of steps, which has a meaning in my problem. But if all else fails, I can use your suggestion as an approximation. Thanks again! $\endgroup$ – mikew Dec 18 '15 at 19:18
  • $\begingroup$ The text book calculation for such results (whether SRW or BM) relies on finding a martingale (which is fine for an arbitrary jump distribution), but also that the walk is equal to $a$ or $b$ when it "reaches" it (which I interpret as "exceeds or equals"). Unless you have a particular distribution in mind (for which there might be some tricks), I think you'll struggle to find a general formula. $\endgroup$ – P.Windridge Dec 18 '15 at 20:36

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