# Random walk with self-transitions taking continuum limit

does anyone have any suggestions regarding how to correctly treat the continuum limit of a random walk that has non-zero self-transition probabilities? To put this concretely, let's say that the walker has a forward-jumping probability of $\alpha$ and a backwards-jumping probability of $\beta$; I am interested in the case where

$$\alpha + \beta < 1$$

I am specifically interested in determining the first-passage time distribution for a random walker subject to these conditions, both on the full interval and on the half-interval subject to a reflection condition at $x=0$. This feels like a problem that has likely been solved somewhere before; however my best search efforts are coming up dry, and so I would appreciate any references or help. Thank you.

• Something like this might be the easiest approach: Let $\gamma=1-\alpha-\beta$ be the probability of self-transition. a) Can you find the first-passage time distribution $p(t)$ for $\gamma=0$? b) For a walk consisting of $t$ steps (non-self-transitions), what distribution of self-transitions would you expect? (The mean would obviously be $\gamma t$). Now combine your answers from a) and b). By the way, the maths.SE has a number of stochastic experts. – lemon Apr 14 '15 at 9:42