Consider the random walk that in each t time unit either goes up or down the amount √t with respective probabilities p and 1− p, where p = 1 2 (1+μ

√t).

(a) Argue that as t → 0 the resulting limiting process is a Brownian motion process with drift rate μ.

(b) Using part

(a) and the results of the gambler’s ruin problem (Section 4.5.1), compute the probability that a Brownian motion process with drift rate μ

goes up A before going down B, A > 0, B > 0.