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.