8. Consider the random walk which in each * time unit either goes up or down the...

8. Consider the random walk which in each Ä* time unit either goes up or down the amount VÄ7 with respective probabilities ñ and 1 - ñ where 77 = 1(1 +//VÄ7).

(a) Argue that as Ä/ 0 the resulting limiting process is a Brownian motion process with drift rate ì.

(b) Using

(a) and the results of the gambler's ruin problem, compute the probability that a Brownian motion process with drift rate ì goes up A before going down Â, A > 0, Â > 0.