1. Jack walks along the x-axis in search of his favorite pizzeria, he starts walking from x = 0 at time t=0. This pizzeria is located at x = 4. For some reasons his motion is described by standard Brownian motion {B(t) : t > 0}. Use the theory in Pinsky and Karlin (8.2) to write down an integral for the probability that Jack reaches the pizzeria during the first 9 time units of his walk. No integral evaluating needed.

Further, we stipulate that B(to + () is likely to be near B(to) = x for small values of t. This is done formally by requiring that limp(y, f x) =0 for y x. (8.2) 1-+0