Brownian Motion, 2nd Edition

14 Skorokhod representation

Let (B_t)_t≥0 be a one-dimensional Brownian motion. Denote by , a, b > 0, the first entry time into the interval (-a, b)^c. We know from Corollary 5.11 that

(14.1)

We will now study the problem which probability distributions can be obtained in this way. More precisely, let X be a random variable with probability distribution function . We want to know if there is a Brownian motion (B_t)_t≥0 and a stopping time τ such that ...