Question

(1) For the standard Brownian motion, (W(t),t2 0], what is the expected first passage time, E(ta), for a > 0, where ta-inf{t : W(t) 2 a]? The following "answers" have been proposed (b) a/2. (c) a (d) 2a (e) None of the above. The correct answer is

Answer 1

A standard Brownian motion is a random process X={Xt:t∈[0,∞)} with state space R that satisfies the following properties:

1. X0=0 (with probability 1).
2. X has stationary increments. That is, for s,t∈[0,∞) with s<t, the distribution of Xt−Xs is the same as the distribution of Xt−s.
3. X has independent increments. That is, for t1,t2,…,tn∈[0,∞) with t1<t2<⋯<tn, the random variables Xt1,Xt2−Xt1,…,Xtn−Xtn−1 are independent.
4. Xt is normally distributed with mean 0 and variance t for each t∈(0,∞).
5. With probability 1, t↦Xt is continuous on [0,∞).