Standard Brownian motion starts at 0 and tau is the first exit from the interval (-2, 3). Compute E[tau], and give the probability that the exit happens at the upper boundary.

Standard Brownian motion starts at 0 and tau is the first exit from the interval (-2, 3). Compute E[tau], and give the probability that the exit happens at the upper boundary.

Approach: Use W itself to get the exit probabilities, then apply optional stopping to the martingale W^2 - t to convert the second moment at exit into the expected time.

6. The process W is a martingale and the stopped process is bounded between -2 and 3, so uniform integrability holds and optional stopping gives 0 = 3p - 2(1 - p), hence the exit time lands on the upper boundary with probability p = 2/5 and on the lower with probability 3/5. The quantity W_t^2 - t is also a martingale, and E[tau] is finite because the exit time from a bounded interval has exponential tails, so optional stopping applies again and gives E[W_tau^2] = E[tau]. Then E[tau] = 9(2/5) + 4(3/5) = 3.6 + 2.4 = 6, which is the product of the two distances, a b with a = 2 and b = 3. The general result E[tau] = ab is the diffusion version of the gambler's ruin expected duration.

Follow-up: For arithmetic Brownian motion with drift mu, what replaces ab as the expected exit time from the same interval?

Key concepts: optional stopping, martingale W^2 - t, exit time, uniform integrability.