X_t = 0.05 t + 0.2 W_t started at zero. Compute the probability that X reaches +1 before it reaches -0.5.

X_t = 0.05 t + 0.2 W_t started at zero. Compute the probability that X reaches +1 before it reaches -0.5.

Approach: Find the exponent making the exponential of the process a martingale, then apply optional stopping at the exit from the two sided interval.

0.7307. The exponential martingale for arithmetic Brownian motion with drift mu and volatility sigma is exp(-theta X_t) with theta = 2 mu / sigma^2, since Ito gives the drift coefficient -theta mu + theta^2 sigma^2 / 2, which vanishes at that theta. Here theta = 2(0.05)/0.04 = 2.5, and exp(-theta x) is the scale function of the diffusion. The exit time from a bounded interval is finite with finite mean, so optional stopping applies and 1 = p exp(-2.5) + (1 - p) exp(1.25), where p is the exit probability at the upper level. With exp(-2.5) = 0.08208 and exp(1.25) = 3.49034, solving gives p = (1 - 3.49034)/(0.08208 - 3.49034) = 0.7307. The same computation with zero drift would give the harmonic answer 0.5/1.5 = 1/3, so the modest positive drift more than doubles the chance of reaching the upper level first.

Follow-up: What is E[tau] for this exit problem, and how does it compare with the driftless case?

Key concepts: exponential martingale, optional stopping, scale function, exit probability.