Let X_t = exp(sigma W_t). Find the SDE satisfied by X, the constant a for which exp(sigma W_t - a t) is a martingale, and E[exp(sigma W_t)].

Let X_t = exp(sigma W_t). Find the SDE satisfied by X, the constant a for which exp(sigma W_t - a t) is a martingale, and E[exp(sigma W_t)].

Approach: Apply Ito to the exponential, read the drift coefficient, then choose the subtracted rate that cancels it exactly.

dX_t = sigma X_t dW_t + (sigma^2/2) X_t dt, a = sigma^2/2, and E[exp(sigma W_t)] = exp(sigma^2 t / 2). Ito applied to f(x) = exp(sigma x) gives f' = sigma f and f'' = sigma^2 f, so df = sigma f dW + (sigma^2/2) f dt, and the second term is the Ito correction produced by the convexity of the exponential. Subtracting a t inside the exponent adds a drift of -a to the same expression, so the choice a = sigma^2/2 kills the dt term and leaves a driftless process, the exponential martingale exp(sigma W_t - sigma^2 t/2), whose expectation is its starting value 1. Rearranging that identity gives the moment generating function E[exp(sigma W_t)] = exp(sigma^2 t/2), and X itself is a geometric Brownian motion with drift sigma^2/2 and volatility sigma.

Follow-up: For which sigma and horizons is exp(sigma W_t - sigma^2 t/2) uniformly integrable on [0, ∞), and what is its almost sure limit?

Key concepts: Ito correction, exponential martingale, moment generating function, geometric Brownian motion.