Solving the SDE: Unveiling the Log-Normal Distribution for Geometric Brownian Motion

Students often struggle with the 'Ito correction' term

-\frac{1}{2}\sigma^2 t

. They erroneously assume the solution is simply

S_0 e^{\mu t + \sigma W_t}

, forgetting that the chain rule for stochastic calculus requires accounting for the second-order derivative

f''(S)

, which does not vanish as it does in standard calculus.

Institutional Reference: Advanced Stochastic Processes