Log-Normality of Geometric Brownian Motion

Apply Itô's lemma to $\ln (S_t)$ to remove the multiplicative stochastic term.

The solved SDE gives $S_t=S_0\exp \left((\mu -{\sigma }^2/2)t+\sigma W_t\right)$.

1. 1. Apply Itô's lemma

   Itô's lemma applied to $\ln (S_t)$ removes the multiplicative structure, giving $d\ln S_t=(\mu -{\sigma }^2/2)dt+\sigma d{W}_t$.

2. 2. Integrate

   Integrating from $0$ to $t$ shows $\ln (S_t)$ is a deterministic drift plus a scaled Brownian motion.

   $$\ln (S_t)=\ln (S_0)+\left(\mu -\frac{{\sigma }^2}{2}\right)t+\sigma W_t\sim N\left(\ln (S_0)+\left(\mu -\frac{{\sigma }^2}{2}\right)t,{\sigma }^{2}t\right)$$

Answer: Normal with mean ln(S_0)+(mu-sigma^2/2)t and variance sigma^2 t