# What is the difference between these two equations for GBMs?

The two equations commonly found online for GBM are:

$\begin{matrix} S_{ t }=S_{ 0 }\exp\left( \left( \mu -\frac { \sigma ^{ 2 } }{ 2 } \right) t+\sigma W_{ t } \right) \\ S_{ t }=S_{ 0 }\exp\left(\mu t+\sigma W_{ t } \right) \end{matrix}$

I found the first one on Wikipedia, and the second one in a Columbia university PDF about simulation of GBMs, Page 4.

The first the solution to: $$dS_t = S_t\left[\mu dt +\sigma dW_t\right]$$ The second is the solution to: $$dS_t = S_t\left[\left(\mu -\frac{\sigma^2}{2}\right)dt + \sigma dW_t\right]$$
The difference is that the first one is a martingale when $\mu$ is equal to zero while the second one is not: $$\mathbb{E}[S_0 exp(\sigma W_t)]= S_0exp(-\frac{\sigma^2}{2}t)$$
The standard form of a geometric Brownian motion is $dS_t = S_t(\mu dt+\sigma dB_t)$, where B is a BM and $\mu$ and $\sigma$ are two real numbers. When you write this process in closed form: it is $S_t = S_0 exp(\mu t + \sigma B_t - \sigma ^2 t/2)$. The process $(S_t e^{-\mu t}, t\geq 0)$ is a martingale.