I want to show that the assumption that stock prices follow geometric Brownian motion is consistent with the weak efficient market hypothesis. I first want to show that the stock price defined by the following SDE is a martingale. \begin{equation} d S_t=\mu S_tdt+\sigma S_td B_t\label{jhblyd} \end{equation} By ITO's formula, \begin{equation*} d\ln {S_t} = \left( {\frac{1}{{{S_t}}}\mu {S_t} + \frac{{ - 1}}{2}\frac{1}{{{S_t}^2}}{\sigma ^2}{S_t}^2} \right)dt + \frac{1}{{{S_t}}}\sigma {S_t}d{B_t} = \left( {\mu - \frac{1}{2}{\sigma ^2}} \right)dt + \sigma d{B_t} %\label{dlnst} \end{equation*} Integrate with respect to $t$, \begin{equation*} \begin{aligned} \ln {S_t} &= \ln S_0+\int_{0}^t\left(\mu-\frac{1}{2}\sigma^2\right)ds+\int_0^t\sigma dB_s\\ & = \ln S_0+\left(\mu-\frac{1}{2}\sigma^2\right)t+\sigma B_t. \end{aligned} \end{equation*} Then, \begin{equation} S_t=S_0e^{(\mu-\frac{1}{2}\sigma^2)t+\sigma B_t}. \end{equation} $S_t$ is a martingale if and only if $\mu=0$.
So when the drift rate is not equal to zero, is that a violation of the weak efficient market hypothesis?