# Different Forms of Geometric Brownian Motion [closed]

If the stock price S follows the geometric brownian motion: $$dS=\mu Sdt+\sigma Sdz$$ $$\frac{dS}S=\mu dt+\sigma dz$$

Where $$dz=\epsilon\sqrt{dt}$$ is a wiener process.

Integrating this to get $$S_T$$ as a function of $$S_0$$

$$\int^T_0\frac{1}Sds=\int^T_0\mu dt +\int^T_0\sigma dz$$ $$=ln(S_T/S_0)=\mu(T-0)+\sigma (z_T-z_0)$$

$$S_T=S_0e^{\mu T+\sigma (z_T-z_0)}$$ Why do others allow $$(z_T-z_0)$$ to be a standard Wiener process?

And why is this not the same as: $$S_T =S_0 e^{(\mu -0.5\sigma^2)T+\sigma \epsilon \sqrt{T}}$$

And what is the difference between the above version, and this version: $$S_T =S_0+dS=S_0+\mu S_0 T+\sigma S_0 \epsilon \sqrt{T}$$?