# Why do we need $dS_t=r S_tdt+\sigma S_tdW_t^Q$?

Suppose $S_t$ is the stock price and follows the dynamics $$dS_t=\mu S_tdt+\sigma S_tdW_t$$. According to Girsanov, we can apply change of measure and obtain $dS_t=r S_tdt+\sigma S_tdW_t^Q$, this implies $\ln S_T = \ln S_0+rT+\sigma W_T^Q$, and therefore $$\mathbb{E}^Q[S_T]=S_0 e^{rT+\frac{1}{2}\sigma^2T}$$, however $\mathbb{E}^Q[S_T]=S_0e^{rT}$ by Fundamental Theorem of Asset Pricing, which is contradicted. Please correct me.

It doesn't imply

$$\ln S_T=\ln S_0+rT+σW^Q_T,$$

it implies

$$\ln S_T=\ln S_0+(r-0.5\sigma^2)T+σW^Q_T.$$

Look up Ito's lemma.

This is covered in just about any book on financial maths including my own Concepts etc.

If $dS_t = r S_t \, dt + \sigma S_t \, dW_t^Q$, $$S_T = S_0 \, e^{\sigma W_T^Q + \left( r - \frac{1}{2} \sigma^2\right) T}\, .$$

Hence $\mathbb{E}\left[ S_T \right] = S_0 \, e^{rT} \,.$