Let $S$ be a GBM with dynamics $dS_t/S_t=rdt+\sigma dW_t$. We want to compute the following expected value: \begin{align*} \mathbb{E}(S_T\log(S_T)). \end{align*} Using a change of measure we can write \begin{align*} \mathbb{E}(S_T\log(S_T)) =S_0\mathbb{\widehat E}(\log(S_T)) \end{align*} where $\mathbb{\widehat P}$ is the measure with $S$ as numeraire. How do we finish this problem? What are the dynamics of $(\log(S_t))_{t\geq 0}$ under $\mathbb{\widehat P}$?


Under the stock numeraire measure, $\frac{B_t}{S_t}$ is a Martingale. We can compute $$d\frac{B_t}{S_t}= \frac{1}{S_t}dB_t -\frac{1}{S_t^2}B_tdS_t+\frac{1}{S_t^3}B_t\sigma^2S_t^2dt\\=\frac{B_t}{S_t}\left(rdt -\mu dt -\sigma dW_t +\sigma^2dt\right)$$ so the growth rate $\mu$ that makes this a Martingale is $$ \mu = r+\sigma^2.$$

So the growth rate of the stock under the stock numeraire measure is $r+\sigma^2$.

Then, applying Ito as usual, you can find that $\log(S_t)$ follows Brownian motion with drift $r+\frac{1}{2}\sigma^2.$ (This is in contrast to $r-\frac{1}{2}\sigma^2$ in the usual case with the bond as numeraire.)


Looking back, I see that I missed the fine print that you called the growth rate of the stock $r$. I hope it's clear that I started in the physical measure, called the stock growth rate $\mu$ and used $r$ to refer to the risk free rate.

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