quant trading strategies - Questions on continuously compounded return vs long term expected return

I have reading a paper from Oliver Grandville on long term expected return. I am trying to reconcile what I am reading in that paper vs what I see under "Application to Stock Market" in Kelly criterion page on Wikipedia.

http://www.cfapubs.org/doi/abs/10.2469/faj.v54.n6.2227

https://en.wikipedia.org/wiki/Kelly_criterion

In his paper, he define the following:

1. $R_{t-1,t} = \frac{S_t-S_{t-1}}{S_{t-1}}$ as the yearly rate of return compounded once per year


2. $X_{t-1,t} = 1 + R_{t-1,t} = \frac{S_t}{S_{t-1}}$ as the yearly dollar return


3. $log \left( X_{t-1,t} \right) = log \left(\frac{S_t}{S_{t-1}}\right)$ as the yearly continuously compounded rate of return


4. $E(X_{t-1,t}) = E(1+R_{t-1,t})$ as the expected value of the yearly dollar return and $V(X_{t-1,t})$ as its variance

He derives that given $log \left( X_{t-1,t} \right) = log \left(\frac{S_t}{S_{t-1}}\right) = \mu + \sigma N(0,1)$ (ie log returns follow normal distribution of mean $\mu$ and variance $\sigma^2)$, we must have that $E(X_{t-1,t}) = exp \left( \mu + \frac{\sigma^2}{2}\right) $.

My questions are:

1. For the wiki article vs this article, the wiki one says expected log return is $R_s = \left( \mu - \frac{\sigma^2}{2}\right)t$. This seems similar to the Grandville article in that if you take $log(E(X_{t-1,t})) = \mu + \frac{\sigma^2}{2}$, but its not exactly the same. Is the because the assumption the wiki article makes is different (ie the stock price moves like Brownian motion?)



2. In the derivation of the max of the expected value in wiki page. Shouldnt it be $G(f) = fE(stock) + (1-f)E(bond)=f*(\mu - \frac{\sigma^2}{2}) + (1-f)*r$, but the vol term is different ($f^2$ vs $f$). Why is that?


3. Finally, the last line of the wiki article mentions that "Remember that $ \mu $ is different from the asset log return $R_s$. Confusing this is a common mistake made by websites and articles talking about the Kelly Criterion." My understanding is that $\mu$ is the expected of the log returns $E(log(X_{t-1,t}))$ whereas $R_s$ is the log of the expected returns $log(E(X_{t-1,t}))$ and that this distinction is important because $E(log(X))$ and $log(E(X))$ are not always equal. Is this understanding correct?