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As Sanjay said, you can apply Itô's Lemma to $f(t,x)=x^2$ and obtain \begin{align*} \mathrm{d} S^2_t=\left(2\mu S_t^2+\sigma^2S_t^2\right)\mathrm{d}t+\left(2\sigma S_t^2\right)\mathrm{d}W_t. \end{align*} Thus, $(S_t^2)$ is again a geometric Brownian motion and hence, for each time point $t$ log-normally distributed with drift $2\mu+\sigma^2$ and volatility $... 3 Classic asset price model in the continuous-time limit using a Wiener process notation can be written as $$dS_t=\mu S_tdt+\sigma S_t dX$$ where$S_t$is the stock price (not the stock return) and$dX$is an independent random variable with normal distribution. If we eliminate the drift ($\mu = 0$) and only focus on randomness as asked in your question we ... 1 For the first question, it is the standard assumption to make for stock returns if no other information is given. That's not to say it's a great assumption, but there it is clearly the only one that can be justified in this context. For the second part, independence of returns tells you that investment for T years has cumulative variance$T \sigma^2$(when ... 2 The first is something of a theoretical question. It's widely held/assumed that stocks follow a BM process, it appears as though the author is setting the table for the subsequent statement. The second is an artifact of applying Ito's lemma...the$dW_tdt$and$dtdt$terms both equal 0, hence fall out, leaving only$dW_t^2\$ = dt. Thus, the variance ...