Dynamics of the square of a lognormal stock

Suppose $S_t$ is a geometric Brownian motion, $dS=\mu S\,dt+\sigma S\,dW$. What stochastic process does $S_t^2$ follow?

Another GBM: $d(S^2)=S^2\big((2\mu+\sigma^2)dt+2\sigma\,dW\big)$. Ito gives $d(S^2)=2S\,dS+(dS)^2$, with $(dS)^2=\sigma^2S^2dt$. So the volatility doubles and the drift becomes $2\mu+\sigma^2$, the extra $\sigma^2$ being the convexity term; directly, $S_t^2=S_0^2e^{(2\mu-\sigma^2)t+2\sigma W_t}$.