Ito integral is generally defined through the sums
$$S_n=\sum_{i=1}^nG(\tau_i)(W(t_i)-W(t_{i-1}))$$
then the limit $\lim_{n\rightarrow\infty}S_n$ must exist in the rms sense. This definition can be easily generalized to cases like
$$S_n=\sum_{i=1}^nG(\tau_i)(W(t_i)-W(t_{i-1}))^\alpha$$
with $\alpha\in\mathbb{R}$ and $\alpha>0$ or
$$S_n=\sum_{i=1}^nG(\tau_i)|(W(t_i)-W(t_{i-1})|$$
where different classes of Wiener processes are considered. From these one can formally define, on the lines of stochastic calculus, $[dW]^\alpha$ or $|dW|$ and the corresponding Ito integral. It is not difficult to show that, at least for some cases, these reduce to the standard form $adW+bdt$ and so are computable.
Given this, I have done the following formal manipulations for stochastic calculus. I consider the square root of a Wiener process and I put
$$dX=[dW]^\frac{1}{2}=(\mu_0+\mu_1|dW|+\mu_2dt)\left(\frac{1-i}{2}{\rm sign}(dW)+\frac{1+i}{2}\right)$$
then I do the square and I get
$$(dX)^2=\mu_0^2{\rm sign}(dW)+dW$$
rather than $(dX)^2=dW$ ($\mu_1=1/2\mu_0$, $\mu_2=-1/8\mu_0^3$): There is a residual Bernoulli process with mean 0 and variance 1 that should be eventually interpreted in the sense of a Ito integral. This can be accomplished by noting that ${\rm sign}(x)=x/|x|$ but this integral would be eventually zero as
$$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^n{\rm sign}(W(t_i)-W(t_{i-1}))=0.$$
My question is this: Is this manipulation correct? Have I got the square root of a Wiener process? If yes, what meaning should I assign to the residual Bernoulli process?
Thanks.
