I am currently working on the understanding of the stochastic nature of the Schroedinger equation. This has a notable history dating back to Nelson's works and relative criticisms. But one can take a different path and, starting from a random walk process with probability
\begin{equation}
P(k;N) = \binom{N}{k}\left(\frac{1}{2}\right)^N,
\end{equation}
one can assume the existence of a "square root" of this process with a complex amplitude
\begin{equation}
A(k;N) = \binom{N}{k}^\frac{1}{2}\left(\frac{1}{2}\right)^{N/2}e^{i\phi(k,N)}
\end{equation}
such that $P(k;N)=|A(k;N)|^2$ and $\phi(k,N)$ are some phases exactly determined. These are quantum amplitudes. Is this anything making sense? Does any literature exist about?
Best Answer
Note: This mostly debunks an answer posted by the OP and now deleted. As a consequence of this unfortunate deletion, the argument below might be a little difficult to follow.
As explained on the MSE page you are referring to, very basic arguments show that the limits of the sums $S_n$ you write in your "answer" cannot exist.
Consider for example the second version of $S_n$ (the one based on absolute values) in the simplest possible situation, that is, on the interval $[0,1]$ and when $G\equiv1$. Note that the Gaussian random variables $W(t_i)-W(t_{i-1})$ are independent with variances $t_i-t_{i-1}$. Hence, for every $i$, $$ E((W(t_i)-W(t_{i-1}))^2)=t_i-t_{i-1}, $$ and, for every $i\ne k$, $$ E(|W(t_i)-W(t_{i-1})|\cdot|W(t_k)-W(t_{k-1})|)=c\cdot\sqrt{t_i-t_{i-1}}\cdot\sqrt{t_k-t_{k-1}}, $$ where it happens that $c=2/\pi$ but this is irrelevant. Again in the simplest case, that is, when $t_i=i/n$ for every $i$, this shows that $$ E(S_n^2)=1+c\cdot(n-1). $$ Hence, in contradiction with what you assert, the sequence $(S_n)_{n\geqslant1}$ diverges in $L^2$.
To sum up, integrals such as $$ \int G\cdot|\mathrm dW|\qquad\text{or}\quad\int G\cdot(\mathrm dW)^{\alpha}, $$ whatever their precise definitions would be, cannot exist due to the (ir)regularity properties of the Brownian paths (except, of course, the second one when $\alpha=1$).