Consider two independent 1-dimensional Brownian motions $W_{t},B_{t}$, with an equidistant partition of the interval $[0,T]$, and $n\Delta≡T$.

How to calculate the expression below? Can we rewrite the limit of the summation when $\Delta$ goes to zero as an Ito integral? $$ S=\lim_{\Delta\to 0} \sum_{i=0}^{n-1}(\frac{1}{\Delta})⋅(W_{(i+1)\Delta}-W_{i\Delta})^{2}⋅(B_{(i+1)\Delta}-B_{i\Delta}) $$

NOTE:

I have done many numerical experiments by Monte-Carlo simulation, I find on interval $[0,1]$ the summation converge to a random variable (I guess it is normal) with mean 0 and standard deviation 1.73 (approximately) by the function "normfit()" in MATLAB.

However, I can not guess the final expression of $S$ because of the "weird" 1.73.