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 as an Ito integral? $$ S=\lim_{\Delta \to 0} \frac{1}{\Delta} \sum_{i=0}^{n-1}(W_{(i+1)\Delta}-W_{i\Delta})^{2}⋅(B_{(i+1)\Delta}-B_{i\Delta}) $$
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 "wierd" 1.73 .