I'm looking at a solution of the following question from Oksendal's book:
"Prove directly from the definition of Ito integrals that $ \int_0^t B_s^2 dB_s = \frac{1}{3} B_t^3 - \int_0^t B_s ds$"
where I believe it means the LHS is defined through the a.s. limit (in this section they use a.s. convergence to define the Ito integral).
The solution breaks up the LHS (prior to taking limit) into three terms, and for the third, it says "by looking at a subsequence, we only need to prove the L^2-convergence." Why exactly is this? I thought maybe it had to do with the result that convergence i.p. is equivalent to each subsequence having a further subsequence converging a.s., and here it is in L^2 so that also works. But I don't see how that is applied here, since they do not begin with an arbitrary subsequence. Also, it seems as if it is true then it is saying L^2 convergence is equivalent to a.s., since I can always apply this "subsequence" argument (which is obviously not true).
Thank you!
If it helps, the third term is
$ \sum_{j=1}^n B_{(j-1)/n}(B_{j/n}-B_{(j-1)/n})^2 - \sum_{j=1}^n B_{(j-1)/n} (\frac{j}{n} - \frac{j-1}{n})$ and the goal is to show it $\rightarrow 0$ a.s..
Edit: As pointed out, I was mistaken and the book does not define Ito integrals in the a.s. sense; they are defined in $L^2$. But then why does the solution prove $L^2$ convergence of the subsequence? Maybe there is an issue in the solution?