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I am trying to construct Ito integral for adapted process. However, I am stuck at some point.

Let $X^n(t)$ be a sequence of simple processes convergent in probability to the process $X(t)$. Then the sequence of their integrals $\int_0^TX^n(t)dB(t)$ also converges in probability to a limit $J$. the random variable $J$ is taken to be the integral $\int_0^TX(t)dB(t)$.

And here an example : $\int_0^tB(t)dB(t)$.

The Ito integral is defined in the following relation $$\int_0^TX^n(t)dB(t)= \sum_{i=0}^{n-1}B(t^n_i)(B(t^n_{i+1})-B(t^n_i))$$ then we have to show that this sequence is converges in probability to J= $1/2B^2(T)-1/2T$.

The question is how to show this (converges in probability to $1/2B^2(T)-1/2T)$?! your help please !

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