More on the Existence and Uniqueness of the solutions of an SDE Proof

Question

An extract from the proof of the existence and uniqueness of the solution of a SDE from Oksendal. I cannot see how holders inequality and the ito isometry are applied.

Answer 1

The estimate by Fatou's lemma says that

$$ Y^{(n)}_t(\omega) \rightarrow X_t(\omega) $$

in $L^2(\Omega \times [0, T])$.

Since $\sigma(t,x)$ is Lipschitz in $x$,

$$ \sigma(t, Y^{(n)}_t(\omega)) \rightarrow \sigma(t, X_t(\omega)) $$

in $L^2(\Omega \times [0, T])$ also. So Ito isometry tells you that

$$ \int_0 ^t \sigma(s, Y^{(n)}_s(\omega)) dB_s \rightarrow \int_0 ^t \sigma(s, X_s(\omega))dB_s $$

in $L^2(\Omega)$.

The second claim about the path-wise Lebesgue integral is similar: it follows from Holder inequality that on a finite measure space ($\Omega \times [0, T]$ in this case), $L^1$-norm is bounded by $L^2$-norm. This shows convergence in $L^1(\Omega)$.