Below is the proof of Lemma 18.5 from Rene Schilling's Brownian motion which states that an Ito process can be approximated uniformly in probability by a simple Ito process.
Now it is stated in the proof of the m-dimensional Ito formula that this Lemma remains valid for $m$-dimensional Ito processes,i.e. the processes $$X_t^j = X_0^j +\sum_{k=1}^d \int_0^t \sigma_{jk}(s)dB_s^k + \int_0^t b_j(s)ds, t\le T, j = 1,\dots, m.$$
In this case, I think the uniform convergence means $P(\sup_{t\le T} \Vert X_t^{\Pi}-X_t\Vert>\epsilon)\to 0$ as $|\Pi|\to 0$ for all $\epsilon>0, T>0$, where $\Vert \cdot \Vert$ is the usual Euclidean norm.
How can we adapt the proof of the $1$-dimensional case to the $m$-dimensional case here? I am especially concerned about the use of Chebysheve and Doob's inequality and the Ito isometry used in the inequalities.
Also, the proof for the drift $b$ uses Jensen's inequality and Markov's inequality as in the answer here : Question about Schilling's Proof in uniform approximation of Ito processes by simple Ito processes..
Is there a simple adaptation of norms here that can generalize into the multidimensional case? I would greatly appreciate some help understanding this.
