The difference of two martingales is still a martingale by linearity of conditional expectation and so

$$Z^n_{t}:=\int_{0}^{t}X_{s}dW_{s}-\sum_{i = 1}^{k_n - 1} X_{t^n_i} (W_{t^n_{i+i} \wedge t} - W_{t^n_{i} \wedge t} \,)$$

is a martingale for each $n$. Indeed, the first term is a martingale by the properties of stochastic integration, while the latter is a martingale transform of a martingale.

Thus we can apply Doob's inequality for $p=2$:

$$\text{E}[| Z^n_T |^p] \leq \text{E}\left[\sup_{0 \leq s \leq T} |Z^n_s|^p\right] \leq \left(\frac{p}{p-1}\right)^p\text{E}[|Z^n_T|^p]$$

to get $L^{2}$-convergence of the supremum as well. For probability-convergence we can use the weaker version

$$\text{P}[\sup_t|Z_t^{n}|\geq C]\leq \frac{\text{E}[|Z_T^{n}|^p]}{C^p}.$$

In terms of the $L_{2}$ convergence one needs a bit more structure. There are many references eg. see here Theorem 5.3.

Now we verify the assumption for the L2 convergence of the elementary processes

$$\phi_{n}(t):=\sum_{i = 1}^{k_n - 1} X_{t^n_i} 1_{[t^n_i,t^n_{i+1}]}$$

to $X_{t}$. For reference of the last one, also see Oksendal SDEs on pags 26-28. When $X_{t}$ is bounded one can use the above simple-functions.

and when $X_{t}$ is unbounded, one can use the following approximation from his step 3

$$\psi_{n}(t):=1_{X_{t}\in [-n,n]}\sum_{i = 1}^{k_n - 1} X_{t^n_i} 1_{[t^n_i,t^n_{i+1}]}-n1_{X_{t}\leq -n}+n1_{X_{t}\geq n}.$$

So if one wants to just have $\phi_{n}$ as the approximation, they would need the following terms to go to zero

$$\int_{0}^{T} E[|\phi_{n}(t)-n|^{2}1_{X(t)>n}]+E[|\phi_{n}(t)+n|^{2}1_{X(t)<-n}]dt.$$

I will try to check if this can be obtained given the above particular assumptions on $X$ or whether there are counterexamples. Other answers are welcome.

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