Let $W$ be a standard one dimensional Brownian motion, and $X$ a continuous process adapted to $W$ such that $\int\_0^T X^2 \, ds < \infty$ almost surely for some $T > 0$.


Define for any sequence of partitions $\mathcal P\_n = \{t\_1^n, \dotsc t\_{k\_n} ^n\}$ of $[0, T]$, the elementary integral process $Y^n$ associated to $\mathcal P\_n$ by


$$Y\_t^n := \sum\_{i = 1}^{k\_n - 1} X\_{t^n\_i} (W\_{t^n\_{i+i} \wedge t} - W\_{t^n\_{i} \wedge t} \,).$$


**Question:** Is it true that for all partitions $\mathcal P\_n$ of $[0, T]$ with mesh going to $0$, we have


$$\sup\_{0 \leq t \leq T} \left\lvert\int\_0^t X\_s \, dW\_s - Y\_t^n\right\rvert \to 0$$


in probability?



Convergence in sup norm of elementary integrals to the Itô integral process
Let
\begin{equation\*}
 X^{(n)}\_t=\sum\_{i=1}^{k\_n-1} X\_{t\_i^n}1\_{(t\_i^n,t\_{i+1}^n]}(t).
\end{equation\*}
Since $X=\{X\_t,0\le t\le T\}$ is a continuous adapted process, then
\begin{gather\*}
 \lim\_{n\to\infty} \sup\_{0\le t\le T}|X^{(n)}\_t - X\_t |=0, \qquad \text{a.s.}, \\
 \lim\_{n\to\infty} \int\_{0}^{T} (X^{(n)}\_t - X\_t )^2\,\mathrm{d}t=0, \qquad \text{a.s.},
\end{gather\*}
and
\begin{equation\*}
 \lim\_{n\to\infty}\mathsf{P}\Big(\Big|\int\_{0}^{T} (X^{(n)}\_t - X\_t )^2\, \mathrm{d}t \Big| >d\Big)=0, \quad \forall d>0. \tag{1}
\end{equation\*}
Due to $(\int\_0^t (X^{n}\_s-X\_s)\mathrm{d}W\_s)^2$ is L-dominated by
\begin{equation\*}
 \langle \int\_0^{\cdot} (X^{(n)}\_s-X\_s)\mathrm{d}W\_s \rangle\_t =
 \int\_{0}^{t} (X^{(n)}\_s - X\_s )^2\,\mathrm{d}s.
\end{equation\*}
Using Lenglart's inequality(cf. S. W. He *et al., Semimartingale Theory and Stochastic Calculus*, Sci. Press and CRC(1992). **Th**9.26, pp240, or
J. Jacod, and A. N. Shiryayev, *Limit Theory for Stochastic Processes*, 2ed. Springer, 2003, **Lemma** I.3.30, pp35.), we get
\begin{align\*}
 &\mathsf{P}\Big(\sup\_{0\le t\le T}\Big[\int\_0^t (X^{(n)}\_s-X\_s) \mathrm{d}W\_s \Big]^2 \ge\epsilon \Big)\\
 &\quad\le \frac{d}{\epsilon}+\mathsf{P}\Big[\int\_{0}^{T} (X^{(n)}\_t - X\_t )^2\,\mathrm{d}t \ge d\Big]. \tag{2}
\end{align\*}
Now, by (1), letting $n\to\infty$ and $d\downarrow 0$ consecutively in (2),
we obtain immediately the following assertion,
\begin{equation\*}
 \lim\_{n\to\infty}\mathsf{P}\Big(\sup\_{0\le t\le T}\Big|\int\_0^t (X^{(n)}\_s-X\_s) \mathrm{d}W\_s \Big|\ge\epsilon \Big)=0, \qquad \forall \epsilon>0. 
\end{equation\*}


