Let
- $(\Omega,\mathcal A,\operatorname P)$ be a probability space
- $(\mathcal F_t)_{t\ge0}$ be a filtration of $\mathcal A$
- $\tau_n:\Omega\to[0,\infty]$ be an $\mathcal F$-stopping time with $$\tau_n\le\tau_{n+1}\tag1$$ for all $n\in\mathbb N_0$ and $$\tau_n\xrightarrow{n\to\infty}\infty\tag2$$
- $X_n:\Omega\to\mathbb R$ be $\mathcal F_{\tau_{n-1}}$-measurable for all $n\in\mathbb N$
- $W$ be a continuous square-integrable $\mathcal F$-martingale on $(\Omega,\mathcal A,\operatorname P)$
- $\mathcal M^2$ be the set of $\operatorname P$-almost surely continuous, $L^2(\operatorname P)$-bounded, $\mathcal F$-martingales $X:\Omega\times[0,\infty)\to\mathbb R$ with $\operatorname P[X_0=0]=1$ equipped with $$\left\|X\right\|_{\mathcal M^2}:=\left\|X_\infty\right\|_{L^2(\operatorname P)}\;\;\;\text{for }X\in\mathcal M^2$$
We can show that $$Z^N:=\sum_{n=1}^NX_n\left(W-W^{\tau_{n-1}}\right)$$ is a continuous $\mathcal F$-martingale for all $N\in\mathbb N$. How can we conclude that $$Z:=\sum_{n\in\mathbb N}X_n\left(W-W^{\tau_{n-1}}\right)$$ (which exists as a pointwise limit) is an $\mathcal F$-martingale too and that $(Z^N)_{N\in\mathbb N}$ converges to $Z$ in $\mathcal M^2$?