Theorem (Itô Stochastic Integral on $\mathcal{E}$ is a R.C. Martingale)

Let $X\in\mathcal{E}$, let $$Y_{t}=\int\limits \mathbb{1}_{[0,t]}X \, dM$$where $M=(M_{t})_{t\ge 0}$ is a right continuous $L^{2}$-martingale. Then $(Y_{t})_{t\ge 0}$ is a right continuous $L^{2}$-martingale.

Theorem (Itô Stochastic Integral on $\Lambda$ is a R.C. Martingale)

Let $M$ be a right continuous $L^{2}$-martingale. Let $X\in\Lambda^{2}(\mathscr{P},M)$. $\forall t\ge 0$ let $$Y_{t}=\int\limits \mathbb{1}_{[0,t]}X \, dM$$Then $(Y_{t})_{t\ge 0}$ is a $L^{2}$-martingale and hence admits a right continuous version.

Theorem (Independence of Bounded RV)

Let $X\in\Lambda^{2}(\mathscr{P},M)$, let $0\le s<t$, let $Z$ be $\mathcal{F}_{s}$-measurable and bounded. Then:

1. $\mathbb{1}_{(s,t]}Z$ is $\mathscr{P}$-measurable (i.e. predictable)
2. $\mathbb{1}_{(s,t]}ZX\in L^{2}(\mathbb{R}^{+}\times\Omega,\mathscr{P},\mu_{M})$
3. $$\int\limits \mathbb{1}_{(s,t]}ZX \, dM =Z\int\limits \mathbb{1}_{(s,t]}X \, dM \text{ a.s.}$$

Theorem (Stopping Time Integral)

Let $M$ be a right continuous $L^{2}$-martingale, let $X\in\Lambda^{2}(\mathscr{P},M)$, and let $(Y_{t})_{t\ge 0}$ be the stochastic integral process $$Y_{t}=\int\limits \mathbb{1}_{[0,t]}X \, dM$$Let $\tau$ be a bounded stopping time. Then $$Y_{\tau}=\int\limits \mathbb{1}_{<span class="wikilink-unresolved" title="Note not published">0,\tau</span>}X \, dM\text{ a.s.}$$