Theorem

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 $$Y_{\tau}=\int\limits \mathbb{1}_{<span class="wikilink-unresolved" title="Note not published">0,\tau</span>}X \, dM\text{ a.s.}$$ ## Proof The proof uses: - Itô Stochastic Integral on Λ is a R.C. Martingale - Predictable Stochastic Intervals - Dominated Convergence Theorem - Itô Isometry v3