Definition

Let $X\in\mathcal{E}$ and let $(M_{t})_{t\ge 0}$ be a right continuous $L^{2}$-martingale. We define $$\int\limits X \, dM =\sum_{i=1}^{n}a_{i}\mathbb{1}_{F_{i}}(M_{t_{i}}-M_{s_{i}})$$to be the Itô Stochastic Integral.

Properties

1. The integral is $\mathbb{R}$-linear $$\int\limits \left( \sum_{i=0}^{n}\mathbb{1}_{(s_{i},t_{i}]\times F_{i}} \right) \, dM=\sum_{i=0}^{n}\int\limits \mathbb{1}_{(s_{i},t_{i}]\times F_{i}} \, dM$$ ## Note
2. $\int\limits X \, dM$ is well-defined i.e. it is independent of whatever chosen representation we have for $X$ as a linear combination of indicator functions. In other words, we can define $X$ as two different sums that add to the same output a.s. but our integral regardless, will evaluate to the same output.
   1. Perhaps this helps to preserve Càdlàg versions of processes and their intervals?
3. $\mathcal{E}$ the set of simple processes is a vector subspace of $L^{2}(\mathbb{R}^{2}\times\Omega,\mathscr{P},\mu_{M})$
4. It follows from the definition that $$\int\limits (\cdot) \, dM :\mathcal{E}\to L^{2}(\Omega,\mathcal{F},P)$$ is $\mathbb{R}$-linear.