Definition

Let $(S_{t})_{t\ge 0}$ be a semimartingale and let $X$ be a continuous adapted process. Then we define the Itô Stochastic Integral of $\mathbb{1}_{[0,t]}X$ w.r.t. $S$ as $$\int\limits \mathbb{1}_{[0,t]}X \, dS =\underbrace{ \int\limits \mathbb{1}_{[0,t]}X \, dM }_{ \text{Itô Stochastic Integral} }+\underbrace{ \int\limits \mathbb{1}_{[0,t]}X \, dA }_{ \text{Pathwise L-S Integral} }$$