Elementary Stochastic Integral of a Simple Adapted Process

Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},P)$ be a filtered probability space, let $(M,\rho)$ be an It^{o} integrator of intensity type with respect to $(\mathcal{F}_t)_{t\ge0}$ (the intensity $\rho$ plays no role in this definition), let $T>0$ be real, and let $H$ be a simple adapted process on $(0,T]$ with representation $\bigl((t_i)_{i=0}^{n},(\xi_i)_{i=0}^{n-1}\bigr)$.

The elementary stochastic integral of $H$ with respect to $M$ over $(0,T]$ is the random variable

This value does not depend on the chosen representation, so the integral is well defined. Indeed, inserting one additional partition point $u\in(t_i,t_{i+1})$ replaces the term $\xi_i(M_{t_{i+1}}-M_{t_i})$ by $\xi_i(M_u-M_{t_i})+\xi_i(M_{t_{i+1}}-M_u)$, which is the same random variable; the refined data is again a representation in the sense of Simple Adapted Process, since $\xi_i$ is $\mathcal{F}_u$-measurable by the filtration inclusion $\mathcal{F}_{t_i}\subseteq\mathcal{F}_u$. Iterating, the sum for any representation equals the sum for the common refinement of two given representations (the partition whose point set is the union of the two point sets). On each refinement interval $(u_j,u_{j+1}]$, the coefficient obtained from either representation equals the common value $H_{u_{j+1}}$ of $H$ on that interval, so the two refined sums coincide term by term.