Elementary predictable integral against a counting-process martingale

Let $M$ be a square-integrable martingale on a finite horizon $[0,T]$ in the all-pairs convention of Continuous-time adapted processes and martingales (in particular, it may be the compensated counting-process martingale of Simple counting-process intensity and compensator contract). Let $0=t_0<t_1<\cdots <t_m=T$ be a deterministic partition, and let each bounded random variable $H_j$ be ${\mathcal{F}}_{t_j}$-measurable. The associated bounded predictable step integrand is $H_0=0$ at time $0$ and $H_s=H_j$ on $(t_j,t_{j+1}]$. For this chosen step representation define, for $0\le t\le T$, $I_t(H):={\sum }_{j=0}^{m-1}{H}_j(M_{t\wedge t_{j+1}}-M_{t\wedge t_j})\mathrm{.}$ This is a finite sum of square-integrable random variables. At this stage the representation is part of the input; representation independence and the isometry are established in Isometry for elementary predictable counting-process integrals ↗. The Axiom of Choice is inherited from the all-pairs martingale and conditional-expectation interface; the finite-sum definition makes no additional choice.

The displayed finite-sum convention agrees with Tian's bounded predictable step-integral construction and Bass's construction for jump martingales. The local martingale and isometry properties are proved separately below.