L² counting-process integral when the limit exists

For the compensator contract on $[0,T]$, let $H$ be a predictable real process with $\mathbb{E}{\int }_0^T{H}_s^{2}d{A}_s<\mathrm{\infty }\mathrm{.}$ When the elementary integrals of a sequence of bounded predictable step processes converging to $H$ in this norm converge in $L^2(P)$, denote the resulting $L^2(P)$ class at time $t$ by ${\int }_0^t{H}_{s}d{M}_s\mathrm{.}$ The definition is conditional on existence of the limit. The L²-extension theorem below proves existence for every such predictable $H$, independence from the approximating sequence, the martingale property, and the isometry. The Axiom of Choice is inherited through the conditional-expectation and $L^2$-completion interfaces used by that theorem.

Bass describes the construction of the jump-martingale stochastic integral as the $L^2$ completion of elementary predictable integrals. This local definition states the domain through the actual compensator $A$ and leaves the existence claim to the following theorem.