L² extension of the predictable counting-process integral

By [F2], choose bounded predictable step processes $H_n$ with $\mathbb{E}{\int }_0^T\mathrm{\mid }{H}_n-H{\mathrm{\mid }}^{2}dA\to 0$. For any $m,n$, [F1] gives $\mathbb{E}\mathrm{\mid }{I}_T(H_n)-I_T(H_m){\mathrm{\mid }}^2=\mathbb{E}{\int }_0^T\mathrm{\mid }{H}_n-H_m{\mathrm{\mid }}^{2}dA$, so terminal integrals are Cauchy in $L^2(P)$; the same estimate on $[0,t]$ gives the Cauchy property at every $t\le T$. [F3] gives a unique full-space limit and, since every $I_t(H_n)$ is ${\mathcal{F}}_t$-measurable, also an ${\mathcal{F}}_t$-measurable limit at each fixed $t$; uniqueness makes these the same $L^2$ class.

If $K_n$ is another step sequence converging to $H$, then $\mathbb{E}{\int }_0^T\mathrm{\mid }{H}_n-K_n{\mathrm{\mid }}^{2}dA\to 0$. Applying [F1] to $H_n-K_n$ makes the terminal limits and each fixed-time limit equal in $L^2$, so the limits in [F5] are independent of approximation.

Norm continuity and [F1] give, for every $t\le T$, $\mathbb{E}\mathrm{\mid }{I}_t(H){\mathrm{\mid }}^2={\lim }_n\mathbb{E}\mathrm{\mid }{I}_t(H_n){\mathrm{\mid }}^2={\lim }_n\mathbb{E}{\int }_0^t{H}_{n,s}^{2}d{A}_s=\mathbb{E}{\int }_0^t{H}_s^{2}d{A}_s$. This proves square integrability and the isometry.

For $s\le t$, [F1] gives $\mathbb{E}[I_t(H_n)\mid {\mathcal{F}}_s]=I_s(H_n)$. By [F4], conditional expectations converge in $L^2$ to $\mathbb{E}[I_t(H)\mid {\mathcal{F}}_s]$, while $I_s(H_n)\to I_s(H)$ in $L^2$ by step 1.1; uniqueness of limits yields $\mathbb{E}[I_t(H)\mid {\mathcal{F}}_s]=I_s(H)$. Hence the extension is a square-integrable martingale, with no claim for predictable integrands of infinite compensator energy.