\mathcal{M}^A_\textup{loc}(\mathbb{R}, {\mathfrak{F}}, {{\mathbf{P}}})$, non-negativity implies $\widetilde{X} \in \mathcal{M}_\textup{loc}(\mathbb{R}, {\mathfrak{F}}, {{\mathbf{P}}})$, c.f. Proposition \[lemma tilde Z local martingale\] in Appendix \[Extensions of non-negative local martingales on sets of interval type\].

Characteristics of Hilbert-space-valued Semimartingales on Sets of Interval Type
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In what follows it is of little additional cost to proceed directly with Hilbert-space-valued processes. We are following closely the monographs [@MP80; @metivier]. For the definition Hilbert-space-valued semimartingales, which can be done parallel to the finite-dimensional case, we refer to Appendix \[A HVS\]. Let $\mathbb{H}$ be a real separable Hilbert space and denote by $\mathcal{N}(\mathbb{H}, \mathbb{H})$ the space of nuclear operators from $\mathbb{H}$ into $\mathbb{H}$.[^4] Let $A \in \mathcal{I}({\mathfrak{F}})$.

\[def A-c\] Let $X\in \mathcal{S}^A(\mathbb{H}, {\mathfrak{F}}, {{\mathbf{P}}})$ and assume that $(B(h), C, \nu)$ is a triplet consisting of the following:

1.  an ${\mathfrak{F}}$-predictable $\mathbb{H}$-valued process $B(h)$ on $A$,

2.  an ${\mathfrak{F}}$-predictable $\mathcal{N}(\mathbb{H}, \mathbb{H})$-valued process $C$ on $A$,

3.  an ${\mathfrak{F}}$-predictable random measure $\nu$ on $\mathbb{R}^+ \times \mathbb{H}$ such that $$\begin{aligned}
    \label{nu dec}
    \nu(\omega, {\operatorname{d}\hspace{-0.003cm}}t, {\operatorname{d}\hspace{-0.003cm}}x) = {\mathbf{1}}_{A \times \mathbb{H}} (\omega, t, x) \nu(\omega, {\operatorname{d}\hspace{-0.003cm}}t, {\operatorname{d}\hspace{-0.003cm}}x).\end{aligned}$$

If there exists an announcing sequence $(\tau_n)_{n \in \mathbb{N}}$ for $A$ such that $(B(h)^{\tau_n}, C^{\tau_n}, \nu^{\tau_n})$, where $$\begin{aligned}
\nu^{\tau_n} (\omega, {\operatorname{d}\hspace{-0.003cm}}t, {\operatorname{d}\hspace{-0.003cm}}x) := {\mathbf{1}}_{{[\hspace{-0.06cm}[}0, \tau_n {]\hspace{-0.06cm}]}\times \mathbb{H}}(\omega, 