with density process $Z$. Denote by $(\tau_n)_{n \in \mathbb{N}}$ an announcing sequence for $A$. Then $X^{\tau_n}$ is an $({\mathfrak{F}}, {{\mathbf{P}}})$-semimartingale with characteristics $(B(h)^{\tau_n}, C^{\tau_n}, \nu^{\tau_n})$, since ${{\mathbf{P}}}$ solves the SMP associated with $(\mathbb{H}; A; \eta; X; B(h), C, \nu)$. Proposition \[pred finie var 0\] in Appendix \[A HVS\] yields that the unique decomposition of $X(h)^{\tau_n}$ is given by $X(h)^{\tau_n} =X_0+ M(h)^{\tau_n} + B(h)^{\tau_n}$. Moreover, the identity ${[\hspace{-0.085cm}[}Z, M(h)^{\tau_n}{]\hspace{-0.085cm}]}^{{\mathbf{P}}}= {[\hspace{-0.085cm}[}Z, M(h){]\hspace{-0.085cm}]}^{{\mathbf{P}}}_{\cdot \wedge \tau_n}$ and again Proposition \[pred finie var 0\] in Appendix \[A HVS\] imply that $$\begin{aligned}
{\langle\hspace{-0.085cm}\langle}Z, M(h)^{\tau_n}{\rangle\hspace{-0.085cm}\rangle}^{{\mathbf{P}}}= {\langle\hspace{-0.085cm}\langle}Z, M(h){\rangle\hspace{-0.085cm}\rangle}^{{{\mathbf{P}}}}_{\cdot \wedge \tau_n},
$$ where ${\langle\hspace{-0.085cm}\langle}Z, M(h)^{\tau_n}{\rangle\hspace{-0.085cm}\rangle}^{{\mathbf{P}}}$ is the unique [càdlàg ]{}${\mathfrak{F}}$-predictable process of finite variation such that the process ${[\hspace{-0.085cm}[}Z, M(h)^{\tau_n}{]\hspace{-0.085cm}]}^{{\mathbf{P}}}- {\langle\hspace{-0.085cm}\langle}Z, M(h)^{\tau_n}{\rangle\hspace{-0.085cm}\rangle}^{{\mathbf{P}}}$ is a local $({\mathfrak{F}}, {{\mathbf{P}}})$-martingale, c.f. Theorem \[GT\] in Appendix  \[Girsanov’s Theorem for Hilbert-Space-valued Semimartingales\]. Furthermore, Proposition II.1.30 in [@JS] yields that $$\begin{aligned}
M^{{\mathbf{P}}}_{\mu^{X^{\tau_n}}}(Z|\widetilde{\mathcal{P}}({\mathfrak{F}})) = {\mathbf{1}}_{{[\hspace{-0.06cm}[}0, \tau_n{]\hspace{-0.06cm}]}} M^{{\mathbf{P}}}_{\mu^X}(Z|\widetilde{\mathcal{P}}({\mathfrak{F}})).\end{aligned}$$ Therefore, the Girsanov-type theorem given by Theorem \[GT\] in Appendix \[Girsanov’s Theorem for Hilbert-Space-valued Semimartingales\] yields that $X^{\tau_n}$ is also an $({\mathfrak{F}}, {{\mathbf{Q}}})$-semimartingale with $({\mathfrak{F}}, {{\mathbf{Q}}})$-characteristics $(\dot{B}(h)^{\tau_n}, C^{\tau_n}, \dot{\nu}^{\tau_n})$. Now the identity ${{\mathbf{Q}}}\circ X_0^{-1} = {{\mathbf{Q}}}_0 \circ X_0^{-1} = {{\mathbf{P}}}\circ X_0^{-1} = \eta$, as ${{\mathbf{P}}}$-a.s. $Z_0 = 1$, yields that ${{\mathbf{Q}}}$ is a solution to the SMP associated with 