where ${{\mathbf{Q}}}_t := Z_t \cdot {{\mathbf{P}}}$ to a measure ${{\mathbf{Q}}}$ which is locally absolutely continuous w.r.t. ${{\mathbf{P}}}$ with density process $Z$. An application of Girsanov’s theorem then yields the existence of a solution to a modified SMP. In view of this observation a contradiction argument yields a sufficient condition for strict local martingality. In the following we formalize this idea.

\[main theorem new 2\] Assume that $(\Omega, \mathcal{F}, {\mathfrak{F}})$ is full. If $Z$ is an $({\mathfrak{F}}, {{\mathbf{P}}})$-martingale, there exists a solution ${{\mathbf{Q}}}$ to the SMP associated with $(\mathbb{H}; A; \eta; X; \dot{B}(h), C, \dot{\nu})$, where on $A$ $$\label{candidtate triplet main theorem}
\begin{split}
\dot{B}(h) = B(h) + &1/Z_- {\mathbf{1}}_{\{Z_- > 0\}}\cdot {\langle\hspace{-0.085cm}\langle}Z, M(h){\rangle\hspace{-0.085cm}\rangle}^{{{\mathbf{P}}}}\textit{ and } \dot{\nu} = Y \cdot \nu,\\ \textit{with } Y &= 1/Z_-{\mathbf{1}}_{\{Z_- > 0\}} M^{{\mathbf{P}}}_{\mu^X} (Z |\widetilde{\mathcal{P}}(\mathfrak{F})),\end{split}$$ for the notation c.f. Appendix \[Girsanov’s Theorem for Hilbert-Space-valued Semimartingales\]. Additionally we have ${{\mathbf{Q}}}\ll_\textup{loc} {{\mathbf{P}}}$ with density process $Z$.

Thanks to the assumption that $(\Omega, \mathcal{F}, {\mathfrak{F}})$ is full, there exists a probability measure ${{\mathbf{Q}}}$ such that for all $t \geq 0$ we have ${{\mathbf{Q}}}= {{\mathbf{Q}}}_t = Z_t \cdot {{\mathbf{P}}}$ on $\mathcal{F}_t$. Since for all $t \geq 0$ it holds that ${{\mathbf{Q}}}_t \ll {{\mathbf{P}}}$ on $\mathcal{F}_t$, we conclude that ${{\mathbf{Q}}}\ll_{\textup{loc}} {{\mathbf{P}}}$ 