$(\mathbb{H}; \tau_n; \eta; X; \dot{B}(h), C, \dot{\nu})$. Since this holds for all $n \in \mathbb{N}$, the claim is proven.

As a corollary we obtain the following sufficient condition for strict local  martingality.

\[coro strict explosion\] Let $(\Omega,\mathcal{F}, {\mathfrak{F}})$ be full. If the SMP associated with $(\mathbb{H}; A; \eta; X; \dot{B}(h), C, \dot{\nu})$, where $\dot{B}(h)$ and $\dot{\nu}$ are given by on $A$, has no solution, then the process $Z$ is a strict local $({\mathfrak{F}}, {{\mathbf{P}}})$-martingale.

In the one-dimensional diffusion setting the *Feller test for explosion*, c.f. [@Feller52] or [@KaraShre], Section 5.5.C, provides a technique to apply Corollary \[coro strict explosion\]. A comprehensive discussion is given by [@MU(2012)].

### The Martingale Property under Uniqueness Assumptions {#main section}

In this section we impose existence and uniqueness assumptions on the modified SMP and derive sufficient conditions for the martingality of $Z$. We denote by $(\tau_n)_{n \in \mathbb{N}}$ an arbitrary announcing sequence for $A$ and set $\bar{A} := \bigcup_{n \in \mathbb{N}} {[\hspace{-0.06cm}[}0, \tau_n \wedge \rho_n{]\hspace{-0.06cm}]}\in \mathcal{I}({\mathfrak{F}})$ and $\tau := \lim_{n \to \infty} \tau_n$.

\[cond main uni\] We define the following conditions:

1.  The SMP associated with $(\mathbb{H}; \bar{A}; \eta; X;\dot{B}(h), C, \dot{\nu})$, where $\dot{B}(h)$ and $\dot{\nu}$ are given by on $\bar{A}$, has a solution ${{\mathbf{Q}}}$ 