the SMP associated with $(\mathbb{H}; \rho; \eta; X; B(h), C, \nu)$ satisfies *uniqueness*, if all solutions coincide on $\mathcal{F}_{\rho-}$.

2.  Let $\Lambda$ be a set of ${\mathfrak{F}}$-stopping times such that for all $\rho\in \Lambda$ we have ${[\hspace{-0.06cm}[}0, \rho{]\hspace{-0.06cm}]}\subseteq A$. We say that the SMP associated with $(\mathbb{H}; A; \eta; X; B(h), C, \nu)$ satisfies *$\Lambda$-uniqueness* if for all $\rho \in \Lambda$ the SMP associated with $(\mathbb{H}; \rho; \eta; X; B(h), C, \nu)$ satisfies uniqueness.

To hope for uniqueness we usually have to assume that $(\mathcal{F}, {\mathfrak{F}})$ is *generated by the candidate process $X$*, i.e. that $$\begin{aligned}
\mathcal{F} = \sigma(X_t, t \geq 0),\ \mathcal{F}^{\hspace{0.03cm}0}_t = \sigma(X_s, s\leq t)\textup{ and } \mathcal{F}_t = \bigcap_{s > t} \mathcal{F}^{\hspace{0.03cm}0}_s.\end{aligned}$$ The choice of the $\sigma$-field $\mathcal{F}_{\rho-}$ in the definition of uniqueness has its origin in classical Markovian settings. More precisely, standard techniques show that all solutions coincide on $\mathcal{F}^{\hspace{0.03cm}0}_{\rho}$ instead of $\mathcal{F}_\rho$. However, for a large class of stopping times $\rho$ it holds that $\mathcal{F}_{\rho-} \subseteq \mathcal{F}^{\hspace{0.03cm}0}_\rho$.

In a finite dimensional Markovian setting a related concept of uniqueness is well-studied, c.f. [@J79], Section 13.3. In a non-Markovian setting we derive explicit conditions implying uniqueness in Appendix \[SMPs and SDEs\]. The following remark is an immediate 