B(h), C, \nu)$ if and only  if

1.  $X|_A$ is ${{\mathbf{P}}}$-indistinguishable from a process in $\mathcal{S}^A(\mathbb{H},{\mathfrak{F}}, {{\mathbf{P}}})$,

2.  $(B(h), C, \nu)|_A$ is ${{\mathbf{P}}}$-indistinguishable from the $({\mathfrak{F}}, {{\mathbf{P}}},A)$-characteristics of $X|_A$,

3.  ${{\mathbf{P}}}\circ X^{-1}_0 = \eta$.

\[SMP subinterval\] ${{\mathbf{P}}}$ is a solution to the SMP associated with $(\mathbb{H}; A; \eta; X; B(h), C, \nu)$ if and only if for all ${\mathfrak{F}}$-stopping times $\rho$ such that ${[\hspace{-0.06cm}[}0, \rho{]\hspace{-0.06cm}]}\subseteq A$, ${{\mathbf{P}}}$ is a solution to the SMP associated with $(\mathbb{H}; \rho; \eta; X; B(h), C, \nu)$.

The implication $\Longleftarrow$ is trivial. The implication $\Longrightarrow$ follows from Corollary \[ordinary semimartingale\] and Proposition \[exist A c\] in Appendix \[facts HSVSM SI\].

As an immediate consequence we obtain the following corollary.

\[coro subinterval\] If ${{\mathbf{P}}}$ is a solution to the SMP associated with $(\mathbb{H}; A; \eta; X; B(h), C, \nu)$, and $\mathcal{I}({\mathfrak{F}})\ni A^* \subseteq A$, then ${{\mathbf{P}}}$ is also a solution to the SMP associated with $(\mathbb{H}; A^*; \eta; X;$ $B^*(h),$ $C^*, \nu^*)$ for any candidate triplet $(B^*(h), C^*, \nu^*)$ which coincides on $A^*$ with $(B(h), C, \nu)$.

Our next goal is to introduce a concept of uniqueness which is in the spirit of the *local uniqueness* concept introduced by [@J79], Section 12.3.e).

1.  We say that 