a semimartingale with known characteristics. This observation leads us to the concept of semimartingale problems.

Hilbert-space-valued Semimartingales and\
Semimartingale Problems on Stochastic Intervals {#Hilbert-Vlaued Semimartingales and Semimartingale Problems on Stochastic Intervals}
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Throughout this section we fix a filtered probability space $(\Omega, \mathcal{F}, {\mathfrak{F}}, {{\mathbf{P}}})$, where ${\mathfrak{F}}= (\mathcal{F}_t)_{t \geq 0}$ is a right-continuous filtration. For the first part of Section \[Processes on Stochastic Intervals\] the probability measure ${{\mathbf{P}}}$ does not play a  role.

Processes on Stochastic Intervals {#Processes on Stochastic Intervals}
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In this first section we introduce the concept of stochastic processes on stochastic intervals, which build the mathematical background to study possibly explosive processes. We partially follow [@HWY; @J79; @JS]. Related topics were also studied by [@emery1982; @Schwartz1981; @sharpe]. We fix a Polish space $E$, which serves as the *state space*. Recall that for two ${\mathfrak{F}}$-stopping times $\tau$ and $\rho$ we set $$\begin{aligned}
{[\hspace{-0.06cm}[}\tau, \rho{]\hspace{-0.06cm}]}:= \big\{(\omega, t) \in \Omega \times \mathbb{R}^+ : \tau(\omega) \leq t \leq \rho(\omega)\big\},\end{aligned}$$ and ${[\hspace{-0.06cm}[}\tau, \rho{[\hspace{-0.06cm}[}, {]\hspace{-0.06cm}]}\tau, \rho{]\hspace{-0.06cm}]}, {]\hspace{-0.06cm}]}\tau, \rho{[\hspace{-0.06cm}[}$ analogeously. Moreover, we denote ${[\hspace{-0.06cm}[}\tau, \tau{]\hspace{-0.06cm}]}=: {[\hspace{-0.06cm}[}\tau{]\hspace{-0.06cm}]}$.

We call a random set $A$ a *set of interval type*, if there exists an increasing sequence of ${\mathfrak{F}}$-stopping times $(\tau_n)_{n \in \mathbb{N}}$ such that $$\begin{aligned}
A 