\in {\mathbb{T}}$, i.e., $L_{i0}(t) > 0$ for all banks $i \in {\mathcal{N}}$ and times $t \in {\mathbb{T}}$. Under Assumption \[ass:initial\], there exists a unique solution of clearing wealths $V: {\mathbb{T}}\to {\mathbb{R}}^{n+1}$ to .

\[rem:regularnetwork\] The assumption that all firms have obligations to the societal node $0$ at all times $t \in {\mathbb{T}}$ guarantees that the financial system is a “regular network” (see [@EN01 Definition 5]) at all times.

The analysis of the discrete-time framework can be extended to a probabilistic setting over the filtered probability space $(\Omega,{\mathcal{F}},({\mathcal{F}}(t))_{t \in {\mathbb{T}}},{\mathbb{P}})$. That is, we can consider the clearing wealths in the same manner assuming the cash flow $c: {\mathbb{T}}\times \Omega \to {\mathbb{R}}^{n+1}$ and nominal liabilities $L: {\mathbb{T}}\times \Omega \to {\mathbb{R}}^{(n+1) \times (n+1)}_+$ be adapted processes. Let ${\mathcal{L}}_t^0({\mathbb{R}}^m)$ be the space of ${\mathcal{F}}_t$-measurable random vectors in ${\mathbb{R}}^m$. Let ${\mathcal{L}}_t^p({\mathbb{R}}^m) \subseteq {\mathcal{L}}_t^0({\mathbb{R}}^m)$ for $p \in (0,\infty]$ be the space of equivalence classes of ${\mathcal{F}}_t$-measurable functions $X: \Omega \to {\mathbb{R}}^m$ such that $\|X\|_p := \left(\int_\Omega \sqrt{\sum_{k = 1}^m X_k(\omega)^2} d{\mathbb{P}}\right)^{1/p} < \infty$ for $p < \infty$ and $\|X\|_{\infty} := \operatorname*{ess\,sup}_{\omega \in \Omega} \sqrt{\sum_{k = 1}^m X_k(\omega)^2}$ for $p = \infty$. The following corollary considers the boundedness and measurability properties of the discrete-time clearing wealths. Though 