distress.

Continuous-time clearing systems {#sec:continuous}
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Consider now a continuous set of clearing times ${\mathbb{T}}$, e.g., ${\mathbb{T}}= [0,T]$ for some (finite) terminal time $T < \infty$ or ${\mathbb{T}}= {\mathbb{R}}_+$. As before, for processes we will use the notation from [@cont2013ito] such that the process $Z: {\mathbb{T}}\to {\mathbb{R}}^n$ has value of $Z(t)$ at time $t \in {\mathbb{T}}$ and history $Z_t := (Z(s))_{s \in [0,t]}$. We will now construct an extension of the continuous-time setting of [@sonin2017] in that we allow for liabilities to change over time and for firms to have stochastic cash flows.

In order to construct a continuous-time model we will begin by considering our network parameters of cash flows and nominal liabilities. Instead of considering $c(t)$ to be the net cash flow at time $t \in {\mathbb{T}}$, we will consider the term $dc(t)$ of marginal change in cash flow at time $t$. Similarly we will consider $dL(t)$ to be the marginal change in nominal liabilities matrix at time $t$; we note that by assumption $dL_{ij}(t) \geq 0$ for all firms $i,j \in {\mathcal{N}}_0$ as, without any payments made, total liabilities should accumulate over time. Our main result in this section (Theorem \[thm:continuous\]) provides existence and uniqueness of the 