\sigma(t,d)\|_1^{op} &\leq D\|c - d\|_1\end{aligned}$$ where $\|\cdot\|_1$ is the 1-norm and $\|\cdot\|_1^{op}$ is the corresponding operator norm. The nominal liabilities $L: {\mathbb{T}}\to {\mathbb{R}}^{(n+1) \times (n+1)}_+$ are deterministic and twice differentiable; for notation we will define $dL(t) = \dot{L}(t) dt$ and $d^2L(t) = \ddot{L}(t) dt^2$. Further, the relative liabilities to society is bounded from below by a level $\delta > 0$, i.e., $\inf_{t \in {\mathbb{T}}} \frac{dL_{i0}(t)}{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(t)} = \delta > 0$ for all banks $i \in {\mathcal{N}}$.

We remark that the assumption on the cash flows can be relaxed so long as the stochastic differential equation has a unique strong solution on ${\mathbb{T}}$ and $\mu,\sigma$ satisfy a local linear growth condition and are locally Lipschitz. This relaxation will be applied in Examples \[ex:differentialEN\] and \[ex:dc-dependence\].

In the prior section on a discrete-time model for clearing wealths, we implicitly assumed a constant time-step between each clearing date of ${{\Delta t}}= 1$ throughout. In order to construct a continuous-time clearing model we will begin by making a discrete-time model with an explicit ${{\Delta t}}>0$ term. In fact, this is immediate from the prior construction with a minor alteration to the cash flow term. Herein we construct the net cash flow at 