+ a_{ij}(t-1,V_{t-1})V_i(t-1)^-}{\max\{\bar p_i(t,V_{t-1}) , V_i(t)^-\}}$.

Using the notation and terms above we can rewrite with respect to the cash flows $c$ and relative exposures $A$ as $$\begin{aligned}
\nonumber V(t) &= V(t-1)^+ + x(t) + \Pi(t,V_{t-1})^{\top}[\bar p(t,V_{t-1}) - V(t)^-]^+ - \bar p(t,V_{t-1})\\
\nonumber &= V(t-1)^+ + x(t) + L(t)^{\top}\vec{1} + A(t-1,V_{t-1})^{\top}V(t-1)^-\\
\nonumber &\qquad - A(t,V_t)^{\top}V(t)^- - L(t)\vec{1} -V(t-1)^-\\
\nonumber &= V(t-1) + x(t) + L(t)^{\top}\vec{1} + A(t-1,V_{t-1})^{\top}V(t-1)^- - A(t,V_t)^{\top}V(t)^- - L(t)\vec{1}\\
\label{eq:discrete-V} &= V(t-1) + c(t) - A(t,V_t)^{\top}V(t)^- + A(t-1,V_t)^{\top}V(t-1)^-.\end{aligned}$$ For the remainder of this paper we will utilize the cash flow $c$ rather than the external (incoming) cash flow $x$. That is, we will consider financial networks defined by the joint parameters $(c,L)$ as given by the state equations  and  for wealths and relative exposures.

With this setup we now wish to extend the existence and uniqueness results of [@EN01] to discrete time.

\[thm:discrete\] Let $(c,L): {\mathbb{T}}\to {\mathbb{R}}^{n+1} \times {\mathbb{R}}^{(n+1) \times (n+1)}_+$ define a dynamic financial network such that every bank has cash flow at least at the level dictated by nominal interbank liabilities, i.e., $c_i(t) \geq \sum_{j \in {\mathcal{N}}} L_{ji}(t) - \sum_{j \in {\mathcal{N}}_0} L_{ij}(t)$, and so that every bank owes to the societal node at all times $t 