---
author:
- 'Tathagata Banerjee[^1]'
- 'Alex Bernstein[^2]'
- 'Zachary Feinstein[^3]'
bibliography:
- 'bibtex2.bib'
title: Dynamic clearing and contagion in financial networks
---

Introduction {#sec:intro}
============

Financial networks and the contagion of bank failures have been widely studied beginning with the seminal work on financial payment networks by Eisenberg & Noe [@EN01]. The 2007-2009 financial crisis and credit crunch showed the severe impacts that systemic crises can have on the financial sector and the economy as a whole. As the costs of such cascading events is tremendous, the modeling of such events is imperative. Recently there have been significant studies on modeling financial systemic risk and financial contagion. Two major classes of models exist for systemic risk, i.e., those based on network models from [@EN01] and those based on a mean field approach [@fouque2013illustrated; @fouque2015meanfield]. Notably, the network model approach generally is considered in only a static, single time, setting while the mean field approach is considered as a differential system. In this paper we will construct a dynamic extension of the interbank network model of [@EN01] thus closing the gap between these two streams of literature.

Interbank networks were studied first in [@EN01] to model the spread of defaults in the financial system. In the Eisenberg-Noe framework, financial firms must satisfy their liabilities by transferring assets. One firm being unable to meet its liabilities due to a shortfall of assets can cause other firms to default on some of their liabilities as well, causing a cascading failure in the financial system. The existence and uniqueness of the clearing payments of this baseline model was proven in [@EN01]. That paper additionally provides methods for numerically computing the realized interbank payments. This baseline model has been extended in multiple directions, including bankruptcy costs, cross-holdings, and fire sales. We refer to [@AW_15; @Staum] for reviews of the prior literature. In regards to bankruptcy costs in financial networks, we refer to [@E07; @RV13; @EGJ14; @GY14; @AW_15; @CCY16; @veraart2017distress]. Cross-holdings have been studied in [@E07; @EGJ14; @AW_15; @GHM12]. Fire sales for a single (representative) illiquid asset have been studied in [@CFS05; @NYYA07; @GK10; @AFM13; @CLY14; @AW_15; @AFM16] and for multiple illiquid assets in [@feinstein2015illiquid; @feinstein2016leverage; @feinstein2017currency]. These network models have been implemented by central banks and regulators for stress testing of and studying cascading failures in the banking systems under their jurisdiction, see, e.g., [@Anand:Canadian; @HK:Modeling; @BEST:2004; @ELS13; @U11; @GHK2011].

Mean field models have also been considered for studying financial contagion and systemic risk. [@fouque2013illustrated] provides a model of agents who revert to the ensemble mean to provide understanding of “systemic risk events” in which many firms fail. Similar mean field diffusion models without controls were studied in, e.g., [@fouque2013stability; @garnier2013a; @garnier2013b]. In contrast, mean field and stochastic games have been proposed for the study of systemic risk in, e.g., [@fouque2015meanfield; @carmona2016delay]. In such models the firms are allowed to borrow from (or lend to) a central bank, the amount of which is optimized to minimize a quadratic cost function. Thus the choice of borrowing and lending provides an optimal control problem beyond the simpler mean field model of [@fouque2013illustrated]. [@NS17] proposes a separate particle system model with mean field interactions.

The current work will focus on adding the time dynamics, which make the mean field models attractive, to the interbank network approach. In fact, the conclusion of [@EN01] provides a discussion of future extensions, one of which is the inclusion of multiple clearing dates. This has been studied directly in [@CC15; @ferrara16]. Additionally, [@KV16] considers a similar approach to model financial networks with multiple maturities. [@feinstein2017currency] further provides another approach to financial networks with multiple maturities by considering each clearing date as a different asset. All of these works, however, only consider clearing at discrete times. [@sonin2017] presents a continuous-time clearing model that exactly replicates the static Eisenberg-Noe framework. In this work we will present both discrete and continuous-time clearing models. However, our emphasis will be on the derivation and the characterization of the continuous-time model. This in part is motivated by the prospect of unification with the mean-field models as well as traditional financial models which typically employ continuous-time models. Additionally, as we will demonstrate, the continuous-time framework no longer requires monotonicity for existence and uniqueness which is generally assumed for static and discrete-time systems. This is valuable for future works that may model network formation and payments as a non-cooperative game; such games may not satisfy the strong monotonicity assumptions usually considered in static and discrete-time systems, but would likely satisfy the sufficient conditions for the continuous-time framework.

The organization of this paper is as follows. In Section \[sec:setting\] we will provide a review of the static Eisenberg-Noe framework. Of particular interest, in this section, we consider the clearing to be in terms of the equity and losses of the firms, as considered in, e.g., [@veraart2017distress; @barucca2016valuation] rather than payments as originally studied in [@EN01]. In Section \[sec:discrete\] we propose a discrete-time formulation for the Eisenberg-Noe model. In discrete time we provide results on existence and uniqueness, as well as a numerical algorithm based on the fictitious default algorithm of [@EN01]. We then extend our model to a continuous-time setting in Section \[sec:continuous\]. For continuous time we consider existence and uniqueness of the clearing solutions, and a numerical algorithm for finding sample paths of this clearing solution, under cash flows modeled by Itô processes. We additionally provide conditions for the discrete-time setting to converge to the continuous-time solution as the time step limits to 0. Section \[sec:discussion\] provides discussion on the financial implications of time dynamics in interbank networks. In particular, we find that the static Eisenberg-Noe clearing solution can be recovered in the continuous-time setting by choosing the network parameters precisely. This allows for a notion of determining the true order of defaults as opposed to the fictitious default order discussed in the static literature based on [@EN01]. However, if the continuous-time network parameters are determined to not follow the rules for recreating the static Eisenberg-Noe setting, then the dynamic and static clearing solutions will generally not coincide. In fact, the set of defaulting and solvent institutions can be altered by rearranging the timing of obligations. As such, using the static Eisenberg-Noe framework for stress testing may result in an incorrect assessment of the health of the financial system. The proofs of the main results are provided in the Appendix.

Static clearing systems {#sec:setting}
=======================

We begin with some simple notation that will be consistent for the entirety of this paper. Let $x,y \in {\mathbb{R}}^n$ for some positive integer $n$, then $$x \wedge y = \left(\min(x_1,y_1),\min(x_2,y_2),\ldots,\min(x_n,y_n)\right)^{\top},$$ $x^- = -(x \wedge 0)$, and $x^+ = (-x)^-$. Further, to ease notation, we will denote $[x,y] := [x_1,y_1] \times [x_2,y_2] \times \ldots \times [x_n,y_n] \subseteq {\mathbb{R}}^n$ to be the $n$-dimensional compact interval for $y - x \in {\mathbb{R}}^n_+$. Similarly, we will consider $x \leq y$ if and only if $y - x \in {\mathbb{R}}^n_+$.

Throughout this paper we will consider a network of $n$ financial institutions. We will denote the set of all banks in the network by ${\mathcal{N}}:= \{1,2,\ldots,n\}$. Often we will consider an additional node $0$, which encompasses the entirety of the financial system outside of the $n$ banks; this node $0$ will also be referred to as society or the societal node. The full set of institutions, including the societal node, is denoted by ${\mathcal{N}}_0 := {\mathcal{N}}\cup \{0\}$. We refer to [@feinstein2014measures; @GY14] for further discussion of the meaning and concepts behind the societal node.

We will be extending the model from [@EN01] in this paper. In that work, any bank $i \in {\mathcal{N}}$ may have obligations $L_{ij} \geq 0$ to any other firm or society $j \in {\mathcal{N}}_0$. We will assume that no firm has any obligations to itself, i.e., $L_{ii} = 0$ for all firms $i \in {\mathcal{N}}$, and the society node has no liabilities at all, i.e., $L_{0j} = 0$ for all firms $j \in {\mathcal{N}}_0$. Thus the *total liabilities* for bank $i \in {\mathcal{N}}$ is given by $\bar p_i := \sum_{j \in {\mathcal{N}}_0} L_{ij} \geq 0$ and relative liabilities $\pi_{ij} := \frac{L_{ij}}{\bar p_i}$ if $\bar p_i > 0$ and arbitrary otherwise; for simplicity, in the case that $\bar p_i = 0$, we will let $\pi_{ij} = \frac{1}{n}$ for all $j \in {\mathcal{N}}_0 \backslash \{i\}$ and $\pi_{ii} = 0$ to retain the property that $\sum_{j \in {\mathcal{N}}_0} \pi_{ij} = 1$. On the other side of the balance sheet, all firms are assumed to begin with some amount of external assets $x_i \geq 0$ for all firms $i \in {\mathcal{N}}_0$. The resultant *clearing payments*, under a no priority of payments assumption, satisfy the fixed point problem in payments $p \in [0,\bar p]$ $$\label{eq:EN-p}
p = \bar p \wedge \left(x + \Pi^{\top}p\right).$$ That is, each bank pays the minimum of what it owes ($\bar p_i$) and what it has ($x_i + \sum_{j \in {\mathcal{N}}} \pi_{ji} p_j$). The resultant vector of *wealths* for all firms is given by $$\label{eq:equity}
V = x + \Pi^{\top}p - \bar p.$$ Noting that payments can be written as a simple function of the wealths ($p = \bar p - V^-$), we provide the following proposition. We refer also to [@veraart2017distress; @barucca2016valuation; @banerjee2017insurance] for similar notions of utilizing clearing wealth instead of clearing payments.

\[prop:EN-e\] A vector $p \in [0,\bar p]$ is a clearing payments in the Eisenberg-Noe setting if and only if $p = [\bar p - V^-]^+$ for some $V \in {\mathbb{R}}^{n+1}$ satisfying the following fixed point problem $$\label{eq:EN-e}
V = x + \Pi^{\top}[\bar p - V^-]^+ - \bar p.$$ Vice versa, a vector $V \in {\mathbb{R}}^{n+1}$ is a clearing wealths (i.e., satisfying ) if and only if $V$ is defined as in for some clearing payments $p \in [0,\bar p]$ as defined in the fixed point problem .

We will prove the first equivalence only, the second follows similarly.

Let $p \in [0,\bar p]$ be a clearing payment vector. Define the wealth vector $V$ by , then it is clear that $V^- = \bar p - p$ by definition as well, i.e., $p = \bar p - V^- \geq 0$. Thus from we immediately recover that the wealth vector $V$ must satisfy .

Let $p = [\bar p - V^-]^+$ for some wealth vector $V \in {\mathbb{R}}^{n+1}$ satisfying . By construction we find $$\begin{aligned}
p &= [\bar p - V^-]^+ = \bar p - \left(x + \Pi^{\top}[\bar p - V^-]^+ - \bar p\right)^-
= \bar p - \left(x + \Pi^{\top}p - \bar p\right)^- = \bar p \wedge \left(x + \Pi^{\top}p\right).\end{aligned}$$ We note that $\bar p \geq \left(x + \Pi^{\top}[\bar p - V^-]^+ - \bar p\right)^-$ can be shown trivially.

Due to the equivalence of the clearing payments and clearing wealths provided in Proposition \[prop:EN-e\], we are able to consider the Eisenberg-Noe system as a fixed point of equity and losses rather than payments. In [@EN01] results for the existence and uniqueness of the clearing payments (and thus for the clearing wealths as well) are provided. In fact, it can be shown that there exists a unique clearing solution in the Eisenberg-Noe framework so long as $L_{i0} > 0$ for all firms $i \in {\mathcal{N}}$. We will take advantage of this result later in this paper. This is a reasonable assumption (as discussed in, e.g., [@GY14]) as obligations to society include, e.g., deposits to the banks.

Discrete-time clearing systems {#sec:discrete}
==============================

Consider now a discrete set of clearing times ${\mathbb{T}}$, e.g., ${\mathbb{T}}= \{0,1,\dots,T\}$ for some (finite) terminal time $T < \infty$ or ${\mathbb{T}}= {\mathbb{N}}$. Such a setting is presented in [@CC15]. 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 $Z_t := (Z(s))_{s = 0}^t$.

(0,9.5) rectangle (6,10) node\[pos=.5\][**Balance Sheet**]{}; (0,9) rectangle (3,9.5) node\[pos=.5\][**Assets**]{}; (3,9) rectangle (6,9.5) node\[pos=.5\][**Liabilities**]{};

(0,7) rectangle (3,9) node\[pos=.5,style=[align=center]{}\][Cash-Flow @ $t = 0$\
$x_i(0)$]{}; (0,6) rectangle (3,7) node\[pos=.5,style=[align=center]{}\][Cash-Flow @ $t = 1$\
$x_i(1)$]{}; (0,2) rectangle (3,6) node\[pos=.5,style=[align=center]{}\][Interbank @ $t = 0$\
$\sum_{j = 1}^n \pi_{ji}(0)p_j(0)$]{}; (0,0) rectangle (3,2) node\[pos=.5,style=[align=center]{}\][Interbank @ $t = 1$\
$\sum_{j = 1}^n \pi_{ji}(1)p_j(1)$]{};

(3,5) rectangle (6,9) node\[pos=.5,style=[align=center]{}\][Cash-Flow @ $t = 0$\
$\sum_{j = 1}^n L_{ij}(0)$]{}; (3,1.5) rectangle (6,5) node\[pos=.5,style=[align=center]{}\][Cash-Flow @ $t = 1$\
$\sum_{j = 1}^n L_{ij}(1)$]{}; (3,0) rectangle (6,1.5) node\[pos=.5,style=[align=center]{}\][Capital\
$V_i(1)$]{};

(0,0) rectangle (3,9); (3,0) rectangle (6,9);

(0,7) – (3,7); (0,6) – (3,6); (0,2) – (3,2); (3,5) – (6,5);

 \

(0,6.5) rectangle (6,7) node\[pos=.5\][**Balance Sheet @ $t = 0$**]{}; (0,6) rectangle (3,6.5) node\[pos=.5\][**Assets**]{}; (3,6) rectangle (6,6.5) node\[pos=.5\][**Liabilities**]{};

(0,4) rectangle (3,6) node\[pos=.5,style=[align=center]{}\][Cash-Flow\
$x_i(0)$]{}; (0,0) rectangle (3,4) node\[pos=.5,style=[align=center]{}\][Interbank\
$\sum_{j = 1}^n \pi_{ji}(0)p_j(0)$]{};

(3,2) rectangle (6,6) node\[pos=.5,style=[align=center]{}\][Cash-Flow\
$\sum_{j = 1}^n L_{ij}(0)$]{}; (3,0) rectangle (6,2) node\[pos=.5,style=[align=center]{}\] (t) [Capital\
$V_i(0)$]{};

(7,6.5) rectangle (13,7) node\[pos=.5\][**Balance Sheet @ $t = 1$**]{}; (7,6) rectangle (10,6.5) node\[pos=.5\][**Assets**]{}; (10,6) rectangle (13,6.5) node\[pos=.5\][**Liabilities**]{};

in [0.5]{}

(7,4+) rectangle (10,5+) node\[pos=.5,style=[align=center]{}\][Cash-Flow\
$x_i(1)$]{}; (7,2+) rectangle (10,4+) node\[pos=.5,style=[align=center]{}\] (t+1) [Carry-Forward\
$V_i(0)^+$]{}; (7,0+) rectangle (10,2+) node\[pos=.5,style=[align=center]{}\][Interbank\
$\sum_{j = 1}^n \pi_{ji}(1)p_j(1)$]{};

(10,1.5+) rectangle (13,5+) node\[pos=.5,style=[align=center]{}\][Cash-Flow\
$\sum_{j = 1}^n L_{ij}(1)$]{}; (13.2,1+) rectangle (16,2+) node\[pos=.5,style=[align=center]{}\][Carry-Forward\
$V_i(0)^- = 0$]{}; (13,1.5+) – (13.2,2+) – (13.2,1+) – cycle; (13,1.5+) – (13.2,2+) – (16,2+) – (16,1+) – (13.2,1+) – cycle; (10,0+) rectangle (13,1.5+) node\[pos=.5,style=[align=center]{}\][Capital\
$V_i(1)$]{}; (6,1) – (7,3+); (6,.25) – (14,.25) – (14,1+);

In this setting, we will consider the external (incoming) cash flow $x: {\mathbb{T}}\to {\mathbb{R}}^{n+1}_+$ and nominal liabilities $L: {\mathbb{T}}\to {\mathbb{R}}^{(n+1) \times (n+1)}_+$ to be functions of the clearing time, i.e., as assets and liabilities with different maturities. The external cash in-flows and nominal liabilities can explicitly depend on the clearing results of the prior times (i.e., $x(t,V_{t-1})$ and $L(t,V_{t-1})$) without affecting the existence and uniqueness results we present, but for simplicity of notation we will focus on the case where the external assets and nominal liabilities are independent of the health and wealth of the firms. Throughout we are considering the discounted cash flows and liabilities so as to simplify notation. In contrast to the static Eisenberg-Noe framework, herein we need to consider the results of the prior times. In particular, if firm $i$ has positive equity at time $t-1$ (i.e., $V_i(t-1) > 0$) then these additional assets are available to firm $i$ at time $t$ in order to satisfy its obligations. Similarly, if firm $i$ has negative wealth at time $t-1$ (i.e., $V_i(t-1) < 0$) then the debts that the firm has not yet paid will roll-forward in time and be due at the next period. For example, consider a network in which obligations come due throughout the day at, e.g., opening, mid-day, and closing, but that all debts must be cleared by the end of the day. In such a way, the current unpaid liabilities may be paid at a future time, but before the terminal time. That is, a firm can be considered in *distress* at a time if it is unable to satisfy its obligations at that time, but only *defaults* if it has negative wealth at the terminal time. Thus in this paper we primarily focus on the intra-day dynamics rather than the inter-day dynamics. See Figure \[fig:discrete-BS\] for a stylized (snapshot of the) balance sheet example for a firm that has positive wealth at time $0$ that rolls forward to time $1$. The full (actualized) balance sheet for this example with only those two time periods is displayed in Figure \[fig:BS\]; we note that the full balance sheet as depicted considers actualized payments rather than the book value of the obligations.

To incorporate the inter-day dynamics in this framework we can “zero out” a firm before the terminal date if it is deemed to default in much the same as in [@banerjee2017insurance]. A broader framework for dealing with various default mechanisms is discussed in Remark \[rem:loans\]. We can further consider the Nash game in which firms decide if they will allow debts to be rolled forward in time. In such a setting, if we include a delay for payment due to, e.g., bankruptcy court so that defaulting firms do not pay any obligations until after the terminal time $T$, then the optimal strategy for all firms (up until the terminal time $T$) would be to always allow other firms to roll all debts forward so as to maximize payments.

\[ass:initial\] Before the time of interest, all firms are solvent and liquid. That is, $V_i(-1) \geq 0$ for all firms $i \in {\mathcal{N}}_0$.

We can now construct the total liabilities and relative liabilities at time $t \in {\mathbb{T}}$ as $$\begin{aligned}
\bar p_i(t,V_{t-1}) &:= \sum_{j \in {\mathcal{N}}_0} L_{ij}(t) + V_i(t-1)^-\\
\pi_{ij}(t,V_{t-1}) &:= \begin{cases} \frac{L_{ij}(t) + \pi_{ij}(t-1,V_{t-2})V_i(t-1)^-}{\bar p_i(t,V_{t-1})} &\text{if } \bar p_i(t,V_{t-1}) > 0 \\ \frac{1}{n} &\text{if } \bar p_i(t,V_{t-1}) = 0, \; j \neq i\\ 0 &\text{if } \bar p_i(t,V_{t-1}) = 0, \; j = i \end{cases} \quad \forall i,j \in {\mathcal{N}}_0.\end{aligned}$$ In this way, coupled with the accumulation of positive equity over time, the clearing wealths must satisfy the following fixed point problem in time $t$ wealths: $$\label{eq:EN-discrete}
V(t) = V(t-1)^+ + x(t) + \Pi(t,V_{t-1})^{\top}\left[\bar p(t,V_{t-1}) - V(t)^-\right]^+ - \bar p(t,V_{t-1}).$$ That is, all firms have a clearing wealth that is the summation of their positive equity at the prior time, the new incoming external cash flow, and the payments made by all other firms minus the total obligations of the firm (including the prior unpaid liabilities). In this way we can construct the wealths of firms forward in time. This can be considered a discrete-time extension of .

We now wish to consider a reformulation of . To accomplish this, we consider a process of cash flows $c$ and functional relative exposures $A$. These we define by $$\begin{aligned}
\nonumber c(t) &:= x(t) + L(t)^{\top}\vec{1} - L(t)\vec{1}\\
\label{eq:discrete-A} a_{ij}(t,V_t) &:= \begin{cases} \pi_{ij}(t,V_{t-1}) &\text{if } \bar p_i(t,V_{t-1}) \geq V_i(t)^-\\ \frac{L_{ij}(t) + \pi_{ij}(t-1,V_{t-2})V_i(t-1)^-}{V_i(t)^-} &\text{if } \bar p_i(t,V_{t-1}) < V_i(t)^-\end{cases} \quad \forall i,j \in {\mathcal{N}}_0.\end{aligned}$$ In the above, $\vec{1} := (1,1,\ldots,1)^{\top}\in {\mathbb{R}}^n$ is the vector of ones. Here we consider $c(t) = x(t) + L(t)^{\top}\vec{1} - L(t)\vec{1} \in {\mathbb{R}}^{n+1}$ to be the vector of book capital levels at time $t$, i.e., the new wealth of each firm assuming all other firms pay in full. We wish to note that the new total liabilities are given by $L(t)\vec{1}$ and the new incoming interbank obligations are given by $L(t)^{\top}\vec{1}$. We can also consider $c_i(t)$ to be the *net cash flow* for firm $i$ at time $t$. Further, we introduce the functional matrix $A: {\mathbb{T}}\times {\mathbb{R}}^{n+1} \to [0,1]^{(n+1) \times (n+1)}$ to be the relative exposure matrix. That is, $a_{ij}(t,V_t)V_i(t)^-$ provides the (negative) impact that firm $i$’s losses have on firm $j$’s wealth at time $t \in {\mathbb{T}}$. This is in contrast to $\Pi$, the relative liabilities, in that it endogenously imposes the limited exposures concept. In this work the two notions will generally coincide, but for mathematical simplicity we introduce this relative exposure matrix. For the equivalence we seek, we define the relative exposures so that $$L(t)^{\top}\vec{1} + A(t-1,V_{t-1})^{\top}V(t-1)^- - A(t,V_t)^{\top}V(t)^- = \Pi(t,V_{t-1})^{\top}[\bar p(t,V_{t-1}) - V(t)^-]^+$$ for any $V(t) \in {\mathbb{R}}^{n+1}$. This formulation is such that if the positive part were removed from the right hand side, the relative exposures $A$ would be defined exactly as the relative liabilities $\Pi$ by construction. In particular, we will define the relative exposures element-wise and pointwise so as to encompass the limited exposures as in . If $\bar p_i(t,V_{t-1}) > 0$ then we can simplify this further as $a_{ij}(t,V_t) = \frac{L_{ij}(t) + a_{ij}(t-1,V_{t-1})V_i(t-1)^-}{\max\{\bar p_i(t,V_{t-1}) , V_i(t)^-\}}$.

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 $t \in {\mathbb{T}}$, 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 we will not utilize this discrete-time result in this paper, we consider it important to discuss random events to more closely match reality. Further, this result will implicitly appear in the construction and analysis of the continuous-time Eisenberg-Noe formulation of the next section.

\[cor:discrete\] Consider the setting of Theorem \[thm:discrete\] where the random network parameters $(c,L)$ adapted to the filtered probability space $(\Omega,{\mathcal{F}},({\mathcal{F}}(t))_{t \in {\mathbb{T}}},{\mathbb{P}})$. If $c(s) \in {\mathcal{L}}_s^p({\mathbb{R}}^{n+1})$ and $L(s) \in {\mathcal{L}}_s^p({\mathbb{R}}^{(n+1) \times (n+1)}_+)$ for all times $s \leq t$ for some $p \in [0,\infty]$, then the unique clearing solution at time $t$ has finite $p$-norm, i.e., $V(t) \in {\mathcal{L}}_t^p({\mathbb{R}}^{n+1})$.

With the construction of the existence and uniqueness of the solution we now want to emphasize the *fictitious default algorithm* from [@EN01] to construct this clearing wealths vector over time. This algorithm is presented for the deterministic setting; if a stochastic setting is desired then Algorithm \[alg:discrete\] provides a method for computing a single sample path. We note that at each time $t$ this algorithm takes at most $n$ iterations. Thus with a terminal time $T$, this algorithm will construct the full clearing solution over ${\mathbb{T}}$ in $nT$ iterations.

\[alg:discrete\] Under the assumptions of Theorem \[thm:discrete\] in a deterministic setting the clearing wealths process $V: {\mathbb{T}}\to {\mathbb{R}}^{n+1}$ can be found by the following algorithm. Initialize $t = -1$ and $V(-1) \geq 0$ as a given. Repeat until $t = \max{\mathbb{T}}$:

1.  Increment $t = t+1$.

2.  \[alg:v\] Initialize $k = 0$, $V^0 = V(t-1) + c(t)$, and $D^0 = \emptyset$. Repeat until convergence:

    1.  Increment $k = k+1$;

    2.  Denote the set of insolvent banks by $D^k := \left\{i \in \{1,2,...,n\} \; | \; V_i^{k-1} < 0\right\}$.

    3.  If $D^k = D^{k-1}$ then terminate and set $V(t) = V^{k-1}$.

    4.  Define the matrix $\Lambda^k \in \{0,1\}^{n \times n}$ so that $\Lambda_{ij}^k = \begin{cases}1 &\text{if } i = j \in D^k \\ 0 &\text{else}\end{cases}$.

    5.  \[alg:vk\] Define $V^k = (I - \Pi(t,V_{t-1})^{\top}\Lambda^k)^{-1}\left(V(t-1) + c(t) + A(t-1,V_{t-1})^{\top}V(t-1)^-\right)$.

\[rem:loans\] Note that in the construction of $V^k$ in step of the fictitious default algorithm we utilize the relative liabilities $\Pi(t,V_{t-1})$ in the matrix inverse rather than the relative exposures $A(t,(V_{t-1},V^k))$. This has the added benefit that this definition of $V^k$ is *not* a fixed point problem, which it would be if the relative exposures matrix at time $t$ were considered. This change is possible since, as discussed in the proof of Theorem \[thm:discrete\], any clearing solution must be in the domain so that the relative liabilities and exposures coincide. This additionally provides the invertibility of this matrix using standard input-output results as discussed in [@EN01; @feinstein2017sensitivity].

We wish to finish up our discussion of the discrete-time Eisenberg-Noe framework by considering some extensions involving loans.

The theoretical framework presented in this paper can be easily extended to incorporate the concepts of loans until some (deterministic) insolvency condition is hit. In particular, we will consider loans made from a central bank or lender of last resort who we will assume are part of the societal node $0$. From this perspective we consider three cases that a firm might be in:

-   [**solvent and liquid**]{} in which case the firm has positive equity and pays off its obligations in full;

-   [**solvent and distressed**]{} in which case the firm has negative equity, but receives an overnight loan (with interest rate set at the risk-free rate for simplicity) to cover all obligations due on that date; and

-   [**insolvent**]{} in which the firm will not receive any loans and is sent to a bankruptcy court.

The determination whether a firm is solvent can be done with an appropriate exogenous solvency function. We will assume that once a firm is deemed insolvent it can never recover to solvency again. Two possible systems for considering insolvent firms are:

1.  [**Receivership:**]{} In such a system, when a firm is deemed insolvent it is placed in receivership so that obligations are payed out on a first-come first-serve basis.

2.  [**Auctions:**]{} In such a system, when a firm is deemed insolvent its future assets are auctioned off in order to pay the future liabilities (in a proportional scheme) at the next time point. This will then affect the cash flows $c$ and nominal liabilities $L$, as such we would need to consider $c(t,V_{t-1})$ and $L(t,V_{t-1})$ to truly consider this case. We refer to [@CC15] for a detailed discussion of the auction model for insolvency. The auction system can be interpreted as an internal mechanism for determining bankruptcy costs in contrast to the exogenous parameter in, e.g., [@RV13].

The existence and uniqueness of the clearing solutions in these scenarios require an additional monotonicity property; we can use the notion a speculative system from [@banerjee2017insurance] to get the desired results. This condition encodes the notion that a firm does not benefit from any firm’s distress.

Continuous-time clearing systems {#sec:continuous}
================================

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 clearing wealths driven by $(dc,dL)$ when $c(t) = \int_0^t dc(s)$ is an Itô process and $L(t) = \int_0^t dL(s)$ is deterministic and continuous (e.g., $dL$ does not include any Dirac delta functions). This setting, and the results on the continuous-time Eisenberg-Noe model, can be extended to the case in which the cash flows and liabilities are additionally functions of the wealths $V$. For simplicity, in this section we will restrict ourselves so that the parameters are independent of the current wealths. In order to construct a continuous-time differential system, we will consider again the discrete-time setting with explicit time steps ${{\Delta t}}$.

\[ass:society\] The cash flows $c$ are defined by the Itô stochastic differential equation $dc(t) = \mu(t,c(t))dt + \sigma(t,c(t))dW(t)$ for $(n+1)$-vector of Brownian motions $W$ over some filtered probability space $(\Omega,{\mathcal{F}},({\mathcal{F}}_t)_{t \in {\mathbb{T}}},{\mathbb{P}})$. Additionally, the drift and diffusion functions $\mu: {\mathbb{T}}\times {\mathbb{R}}^{n+1} \to {\mathbb{R}}^{n+1}$ and $\sigma: {\mathbb{T}}\times {\mathbb{R}}^{n+1} \to {\mathbb{R}}^{(n+1) \times (n+1)}$ are jointly continuous and satisfy the linear growth and Lipschitz continuous conditions, i.e., there exist constants $C,D > 0$ such that for all times $t \in {\mathbb{T}}$ and cash flows $c,d \in {\mathbb{R}}^{n+1}$ $$\begin{aligned}
\|\mu(t,c)\|_1 + \|\sigma(t,c)\|_1^{op} &\leq C(1 + \|c\|_1)\\
\|\mu(t,c) - \mu(t,d)\|_1 + \|\sigma(t,c) - \sigma(t,d)\|_1^{op} &\leq D\|c - d\|_1\end{aligned}$$ 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 time $t$ to be given by ${\Delta}c(t,{{\Delta t}}) := \int_{t-{{\Delta t}}}^t dc(s)$ and the nominal liabilities at time $t$ are similarly provided by ${\Delta}L(t,{{\Delta t}}) := \int_{t-{{\Delta t}}}^t dL(s)$ where both $dc$ and $dL$ are discussed above (additionally, we set $dc(-t) = 0$ and $dL(-t) = 0$ for any times $t < 0$). The choice of notation for ${\Delta}c$ and ${\Delta}L$ are to make explicit the “change” inherent in the construction.

With these parameters we can construct the ${{\Delta t}}$-discrete-time clearing process $V(t,{{\Delta t}})$ and exposure matrix $A(t,{{\Delta t}},V_t({{\Delta t}}))$ by: $$\begin{aligned}
\label{eq:discrete-Vdt} \begin{split}V(t,{{\Delta t}}) &= V(t-{{\Delta t}},{{\Delta t}}) + {\Delta}c(t,{{\Delta t}}) - A(t,{{\Delta t}},V_t({{\Delta t}}))^{\top}V(t,{{\Delta t}})^-\\ &\qquad + A(t-{{\Delta t}},{{\Delta t}},V_{t-{{\Delta t}}}({{\Delta t}}))^{\top}V(t-{{\Delta t}},{{\Delta t}})^-\end{split}\\
\label{eq:discrete-Adt} \begin{split}a_{ij}(t,{{\Delta t}},V_t({{\Delta t}})) &= \frac{{\Delta}L_{ij}(t,{{\Delta t}})+a_{ij}(t-{{\Delta t}},{{\Delta t}},V_{t-{{\Delta t}}}({{\Delta t}}))V_i(t-{{\Delta t}},{{\Delta t}})^-}{\max\{\sum_{k \in {\mathcal{N}}_0} {\Delta}L_{ik}(t,{{\Delta t}})+V_i(t-{{\Delta t}},{{\Delta t}})^- , V_i(t,{{\Delta t}})^-\}}1_{\{i \neq 0\}}\\ &\qquad + \frac{1}{n}1_{\{i = 0,\; j \neq 0\}} \qquad \forall i,j \in {\mathcal{N}}_0.\end{split}\end{aligned}$$ Here we assume that $V(t) = V(-1) \geq 0$ for every time $t < 0$ as in Assumption \[ass:initial\]. This construction can be computed either in continuous time $t \in {\mathbb{T}}$ with sliding intervals of size ${{\Delta t}}$ or at the discrete times $t \in \{0,{{\Delta t}},...,T\}$. The existence and uniqueness of this system follow exactly as in Theorem \[thm:discrete\] under Assumption \[ass:society\].

\[cor:discrete-cont\] Let $(dc,dL): {\mathbb{T}}\to {\mathbb{R}}^{n+1} \times {\mathbb{R}}^{(n+1) \times (n+1)}_+$ define a dynamic financial network satisfying Assumption \[ass:society\] such that every bank has cash flow at least at the level dictated by nominal interbank liabilities, i.e., ${\Delta}c_i(t,{{\Delta t}}) \geq \sum_{j \in {\mathcal{N}}} {\Delta}L_{ji}(t,{{\Delta t}}) - \sum_{j \in {\mathcal{N}}_0} {\Delta}L_{ij}(t,{{\Delta t}})$ for all banks $i \in {\mathcal{N}}_0$, times $t \in {\mathbb{T}}$, and step-sizes ${{\Delta t}}> 0$. Under Assumption \[ass:initial\], there exists a unique solution of clearing wealths $V: {\mathbb{T}}\times {\mathbb{R}}_{++} \to {\mathbb{R}}^{n+1}$ to . Further, the clearing wealths are jointly continuous in time and step-size.

Now we want to consider the limiting behavior of this discrete-time system as ${{\Delta t}}$ tends to 0. To do so, first, we will consider the formulation of the relative exposures $a_{ij}$ from bank $i$ to $j$. From Corollary \[cor:discrete-cont\] and Assumption \[ass:society\], we know that for any time $t \in {\mathbb{T}}$ and bank $i \in {\mathcal{N}}$ it must follow that $\sum_{k \in {\mathcal{N}}_0} {\Delta}L_{ik}(t,{{\Delta t}}) + V_i(t-{{\Delta t}},{{\Delta t}})^- \geq V_i(t,{{\Delta t}})^-$ for ${{\Delta t}}> 0$ small enough due to the joint continuity of the wealths in time and step-size. Thus in the limiting case, as ${{\Delta t}}\searrow 0$, we find that we can consider the relative liabilities rather than the relative exposures, i.e., for ${{\Delta t}}$ small enough $$\label{eq:discrete-Adt-limiting}
\begin{split}a_{ij}(t,{{\Delta t}},V_t({{\Delta t}})) &= \frac{{\Delta}L_{ij}(t,{{\Delta t}})+a_{ij}(t-{{\Delta t}},{{\Delta t}},V_{t-{{\Delta t}}}({{\Delta t}}))V_i(t-{{\Delta t}},{{\Delta t}})^-}{\sum_{k \in {\mathcal{N}}_0} {\Delta}L_{ik}(t,{{\Delta t}})+V_i(t-{{\Delta t}},{{\Delta t}})^-}1_{\{i \neq 0\}}\\
&\qquad + \frac{1}{n}1_{\{i = 0,\; j \neq 0\}} \qquad \forall i,j \in {\mathcal{N}}_0.\end{split}$$ Rearranging these terms we are able to deduce that, for any firm $i \in {\mathcal{N}}$, $$\label{eq:discrete-Adt-limiting2}
\begin{split}&[a_{ij}(t,{{\Delta t}},V_t({{\Delta t}})) - a_{ij}(t-{{\Delta t}},{{\Delta t}},V_{t-{{\Delta t}}}({{\Delta t}}))] V_i(t-{{\Delta t}},{{\Delta t}})^- \\ &\qquad = {\Delta}L_{ij}(t,{{\Delta t}}) - a_{ij}(t,{{\Delta t}},V_t({{\Delta t}})) \sum_{k \in {\mathcal{N}}_0} {\Delta}L_{ik}(t,{{\Delta t}}).\end{split}$$

Coupled with the assumption that the societal node always has positive wealth, we are thus able to consider the limiting behavior of as the step-size ${{\Delta t}}$ tends to 0. To do so, consider $$\begin{aligned}
\begin{split}V(t,{{\Delta t}}) &= V(t-{{\Delta t}},{{\Delta t}}) + {\Delta}c(t,{{\Delta t}}) - A(t,{{\Delta t}},V_t({{\Delta t}}))^{\top}V(t,{{\Delta t}})^-\\ &\qquad + A(t-{{\Delta t}},{{\Delta t}},V_{t-{{\Delta t}}})^{\top}V(t-{{\Delta t}},{{\Delta t}})^-\end{split}\\
\begin{split} &= V(t-{{\Delta t}},{{\Delta t}}) + {\Delta}c(t,{{\Delta t}})\\ &\qquad - A(t,{{\Delta t}},V_t({{\Delta t}}))^{\top}V(t,{{\Delta t}})^- + A(t,{{\Delta t}},V_t({{\Delta t}}))^{\top}V(t-{{\Delta t}},{{\Delta t}})^-\\ &\qquad - A(t,{{\Delta t}},V_t({{\Delta t}}))^{\top}V(t-{{\Delta t}},{{\Delta t}})^- + A(t-{{\Delta t}},{{\Delta t}},V_{t-{{\Delta t}}})^{\top}V(t-{{\Delta t}},{{\Delta t}})^-\end{split}\\
\begin{split} &= V(t-{{\Delta t}},{{\Delta t}}) + {\Delta}c(t,{{\Delta t}}) - A(t,{{\Delta t}},V_t({{\Delta t}}))^{\top}[V(t,{{\Delta t}})^- - V(t-{{\Delta t}},{{\Delta t}})^-]\\ &\qquad - {\Delta}L(t,{{\Delta t}})^{\top}\vec{1} + A(t,{{\Delta t}},V_t({{\Delta t}}))^{\top}{\Delta}L(t,{{\Delta t}}) \vec{1}.\end{split}\end{aligned}$$ Consider the notation for the matrix of distressed firms from the fictitious default algorithm (Algorithm \[alg:discrete\]), i.e., $\Lambda(V) \in \{0,1\}^{(n+1) \times (n+1)}$ is the diagonal matrix of banks in distress $$\Lambda_{ij}(V) = \begin{cases}1 &\text{if } i = j \neq 0 \text{ and } V_i < 0 \\ 0 &\text{else}\end{cases} \quad \forall i,j \in {\mathcal{N}}_0.$$ We are able to set $\Lambda_{00}(V) = 0$ without loss of generality since, by assumption, the outside node $0$ has no obligations into the system. Thus, as with , by continuity of the clearing wealths and ${{\Delta t}}$ small enough, we can conclude that except at specific event times (to be considered later, see Algorithm \[alg:continuous\]) it follows that $\Lambda(V(t,{{\Delta t}})) = \Lambda(V(t-{{\Delta t}},{{\Delta t}}))$. Thus, with this added notation we can reformulate the clearing wealths equation as $$\begin{aligned}
\begin{split}V(t,{{\Delta t}}) &= V(t-{{\Delta t}},{{\Delta t}}) + A(t,{{\Delta t}},V_t({{\Delta t}}))^{\top}\Lambda(V(t,{{\Delta t}})) [V(t,{{\Delta t}}) - V(t-{{\Delta t}},{{\Delta t}})] + {\Delta}c(t,{{\Delta t}}) \\ &\qquad - {\Delta}L(t,{{\Delta t}})^{\top}\vec{1} + A(t,{{\Delta t}},V_t({{\Delta t}}))^{\top}{\Delta}L(t,{{\Delta t}}) \vec{1}.\end{split}\end{aligned}$$ For the construction of a differential form we can consider the equivalent formulation $$\begin{aligned}
\label{eq:discrete-Vdt-limiting}
\begin{split} V(t,{{\Delta t}}) &- V(t-{{\Delta t}},{{\Delta t}}) =\\
&[I - A(t,{{\Delta t}},V_t({{\Delta t}}))^{\top}\Lambda(V(t,{{\Delta t}}))]^{-1} \left(\begin{array}{l}{\Delta}c(t,{{\Delta t}}) - {\Delta}L(t,{{\Delta t}})^{\top}\vec{1}\\ \quad + A(t,{{\Delta t}},V_t({{\Delta t}}))^{\top}{\Delta}L(t,{{\Delta t}}) \vec{1}\end{array}\right).\end{split}\end{aligned}$$ Note that $I - A(t,{{\Delta t}},V_t({{\Delta t}}))^{\top}\Lambda(V(t,{{\Delta t}}))$ is invertible by standard input-output results and as proven in Proposition \[prop:Leontief\].

Utilizing and and taking the limit as ${{\Delta t}}\searrow 0$, we are thus able to construct the joint differential system: $$\begin{aligned}
\label{eq:continuous-V} dV(t) &= [I - A(t)^{\top}\Lambda(V(t))]^{-1} \left(dc(t) - dL(t)^{\top}\vec{1} + A(t)^{\top}dL(t) \vec{1}\right)\\
\label{eq:continuous-A} da_{ij}(t) &= \begin{cases} \frac{d^2L_{ij}(t) - a_{ij}(t)\sum_{k \in {\mathcal{N}}_0} d^2L_{ik}(t)}{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(t)} &\text{if } i \in {\mathcal{N}}, \; V_i(t) \geq 0 \\ \frac{dL_{ij}(t) - a_{ij}(t)\sum_{k \in {\mathcal{N}}_0} dL_{ik}(t)}{V_i(t)^-} &\text{if } i \in {\mathcal{N}}, \; V_i(t) < 0 \\ 0 &\text{if } i = 0\end{cases} \quad \forall i,j \in {\mathcal{N}}_0\end{aligned}$$ with initial conditions $V(0) \geq 0$ given and $a_{ij}(0) = \frac{dL_{ij}(0)}{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(0)}1_{\{i \neq 0\}} + \frac{1}{n}1_{\{i = 0,\; j \neq 0\}}$ for all firms $i,j \in {\mathcal{N}}_0$. As in , $I - A(t)^{\top}\Lambda(V(t))$ is invertible by standard input-output results and as proven in Proposition \[prop:Leontief\]. The first case in is constructed by noting that $a_{ij}(t) = \frac{dL_{ij}(t)}{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(t)}$ if $V_i(t) \geq 0$ and $i \in {\mathcal{N}}$ and $da_{0j}(t) = 0$ for any firm $j \in {\mathcal{N}}_0$ for all times $t$; the second case in follows from and taking the limit as ${{\Delta t}}\searrow 0$. Note that this differential system is discontinuous, with events at times when firms cross the 0 wealth boundary, i.e., when $\Lambda(V(t)) \neq \Lambda(V(t^-))$. As such, we will consider the differential system on the inter-event intervals, then update the differential system between these intervals. This is made more explicit in the proof of Theorem \[thm:continuous\] and in Algorithm \[alg:continuous\]. As with the discrete-time system , the relative exposures follow the incoming proportional obligations if a firm has a surplus wealth. When a firm is in distress, the relative exposures follow a path that provides the average relative obligations between new liabilities and the prior unpaid liabilities.

As in the discrete-time section we consider the debt to roll forward in this case. In this way we encode the notion of either intra-day dynamics in this model or when bankruptcy court would not settle debts before the terminal time $T$ for the system. To allow for insolvencies, we can consider some (deterministic) mechanism to determine when a bank becomes insolvent and restart the differential system with updated parameters from that time point, e.g., using an instantaneous auction as in [@CC15]; see also Remarks \[rem:loans\] and \[rem:cont-loans\].

We will complete our discussion of the construction of this differential system by providing some properties on the relative liabilities and exposures matrix $A$. Notably, these properties are those that would be expected from the discrete-time setting for the relative exposures. Namely, as a firm recovers from a distressed state its relative liabilities return to be only the fraction of incoming liabilities, that the relative exposures are bounded from below by 0 (and to society by $\delta$ as provided in Assumption \[ass:society\]), and the relative exposure matrix is row stochastic at all times.

\[prop:A\] Let $(dc,dL): {\mathbb{T}}\to {\mathbb{R}}^{n+1} \times {\mathbb{R}}^{(n+1) \times (n+1)}_+$ define a dynamic financial network satisfying Assumption \[ass:society\]. Let $(V,A): {\mathbb{T}}\to {\mathbb{R}}^{n+1} \times {\mathbb{R}}^{(n+1) \times (n+1)}$ be any solution of the differential system and satisfying Assumption \[ass:initial\]. The relative exposure matrix $A(t)$ satisfies the following properties:

1.  \[prop:A1\] For any bank $i \in {\mathcal{N}}$, if $V_i(t) \nearrow 0$ as $t \nearrow \tau$ then $\lim_{t \nearrow \tau} a_{ij}(t) = \frac{dL_{ij}(\tau)}{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(\tau)}$.

2.  \[prop:A2\] For all times $t \in {\mathbb{T}}$ and for any bank $i \in {\mathcal{N}}$, the elements $a_{ij}(t) \geq 0$ for all banks $j \in {\mathcal{N}}$ and $a_{i0}(t) \geq \delta$;

3.  \[prop:A3\] For all times $t \in {\mathbb{T}}$ and for any bank $i \in {\mathcal{N}}_0$, the row sums $\sum_{k \in {\mathcal{N}}_0} a_{ik}(t) = 1$;

With this differential construction and , we seek to prove existence and uniqueness of the clearing solutions. For notational simplicity, define the space of relative exposure matrices $${\mathbb{A}}:= \left\{A \in [0,1]^{(n+1) \times (n+1)} \; | \; A\vec{1} = \vec{1}, \; a_{ii} = 0, \; a_{i0} \geq \delta \; \forall i \in {\mathcal{N}}, \; a_{0j} = \frac{1}{n} \; \forall j \in {\mathcal{N}}\right\}.$$ From Proposition \[prop:A\], we have already proven that if $(V,A): {\mathbb{T}}\to {\mathbb{R}}^{n+1} \times {\mathbb{R}}^{(n+1) \times (n+1)}$ is a solution to the continuous-time Eisenberg-Noe system then $A(t) \in {\mathbb{A}}$ for all times $t \in {\mathbb{T}}$.

\[thm:continuous\] Let ${\mathbb{T}}= [0,T]$ be a finite time period and let $(dc,dL): {\mathbb{T}}\to {\mathbb{R}}^{n+1} \times {\mathbb{R}}^{(n+1) \times (n+1)}_+$ define a dynamic financial network satisfying Assumption \[ass:society\]. There exists a unique strong solution to the clearing wealths and relative exposures $(V,A)$ satisfying and if $V(0) \in {\mathbb{R}}^{n+1}_{++}$.

\[rem:dc\] The restrictions on the cash flows $dc$ made in Assumption \[ass:society\] can be relaxed to depend explicitly on the wealths and relative exposures, i.e., $$dc(t) = \mu(t,c(t),V(t),A(t))dt + \sigma(t,c(t),V(t),A(t))dW(t).$$ This would still guarantee a unique strong solution of the clearing wealths and relative exposures as in Theorem \[thm:continuous\] so long as $\mu,\sigma$ satisfy a local linear growth condition, local Lipschitz condition, and $c(t)$ can be bounded above and below by elements of ${\mathcal{L}}_t^2({\mathbb{R}}^{n+1})$ for all time $t$.

We now present an algorithm for numerically computing an approximation of a single sample path for the continuous-time Eisenberg-Noe clearing system. To do so we consider Euler’s method for differential equations with an event finding algorithm.

\[alg:continuous\] Under the assumptions of Theorem \[thm:continuous\] for a fixed event $\omega \in \Omega$ the clearing wealths process $V: {\mathbb{T}}\to {\mathbb{R}}^{n+1}$ and relative exposures $A: {\mathbb{T}}\to {\mathbb{A}}$ can be found by the following algorithm. Fix a step-size ${{\Delta t}}_0 > 0$. Initialize $t = 0$, $V(0) \geq 0$ given, $a_{ij}(0) = \frac{dL_{ij}(0)}{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(0)}1_{\{i \neq 0\}} + \frac{1}{n}1_{\{i = 0,\; j \neq 0\}}$, and $\Lambda = \{0\}^{(n+1) \times (n+1)}$. Repeat until $t \geq T$:

1.  Initialize $\Lambda^0 \neq \Lambda$ and ${{\Delta t}}= {{\Delta t}}_0$.

2.  Sample $Z \sim N(0,I)$.

3.  Repeat until $\Lambda^0 = \Lambda$:

    1.  Set $\Lambda^0 = \Lambda$.

    2.  Compute $$\begin{aligned}
        \bar\mu(t) &= (I - A(t)^{\top}\Lambda)^{-1} \left(\mu(t,c(t)) - \dot{L}(t)^{\top}\vec{1} + A(t)^{\top}\dot{L}(t) \vec{1}\right)\\
        \bar\sigma(t) &= (I - A(t)^{\top}\Lambda)^{-1} \sigma(t,c(t)) Z.\end{aligned}$$

    3.  \[alg:dt\] Loop through each bank $i \in {\mathcal{N}}$:

        1.  \[alg:dt1\] If $V_i(t) > 0$, $\bar\mu_i(t) < 0$, and $\bar\sigma_i(t)^2 - 4\bar\mu_i(t)V_i(t) \geq 0$ then $${{\Delta t}}= \min\left\{{{\Delta t}}\; , \; \left(\frac{-\bar\sigma_i(t) - \sqrt{\bar\sigma_i(t)^2 - 4\bar\mu_i(t)V_i(t)}}{2\bar\mu_i(t)}\right)^2\right\}.$$

        2.  \[alg:dt4\] If $V_i(t) < 0$, $\bar\mu_i(t) \neq 0$, and $\bar\sigma_i(t)^2 - 4\bar\mu_i(t)V_i(t) \geq 0$ then $${{\Delta t}}= \min\left\{{{\Delta t}}\; , \; \left(\frac{-\bar\sigma_i(t) + \sqrt{\bar\sigma_i(t)^2 - 4\bar\mu_i(t)V_i(t)}}{2\bar\mu_i(t)}\right)^2\right\}.$$

        3.  \[alg:dt25\] If $\bar\mu_i(t) = 0$ and $V_i(t)\bar\sigma_i(t) < 0$ then ${{\Delta t}}= \min\left\{{{\Delta t}}\; , \; V_i(t)^2 / \bar\sigma_i(t)^2\right\}$.

        4.  \[alg:dt7\] If $\bar\mu_i(t)\bar\sigma_i(t) < 0$ then ${{\Delta t}}= \min\left\{{{\Delta t}}\; , \; \bar\sigma_i(t)^2 / \bar\mu_i(t)^2\right\}$.

    4.  Compute $\Delta V(t) = \bar\mu(t){{\Delta t}}+ \bar\sigma(t) \sqrt{{{\Delta t}}}$.

    5.  Define the matrix $\Lambda \in \{0,1\}^{(n+1) \times (n+1)}$ such that $$\Lambda_{ij} = \begin{cases}0 &\text{if } i = j \neq 0, \; V_i(t) > 0 \text{ or } [V_i(t) = 0, \; \Delta V_i(t) \geq 0] \\ 1 &\text{if } i = j \neq 0, \; V_i(t) < 0 \text{ or } [V_i(t) = 0, \; \Delta V_i(t) < 0] \\ 0 &\text{else} \end{cases}$$

4.  Define the matrix $\bar\Lambda \in \{0,1\}^{(n+1) \times (n+1)}$ so that $\bar\Lambda = \begin{cases}1 &\text{if } i = j \neq 0, \; V_i(t) < 0 \\ 0 &\text{else}\end{cases}$.

5.  Set $$\begin{aligned}
    c(t + {{\Delta t}}) &= c(t) + \mu(t,c(t)){{\Delta t}}+ \sigma(t,c(t))\sqrt{{{\Delta t}}}Z\\
    V(t + {{\Delta t}}) &= V(t) + \Delta V(t)\\
    A(t + {{\Delta t}}) &= \bar\Lambda\left[A(t) + \operatorname{diag}(V(t)^-)^{-1}[\dot{L}(t) - A(t) \ast (\dot{L}(t)\mathbbm{1})]{{\Delta t}}\right]\\
    &\qquad + (I - \bar\Lambda)\operatorname{diag}(\dot{L}(t)\vec{1})^{-1}\dot{L}(t).\end{aligned}$$ where $\mathbbm{1} = \{1\}^{(n+1) \times (n+1)}$ and $\ast$ denotes the element-wise multiplication operator.

6.  Increment $t = t + {{\Delta t}}$.

If $t > T$ then set $$\begin{aligned}
c(T) &= c(t-{{\Delta t}}) + \frac{c(t) - c(t-{{\Delta t}})}{{{\Delta t}}}(T - [t-{{\Delta t}}])\\
V(T) &= V(t-{{\Delta t}}) + \frac{V(t) - V(t-{{\Delta t}})}{{{\Delta t}}}(T - [t-{{\Delta t}}])\\
A(T) &= A(t-{{\Delta t}}) + \frac{A(t) - A(t-{{\Delta t}})}{{{\Delta t}}}(T - [t-{{\Delta t}}]).\end{aligned}$$

In the above event-finding algorithm for the continuous-time Eisenberg-Noe system, the main concern is that we do not increment time too far in any step so as to pass over an event (e.g., a solvent bank becoming a distressed bank). This is accomplished in the loop described in step . In particular, - guarantee that $V_i(t) + \bar\mu_i(t){{\Delta t}}+ \bar\sigma_i(t)\sqrt{{{\Delta t}}}$ is nonnegative if $V_i(t) > 0$ and nonpositive if $V_i(t) < 0$. The additional condition in guarantees that the direction of $\bar\mu_i(t){{\Delta t}}+ \bar\sigma_i(t)\sqrt{{{\Delta t}}}$ is maintained as ${{\Delta t}}$ shrinks, i.e., if ${{\Delta t}}$ is too large then the direction of the change in wealth could be impacted by choosing a smaller (and thus more accurate) step-size. While not strictly necessary, we include step  as it improves the accuracy of the algorithm.

\[rem:cont-loans\] As with the discrete-time setting discussed in Remark \[rem:loans\], we can introduce the concept of loans from a central bank to the continuous-time Eisenberg-Noe system. To do so we would need to introduce stopping times associated with each bank becoming insolvent. Notably, the receivership setting would act the same as our described continuous-time Eisenberg-Noe system after insolvencies occur. In contrast, a pure auction model would eliminate all need for continuous-time contagion. At the time of the auction a static system would be considered, e.g., the static Eisenberg-Noe clearing, based on the results of the auction; this would update the cash flow parameters for each firm going forward, but no dynamic contagion would need to be modeled.

We wish to conclude our discussion of the continuous-time Eisenberg-Noe system by providing a result on how the unique solution to the discrete-time solution converges to the continuous-time solution as ${{\Delta t}}\searrow 0$. That is, we wish to consider how the unique clearing wealths and relative exposures solving the discrete-time systems and converge to those in continuous-time Eisenberg-Noe system and as the step-size decreases to 0.

\[lemma:limiting\] Consider the setting of Corollary \[cor:discrete-cont\] and Theorem \[thm:continuous\]. Then the continuous-time clearing solutions at any time $t \in {\mathbb{T}}$ is the limit of the discrete-time solution as the step-size tends to 0, i.e., $(V(t),A(t)) = \lim_{{{\Delta t}}\searrow 0} (V(t,{{\Delta t}}),A(t,{{\Delta t}},V_t({{\Delta t}})))$ where $(V(\cdot,{{\Delta t}}),A(\cdot,{{\Delta t}}))$ satisfy and and $(V,A)$ satisfy and .

Discussion {#sec:discussion}
==========

In this section we will consider the implications of time on the clearing solutions in the Eisenberg-Noe setting. Specifically, we will focus on the continuous-time formulation, though all conclusions hold in the discrete-time setting as well. Notably, we deduce rules so as to recreate the static Eisenberg-Noe clearing solution via our continuous-time differential system, which (independently) replicates the results from [@sonin2017]. Further, we consider the implications of time dynamics on the health of the financial system by determining bounds on how different the static clearing solution and a dynamic solution might be. This demonstrates the importance of time dynamics on accurately assessing the health and wealth of the financial system.

The static model as a differential system {#sec:discussion-EN}
-----------------------------------------

Herein we will consider the case in which the relative liabilities are constant through time. That is, we consider the setting in which $dL_{ij}(s)/\sum_{k \in {\mathcal{N}}_0} dL_{ik}(s) = dL_{ij}(t)/\sum_{k \in {\mathcal{N}}_0} dL_{ik}(t)$ for all times $s,t \in {\mathbb{T}}$ and firms $i,j \in {\mathcal{N}}_0$ so long as $\sum_{k \in {\mathcal{N}}_0} dL_{ik}(s), \sum_{k \in {\mathcal{N}}_0} dL_{ik}(t) > 0$. The key implication of this assumption is that the relative exposures matrix in  can be found explicitly to equal the relative liabilities $$a_{ij}(t) = \pi_{ij} := \begin{cases} \frac{dL_{ij}(s_i)}{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(s_i)} &\text{if } s_i < \sup{\mathbb{T}}\\ \frac{1}{n} &\text{if } s_i = \sup{\mathbb{T}}, \; j \neq i\\ 0 &\text{if } s_i = \sup{\mathbb{T}}, \; j = i\end{cases}$$ for all times $t$ and banks $i,j \in {\mathcal{N}}_0$ where $s_i \in \{t \in {\mathbb{T}}\; | \; \sum_{k \in {\mathcal{N}}_0} dL_{ik}(t) > 0\}$ chosen arbitrarily strictly less than $\sup{\mathbb{T}}$ (and $s_i = \sup{\mathbb{T}}$ if the supremum is taken over the empty set).

Further, expanding and solving the differential system , we deduce that the continuous-time clearing wealths must satisfy the fixed point problem $$\label{eq:continuous-static}
V(t) = V(0) + \int_0^t dc(s) - \Pi^{\top}V(t)^-$$ at all time $t \in {\mathbb{T}}$. Therefore, if $\int_0^t dc(s) \geq \int_0^t dL(s)^{\top}\vec{1} - \int_0^t dL(s)\vec{1}$ at some time $t$, it follows that $V(t)$ are the *static* clearing wealths to the Eisenberg-Noe system with aggregated data with nominal liabilities matrix defined by $\int_0^t dL(s)$ and (incoming) external cash flow given by $\int_0^t dc(s) - \left(\int_0^t dL(s)^{\top}\vec{1} - \int_0^t dL(s)\vec{1}\right)$. Importantly, this means that, if the relative liabilities are kept constant over time, taking aggregated data and considering the static Eisenberg-Noe framework will produce the same *final* clearing wealths as the dynamic Eisenberg-Noe setting presented in this paper. However, though the set of defaulting banks is the same as in the static setting, the order of defaults need not strictly follow the order given in the fictitious default algorithm of [@EN01].

\[defn:order-default\] A bank is called a ***$k$th-order default*** in the static Eisenberg-Noe setting if it is determined to be in default in the $k$th iteration of the fictitious default algorithm (see, e.g., [@EN01 Section 3.1] or the inner loop of Algorithm \[alg:discrete\]).

We note that the first-order defaults are exactly those firms that have negative wealth even if it has no negative exposure to other firms (i.e., all other firms satisfy their obligations in full).

\[prop:order-defaults\] Let $(x,\bar{L}) \in {\mathbb{R}}_+^{n+1} \times {\mathbb{R}}^{(n+1) \times (n+1)}_+$ denote the static incoming external cash flow and nominal liabilities. Define a dynamic system over the time period ${\mathbb{T}}= [0,T]$ such that $V(0) \in [0,x]$, $dL(t) = \frac{1}{T}\bar{L}dt$, and $dc(t) = \frac{1}{T}\left(x - V(0) + \bar{L}^{\top}\vec{1} - \bar{L}\vec{1}\right)dt$. The clearing wealths at the terminal time $V(T)$ are equal to those given in the static setting. Additionally, no firm will ever recover from distress in the dynamic setting. Finally, the first $k$th-order default will occur only after the first $(k-1)$th-order default in the static fictitious default algorithm; in particular, the first firm to become distressed will be a first-order default in the static fictitious default algorithm.

The fact that the clearing wealths $V(T)$ are equal to the static Eisenberg-Noe clearing wealths (as defined in Proposition \[prop:EN-e\]) follows from and the logic given in the proof of Lemma \[lemma:limiting\]. Additionally, since $dc(t)$ is constant in time and firms are beginning in a solvent state, over time the unpaid liabilities may accumulate as a negative factor on bank balance sheets, but there is no outlet to allow for a firm to recover from distress. Finally, by definition, a $k$th-order default is only driven into distress through the failure of the $(k-1)$th-order defaults (and *not* solely by the $(k-2)$th-order defaults). Therefore, by way of contradiction, if a $k$th-order default were to occur before any $(k-1)$th-order default then such a firm must default without regard to what happens to the $(k-1)$th-order defaults, i.e., this firm must be a $(k-1)$th-order default. By this same logic, the first firm to become distressed must be a first-order default.

The notion of real defaulting times differing from the order introduced by the fictitious default algorithm of [@EN01] is unsurprising. Consider a financial system with two subgraphs that are only connected through their obligations to the societal node. By construction, the default of a firm in one subgraph will have no impact on the firms in the other subgraph. Thus we can construct a network so that all defaults in one subgraph (including higher order defaults as defined in Definition \[defn:order-default\]) occur before any first-order defaults in the other subgraph.

Notably, Proposition \[prop:order-defaults\] states that, provided the aggregate data (until the terminal time) is kept constant, the clearing wealths at the terminal time will be path-independent in this setting. We will demonstrate this with an illustrative example demonstrating this setting in a small 4 bank (plus societal node) system. In particular, we will consider the cash flows $c$ to be defined as a Brownian bridge so as to provide the appropriate aggregate data at the terminal time.

\[ex:differentialEN\] Consider a financial system with four banks, each with an additional obligation to an external societal node. Consider the time interval ${\mathbb{T}}= [0,1]$ with aggregated data such that the initial wealths are given by $V(0) = (100,1,3,2,5)^{\top}$, cash flows $dc$ are such that $\int_0^1 dc(s) = \bar{L}^{\top}\vec{1} - \bar{L}\vec{1}$, and where the nominal liabilities matrix $dL = \bar{L}dt$ is defined by $$\bar{L} = \left(\begin{array}{ccccc}0 & 0 & 0 & 0 & 0 \\ 3 & 0 & 7 & 1 & 1 \\ 3 & 3 & 0 & 3 & 3 \\ 3 & 1 & 1 & 0 & 1 \\ 3 & 1 & 2 & 1 & 0 \end{array}\right).$$ The *static* Eisenberg-Noe clearing wealths, with nominal liabilities $\bar{L}$ and external assets $V(0)$, are found to be $V(1) \approx (109.38,-6.81,-3.03,-0.32,1.62)^{\top}$. Further, from the static fictitious default algorithm, we can determine that bank 1 is a first-order default, bank 2 is a second-order default, and bank 3 is a third-order default. Consider now three dynamic settings which are differentiated only by the choice of the cash flows $dc$:

1.  Consider the deterministic setting introduced in Proposition \[prop:order-defaults\], i.e., $dc(t) = [\bar{L}^{\top}\vec{1} - \bar{L}\vec{1}]dt$ for all times $t \in {\mathbb{T}}$.

2.  Consider a Brownian bridge with low volatility, i.e., $dc(t) = \frac{\bar{L}^{\top}\vec{1} - \bar{L}\vec{1} - c(t)}{1-t}dt + dW(t)$ for vector of independent Brownian motions $W$ and with $c(0) = 0$.

3.  Consider a Brownian bridge with high volatility, i.e., $dc(t) = \frac{\bar{L}^{\top}\vec{1} - \bar{L}\vec{1} - c(t)}{1-t}dt + 5dW(t)$ for vector of independent Brownian motions $W$ and with $c(0) = 0$.

[0.45]{} ![Example \[ex:differentialEN\]: Comparison of clearing wealths under deterministic and random cash flows that aggregate to the same terminal values.[]{data-label="fig:EN"}](EN-BB0.eps "fig:"){width="\linewidth"}

 

[0.45]{} ![Example \[ex:differentialEN\]: Comparison of clearing wealths under deterministic and random cash flows that aggregate to the same terminal values.[]{data-label="fig:EN"}](EN-BB1.eps "fig:"){width="\linewidth"}

 

[0.45]{} ![Example \[ex:differentialEN\]: Comparison of clearing wealths under deterministic and random cash flows that aggregate to the same terminal values.[]{data-label="fig:EN"}](EN-BB5.eps "fig:"){width="\linewidth"}

A single sample path for each dynamic setting is provided. In each plot we reduce the equity of the societal node by 100 so that it begins with an initial wealth of 0, but more importantly so that it can easily be displayed on the same plot as the other 4 institutions. First, we point out that, as indicated by the circles at the terminal time in each plot, the terminal wealths of the continuous-time setting match up with the clearing wealths in the static model. We further note that in the deterministic setting (Figure \[fig:EN-BB0\]) and the low volatility setting (Figure \[fig:EN-BB1\]) the order of defaults is maintained. However, in the high volatility setting (Figure \[fig:EN-BB5\]) the order of defaults given by the fictitious default algorithm no longer holds.

The implications of time dynamics {#sec:discussion-time}
---------------------------------

Now we will consider the case in which the relative liabilities change over time. As in the prior discussion, we will focus on the setting in which the aggregate cash flows and interbank liabilities correspond to a static Eisenberg-Noe model. As the liabilities are now changing over time there is an inherent prioritization in the obligations due to the rolling forward of unpaid debts. Any earlier obligations are more likely to be paid, and accumulate to be paid proportionally with any new obligations. As such, by altering only the rate at which the liabilities are due, the terminal wealths and also the set of defaulting firms can be modified. Proposition \[prop:defaults\] provides analysis on which banks will always be solvent and which will always be in default at the terminal time. In particular, the results of Proposition \[prop:defaults\] show that the *static* Eisenberg-Noe model applied to aggregate data can produce a viewpoint on the health of the financial system that is either incorrectly optimistic or pessimistic; without explicitly knowing the dynamics of the cash flows and liabilities, only rough estimates can be considered. This is in contrast to, e.g., [@GY14] in which data from the European Banking Authority’s 2011 stress test was utilized to assess the health of the European financial system without time dynamics.

\[defn:order-solvent\] In the static Eisenberg-Noe setting a bank is called a ***first-order solvency*** if it has positive wealth even under the maximum negative exposure (i.e., no other firms pay at all).

Note that, by assumption, the societal node $0$ will always be a first-order solvent institution.

\[prop:defaults\] Let $(x,\bar{L}) \in {\mathbb{R}}_+^{n+1} \times {\mathbb{R}}^{(n+1) \times (n+1)}_+$ denote the static incoming external cash flow and nominal liabilities. Define a dynamic system over the time period ${\mathbb{T}}= [0,T]$ such that $V(0) \in [0,x]$, $\int_0^T dL(t) = \bar{L}$, and $\int_0^T dc(t) = x - V(0) + \bar{L}^{\top}\vec{1} - \bar{L}\vec{1}$. At time $T$, those banks that are first-order defaults in the static setting will be in default in the dynamic setting. Similarly, those banks that are first-order solvencies in the static setting will be solvent in the dynamic setting at the terminal time.

This result follows from the definition of a first-order default or solvency as such firms allow us to disregard all interbank dynamics.

To conclude this discussion, we will consider two examples with the same aggregate values as given in Example \[ex:differentialEN\]. The first example considers the case in which the nominal liabilities are shifted in time so as to have the maximum possible number of banks be solvent or, vice versa, the maximum number of banks be in default at the terminal time. The second example considers a fixed structure for the nominal liabilities in time (but non-constant relative liabilities), thus demonstrating the path-dependence of the clearing wealths on the cash flows.

\[ex:dL-dependence\] Consider the financial system described in Example \[ex:differentialEN\] over the time interval ${\mathbb{T}}= [0,1]$ with aggregated data such that the initial wealths $V(0) = (100,1,3,2,5)^{\top}$ and where the aggregate nominal liabilities matrix is defined by $\bar{L}$. Further, consider the cash flows $dc(t) = dL(t)^{\top}\vec{1} - dL(t)\vec{1}$ for all times $t \in {\mathbb{T}}$ where $dL$ is either:

1.  prioritizing the defaulting firms: $dL(t) = 5\bar{L}\left(E_0 1_{\{t \in (0.8,1]\}} + \sum_{i \in {\mathcal{N}}}E_i 1_{\{t \in (0.2(i-1),0.2i]\}}\right)$, or

2.  prioritizing society: $dL(t) = 5\bar{L}\left(E_0 1_{\{t \in [0,0.2)\}} + \sum_{i \in {\mathcal{N}}}E_i 1_{\{t \in (0.2i,0.2(i+1)]\}}\right)$

where the collection of matrices $E_i \in \{0,1\}^{(n+1) \times (n+1)}$ are such that $(E_i)_{ii} = 1$ and all other elements are set to $0$.

[0.45]{} ![Example \[ex:dL-dependence\]: Comparison of clearing wealths under different ordering of the nominal liabilities in time that aggregate to the same terminal values.[]{data-label="fig:dL"}](dL-1Default.eps "fig:"){width="\linewidth"}

 

[0.45]{} ![Example \[ex:dL-dependence\]: Comparison of clearing wealths under different ordering of the nominal liabilities in time that aggregate to the same terminal values.[]{data-label="fig:dL"}](dL-1Solvent.eps "fig:"){width="\linewidth"}

As in Figure \[fig:EN\], the circles at the terminal time in both plots denote the clearing wealths under the static Eisenberg-Noe setting. It is clear in both examples that the terminal dynamic clearing wealths now are *not* equal to the static wealths. Further, by choosing the liabilities to be introduced in the order provided we provide the settings so that only the first-order defaults, Bank 1, have negative terminal wealth (Figure \[fig:dL-1default\]) or so that only the first-order solvencies, the societal firm, have positive terminal wealth (Figure \[fig:dL-1solvent\]). In Figure \[fig:dL-1default\], we notice that firms 2 and 3 have a terminal wealth of $0$, so although they are not defaulting, they do not have any positive equity either. Further, it is clear that though all financial firms have improved their wealth given this ordering of the nominal liabilities, the societal wealth is decreased (though to a lesser amount than the aggregate improvement for the banks) in comparison to the static results. In contrast, in the second scenario in which obligations to society are first (Figure \[fig:dL-1solvent\]), the societal wealths are greater than those provided in the static setting but all banks have less wealth. Notice further that, even after the obligations to society have “ended” at time 0.2 the societal wealth still increases. This occurs as the banks in distress receive money as their incoming liabilities come due and thus they have cash to immediately transfer to cover the prior unpaid obligations to, e.g., society. Finally, this numerically verifies the results of Proposition \[prop:defaults\] and demonstrates the importance of understanding the order of obligations for an accurate measure of the health of the financial system.

\[ex:dc-dependence\] Consider the financial system described in Example \[ex:differentialEN\] over the time interval ${\mathbb{T}}= [0,1]$ with aggregated data such that the initial wealths $V(0) = (100,1,3,2,5)^{\top}$ and where the aggregate nominal liabilities matrix is defined by $\bar{L}$. Further, consider the nominal liabilities determined by $$dL(t) = \bar{L}\left(\begin{array}{l}E_0 + \frac{1}{0.237}E_1 1_{\{t \in [0.145,0.382]\}} + \frac{1}{0.178}E_2 1_{\{t \in [0.331,0.509]\}}\\ \quad + \frac{1}{0.439}E_3 1_{\{t \in [0.301,0.740]\}} + \frac{1}{0.105}E_4 1_{\{t \in [0.673,0.778]\}}\end{array}\right)$$ where the collection of matrices $E_i \in \{0,1\}^{(n+1) \times (n+1)}$ are such that $(E_i)_{ii} = 1$ and all other elements are set to $0$. Finally, consider the cash flows determined by a Brownian bridge with volatility of 2, i.e., $dc(t) = \frac{\bar{L}^{\top}\vec{1} - \bar{L}\vec{1} - c(t)}{1-t}dt + 2dW(t)$ for vector of independent Brownian motions $W$ and with $c(0) = 0$.

![Example \[ex:dc-dependence\]: Empirical distribution of the terminal societal wealths under random cash flows. The $\times$ marks the societal wealth under the static Eisenberg-Noe framework with aggregated data.[]{data-label="fig:dc"}](dc-10000.eps){width=".5\linewidth"}

Figure \[fig:dc\] depicts the empirical distribution of the terminal societal wealths under 10,000 samples of the Brownian bridge cash flows. The black curve depicts the kernel density for this empirical distribution. The $\times$ illustrates the societal wealth under the static Eisenberg-Noe framework considering the aggregated data (as provided in Example \[ex:differentialEN\]). The key takeaway of this figure is the payments to society range from 8.12 to 10.20 out of an obligated 12, i.e., society can experience anywhere from 16% to 32% shortfall in payments depending on the sample path. This also implies that society can experience anywhere from a 13.4% decrease to an 8.8% increase over the payments found under the static Eisenberg-Noe model. Similar results can be shown for the other firms in the system as well. Notably, firms 2, 3, and 4 all have simulations in which they are solvent at the terminal time and simulations in which they are defaulting on their obligations. Recall none of these three firms are first-order defaults or first-order solvencies. Empirically, firm 2 (a second-order default) is found to default in approximately 98% of the simulations; firm 3 (a third-order default) is found to default in approximately 3.6% of simulations; firm 4 (which does not default in the static setting) is found to default in just 0.03% of the provided simulations (i.e., 3 out of the 10,000 simulations). Therefore, if relative liabilities are not constant over time, the order of the cash flows can have a significant impact on the health of the system.

Conclusion {#sec:conclusion}
==========

In this paper we considered an extension of the financial contagion model of [@EN01] to allow for cash flows and obligations to be dynamic in time. We presented this model in both discrete and continuous time, thus extending the frameworks of [@CC15; @ferrara16; @KV16] which consider only discrete-time clearing. Notably, we determine conditions for existence and uniqueness of the clearing solutions under deterministic and Itô settings. In this way, we have written a dynamical system for the Eisenberg-Noe contagion model that may include an inherent prioritization scheme. Specifically, we determine that if the relative liabilities are constant over time then the dynamic Eisenberg-Noe model presented herein will reproduce the static system at the terminal time in a path-independent manner. Notably, in such a setting, we are able to determine the true defaulting order rather than the fictitious order found in the fictitious default algorithm that is widely used in computing static clearing models. If, however, the relative liabilities are not constant over time, then we determine that the static Eisenberg-Noe model may report an incorrectly optimistic or pessimistic picture of the financial system.

Three clear extensions of this model are apparent to us, and which we foresee creating further divergence between static and dynamic models. The first extension is the inclusion of illiquid assets and fire sales. In the static models, e.g., [@CFS05; @AFM16; @feinstein2015illiquid], there is no first mover advantage to liquidating assets as all firms receive the same price. However, in a dynamic model there may be advantage to liquidating early in order to receive a higher price, but which may precipitate a larger fire sale amongst the other firms. The second extension is the inclusion of contingent payments and credit default swaps. In the static setting this has recently been considered by [@banerjee2017insurance; @SSB16; @SSB16b]. By considering the network dynamics to be dependent on the history of clearing wealths, many of the difficulties reported in the static works are likely to be resolved naturally; we refer to [@banerjee2017insurance] which provides an initial discussion of this extension. The final extension, for which we believe the proposed dynamic model will be especially useful, is in considering strategic or dynamic actions by the market participants, e.g., incorporating bankruptcy costs and strategic decisions on rolling forward of debt. We feel that the continuous-time framework will be particularly suitable for these extensions as it allows us to construct unique clearing solutions without requiring strong monotonicity assumption.

Proof of results in Section \[sec:discrete\] {#sec:discrete-proofs}
============================================

We will prove this result inductively. First consider time $t = 0$. Recall from Assumption \[ass:initial\] that $V(-1) \geq 0$. The clearing wealths at time $0$ follow the fixed point equation $$V(0) = \Phi(0,V(0)) := V(-1) + c(0) - A(0,V_0)^{\top}V(0)^-.$$ Note that, by construction, $A(0,V_0)^{\top}V(0)^- \leq L(0)^{\top}\vec{1}$. Therefore any clearing solution must fall within the compact range $[V(-1) + c(0) -L(0)^{\top}\vec{1},V(-1) + c(0)] \subseteq {\mathbb{R}}^{n+1}$. It is clear from the definition that $\Phi(0,\cdot)$ is a monotonic operator, and thus there exists a greatest and least clearing solution $V^\uparrow(0) \geq V^\downarrow(0)$ by Tarski’s fixed point theorem [@Z86 Theorem 11.E], both of which must fall within this domain. Further, $a_{ij}(0,V_0) = \frac{L_{ij}}{\sum_{k \in {\mathcal{N}}_0} L_{ik}}$ (for $i \in {\mathcal{N}}$ and $j \in {\mathcal{N}}_0$) for any wealth $V(0)$ in this domain since $V(-1) + c(0) - L(0)^{\top}\vec{1} \geq -L(0) \vec{1} = -\bar p(0,V_{-1})$. We will prove uniqueness as it is done in [@EN01] by noting additionally that we can assume that the societal node will always have positive equity (i.e., $V^\downarrow(0) \geq 0$). First, we will show that the positive equities are the same for every firm no matter which clearing solution is chosen, i.e., $V_i^\uparrow(0)^+ = V_i^\downarrow(0)^+$ for every firm $i \in {\mathcal{N}}_0$. By definition $V^\uparrow(0) \geq V^\downarrow(0)$ and using $\sum_{j \in {\mathcal{N}}_0} a_{ij}(0) = 1$ for every firm $i \in {\mathcal{N}}_0$ we recover $$\begin{aligned}
\sum_{i \in {\mathcal{N}}_0} V_i^\uparrow(0)^+ &= \sum_{i \in {\mathcal{N}}_0} \left[V_i^\uparrow(0) + V_i^\uparrow(0)^-\right]\\
&= \sum_{i \in {\mathcal{N}}_0} \left[V_i(-1) + c_i(0) - \sum_{j \in {\mathcal{N}}} a_{ji}(0,V_0^\uparrow) V_j^\uparrow(0)^- + V_i^\uparrow(0)^-\right]\\
&= \sum_{i \in {\mathcal{N}}_0} \left[V_i(-1) + c_i(0)\right] - \sum_{j \in {\mathcal{N}}} V_j^\uparrow(0)^- \sum_{i \in {\mathcal{N}}_0} a_{ji}(0,V_0^\uparrow) + \sum_{i \in {\mathcal{N}}_0} V_i^\uparrow(0)^-\\
&= \sum_{i \in {\mathcal{N}}_0} \left[V_i(-1) + c_i(0)\right] = \sum_{i \in {\mathcal{N}}_0} V_i^\downarrow(0)^+.\end{aligned}$$ Therefore it must be the case that $V_i^\uparrow(0)^+ = V_i^\downarrow(0)^+$ for all firms $i \in {\mathcal{N}}_0$. Since we assume that the societal node will always have positive equity, it must be the case that $V_0^\uparrow(0) = V_0^\downarrow(0)$. Now since we assume that each node $i \in {\mathcal{N}}$ owes to the societal node, if any firm $i \in {\mathcal{N}}$ is such that $0 \geq V_i^\uparrow(0) > V_i^\downarrow(0)$ then it must be that $V_0^\uparrow(0) > V_0^\downarrow(0)$, which is a contraction.

Continuing with the inductive argument, assume that the history of clearing wealths $V_{t-1}$ up to time $t-1$ is fixed and known. The clearing wealths at time $t$ follow the fixed point equation $$V(t) = \Phi(t,V(t)) := V(t-1) + c(t) - A(t,V_t)^{\top}V(t)^- + A(t-1,V_{t-1})^{\top}V(t-1)^-.$$ Note that, by construction, $A(t,V_t)^{\top}V(t)^- \leq L(t)^{\top}\vec{1} + A(t-1,V_{t-1})^{\top}V(t-1)^-$. Therefore any clearing solution must fall within the compact range $[V(t-1) + c(t) - L(t)^{\top}\vec{1} , V(t-1) + c(t) + A(t-1,V_{t-1})^{\top}V(t-1)^-] \subseteq {\mathbb{R}}^{n+1}$. Further, $a_{ij}(t,V_t) = \frac{L_{ij} + a_{ij}(t-1,V_{t-1})V_i(t-1)^-}{\sum_{k \in {\mathcal{N}}_0} L_{ik} + V_i(t-1)^-}$ (for $i \in {\mathcal{N}}$ and $j \in {\mathcal{N}}_0$) for any wealth $V(t)$ in this domain since $V(t-1) + c(t) - L(t)^{\top}\vec{1} \geq -V(t-1)^- - L(t) \vec{1} = -\bar p(t,V_{t-1})$. Thus we can apply the same logic as in the time $0$ case to recover existence and uniqueness of the clearing wealths $V(t)$ at time $t$.

This follows immediately from the proof of Theorem \[thm:discrete\] using induction and noting that the lattice upper and lower bounds for the domain and range spaces of $\Phi(s,\cdot)$ are subsets of ${\mathcal{L}}_s^p({\mathbb{R}}^{n+1})$. Therefore any clearing solution $V(t)$ is bounded above and below by an element of ${\mathcal{L}}_t^p({\mathbb{R}}^{n+1})$ and the result is proven.

Proof of results in Section \[sec:continuous\] {#sec:continuous-proof}
==============================================

Existence and uniqueness of the clearing solutions follows from Theorem \[thm:discrete\]. To prove continuity we will employ an induction argument. To do so, we will consider the reduced domain $V: {\mathbb{T}}\times [\epsilon,\infty) \to {\mathbb{R}}^{n+1}$ for some $\epsilon > 0$. That is, we restrict the step-size ${{\Delta t}}\geq \epsilon$. As we will demonstrate that the continuity argument holds for any $\epsilon > 0$ then the desired result must hold as well. Before continuing, consider an expanded version of the recursive formulation of , i.e., $$\label{eq:discrete-Vdt-expand}
V(t,{{\Delta t}}) = V(-1) + \int_0^t dc(s) - A(t,{{\Delta t}},V_t({{\Delta t}}))^{\top}V(t,{{\Delta t}})^-
$$ for all times $t \in {\mathbb{T}}$. Fix the minimal step-size $\epsilon > 0$. Note that the relative exposures satisfy $a_{ij}(t,{{\Delta t}},V_t({{\Delta t}})) := \frac{\int_0^t dL_{ij}(s)}{\sum_{k \in {\mathcal{N}}_0} \int_0^t dL_{ik}(s)}$ for any time $t \in [0,\epsilon)$ by the assumption that $V(-1) \geq 0$. Thus we can conclude $V: [0,\epsilon) \times [\epsilon,\infty) \to {\mathbb{R}}^{n+1}$ is continuous by an application of [@feinstein2014measures Proposition A.2]. Now, by way of induction, assume that $V: [0,s) \times [\epsilon,\infty) \to {\mathbb{R}}^{n+1}$ is continuous for some $s > 0$. Again, by [@feinstein2014measures Proposition A.2], we are able to immediately conclude that $V: [0,s+\epsilon) \cap {\mathbb{T}}\times [\epsilon,\infty) \to {\mathbb{R}}^{n+1}$ is continuous. As we are able to always extend the continuity result by $\epsilon > 0$ in time, the result is proven.

1.  Consider firm $i \in {\mathcal{N}}$. By assumption we have that $a_{ij}(t)$ for $t \nearrow \tau$ solves the first order differential equation: $$\frac{da_{ij}(t)}{dt}+\frac{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(t)/dt}{V_i(t)^-}a_{ij}(t)=\frac{dL_{ij}(t)/dt}{V_i(t)^-}.$$ For sake of simplicity, let this differential equation start at time $0$ with $V_i(0) < 0$ and some initial value $a_{ij}(0)$. Then this differential equation can be solved via the integrating factor $\nu(t) := \int_0^t \frac{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(s)}{V_i(s)^-}ds $. Thus for $t \nearrow \tau$ it follow that $$a_{ij}(t)= e^{-\nu(t)}\left[\int_0^t e^{\nu(s)} \frac{dL_{ij}(s)}{V_i(s)^-} + a_{ij}(0)\right].$$ Therefore, utilizing L’Hôspital’s rule, $$\begin{aligned}
    \lim_{t \nearrow \tau} a_{ij}(t) &= \lim_{t \nearrow \tau} e^{-\nu(t)}\left[\int_0^t e^{\nu(s)} \frac{dL_{ij}(s)}{V_i(s)^-} + a_{ij}(0)\right]\\
    &= \lim_{t \nearrow \tau} \frac{e^{\nu(t)}\frac{dL_{ij}(t)}{V_i(t)^-}}{e^{\nu(t)} \frac{d}{dt} \nu(t)} = \lim_{t \nearrow \tau} \frac{dL_{ij}(t)/V_i(t)^-}{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(t)/V_i(t)^-}
    = \frac{dL_{ij}(\tau)}{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(\tau)}.\end{aligned}$$

2.  First, if $V_i(t) \geq 0$ then by construction (and the above result) it follows that $a_{ij}(t) = \frac{dL_{ij}(t)}{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(t)} \geq 0$ for any $i,j \in {\mathcal{N}}_0$ and $a_{i0}(t) \geq \delta$ by this construction. Consider now the case for $V_i(t) < 0$ and assume $a_{ij}(t) < 0$. Let $\tau = \sup\{s \leq t \; | \; V_i(s) = 0\}$. Since $a_{ij}(\tau) \in [0,1]$ by construction and the relative exposures are continuous, this implies there exists some time $s \in [\tau,t)$ such that $a_{ij}(s) = 0$. By the definition of the relative exposures, this must follow that $da_{ij}(s) \geq 0$ for any time $a_{ij}(s) \leq 0$ (with $da_{ij}(s) > 0$ if $a_{ij}(s) < 0$), thus $a_{ij}(t) < 0$ can never be reached. Further, assume $a_{i0}(t) < \delta$. By Assumption \[ass:society\], if $a_{i0}(s) \leq \frac{dL_{i0}(s)}{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(s)}$ then $da_{i0}(s) \geq 0$. In particular, if $a_{i0}(s) \leq \delta$ then $da_{i0}(s) \geq 0$ (with $da_{i0}(s) > 0$ if $a_{i0}(s) < \delta$). Thus, by the same contradiction found in the case for $j \in {\mathcal{N}}$, we are able to bound $a_{i0}(t) \geq \delta$.

3.  First, if $i = 0$ then $\sum_{j \in {\mathcal{N}}_0} a_{0j}(t) = 1$ by property that $a_{0j}(t) = \frac{1}{n}1_{\{j \neq 0\}}$ for all times $t$. Now consider $i \in {\mathcal{N}}$, if $V_i(t) \geq 0$ then by construction (and the above result) it follows that $\sum_{j \in {\mathcal{N}}_0} a_{ij}(t) = \sum_{j \in {\mathcal{N}}_0} \frac{dL_{ij}(t)}{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(t)} = 1$. Consider now the case for $V_i(t) < 0$ and let $\tau = \sup\{s \leq t \; | \; V_i(s) = 0\}$. Since $\sum_{j \in {\mathcal{N}}_0} a_{ij}(\tau) = 1$ by prior results, we will assume that $\sum_{j \in {\mathcal{N}}_0} a_{ij}(t) = 1$ to deduce $$\begin{aligned}
    \sum_{j \in {\mathcal{N}}_0} da_{ij}(t) &= \sum_{j \in {\mathcal{N}}_0} \frac{dL_{ij}(t) - a_{ij}(t) \sum_{k \in {\mathcal{N}}_0} dL_{ik}(t)}{V_i(t)^-}\\
    &= \frac{\sum_{j \in {\mathcal{N}}_0} dL_{ij}(t)}{V_i(t)^-} - \frac{\left(\sum_{j \in {\mathcal{N}}_0} a_{ij}(t)\right) \left(\sum_{k \in {\mathcal{N}}_0} dL_{ik}(t)\right)}{V_i(t)^-} = 0.\end{aligned}$$ Therefore based on the initial conditions, $a_{ij}(t)$ must evolve so that it maintains the constant row sum of $1$.

Recall that the initial values to the Eisenberg-Noe differential system are $V_i(0) > 0$ and $a_{ij}(0) = \frac{dL_{ij}(0)}{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(0)}1_{\{i \neq 0\}} + \frac{1}{n}1_{\{i = 0,\; j \neq 0\}}$ for all banks $i,j \in {\mathcal{N}}_0$. For ease of notation, consider $\tau_0 := 0$ and recursively define the stopping times $$\tau_{m+1} := \inf\{t \in (\tau_m,T] \; | \; V_i(\tau_m)V_i(t) < 0 \text{ or } [V_i(\tau_m) = 0, \; dV_i(\tau_m)V_i(t) < 0]\}.$$ That is, $\tau_m \in {\mathbb{T}}$ is the time of the $m$th change in $\Lambda(V)$. Without loss of generality, we will assume that $\tau_m = T$ if the infimum is taken over an empty set. We note that the times $\tau_m$ are all stopping times with respect to the natural filtration.

With these times, note that in particular, on the interval $(\tau_m,\tau_{m+1}]$ we can consider the set of distressed banks to be constant; to simplify, and slightly abuse, notation we can thus consider a constant matrix of distressed firms $\Lambda(\tau_m)$ in the interval $(\tau_m,\tau_{m+1}]$. We will now construct the unique strong solution forward in time over these time intervals, noting that we update $\Lambda$ and $\tau_{m+1}$ once the next event is found.

First, by construction, on $[0,\tau_1]$ there exists a unique solution to the differential system provided by $V(t) = V(0) + c(t)$ and $a_{ij}(t) = \frac{dL_{ij}(t)}{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(t)} 1_{\{i \neq 0\}} + \frac{1}{n}1_{\{i = 0,\; j \neq 0\}}$ for all banks $i,j \in {\mathcal{N}}_0$. Assume there exists a strong solution in the time interval $[0,\tau_m]$ for $\tau_m < T$. Now we want to prove the existence and uniqueness for the clearing wealths and relative exposures on the interval $(\tau_m,\tau_{m+1}]$. Expanding $dc(t)$ based on its differential form allows us to consider as $$\begin{aligned}
dV(t) &= [I-A(t)^{\top}\Lambda(\tau_m)]^{-1}(\mu(t,c(t))-[\dot{L}(t)^{\top}- A(t)^{\top}\dot{L}(t)]\vec{1})dt\\
&\quad +[I-A(t)^{\top}\Lambda(\tau_m)]^{-1}\sigma(t,c(t)) dW(t)\\
&= \bar\mu(t,c(t),A(t),V(t))dt+\bar\sigma(t,c(t),A(t),V(t))dW(t).\end{aligned}$$ Let us first consider the linear growth condition for $dV$. Utilizing the $1$-norm and where $\|\cdot\|_1^{op}$ denotes the corresponding operator norm, let $A \in {\mathbb{A}}$ and $V \in {\mathbb{R}}^{n+1}$, then $$\begin{aligned}
\|\bar\mu(t,c,A,V)\|_1 &+ \|\bar\sigma(t,c,A,V)\|_1^{op}\\
&\leq \|(I - A^{\top}\Lambda(\tau_m))^{-1}\|_1^{op} \left(\|\mu(t,c)\|_1 + \|[\dot{L}(t)^{\top}- A^{\top}\dot{L}(t)]\vec{1}\|_1 + \|\sigma(t,c)\|_1^{op}\right)\\
&\leq \sum_{k = 0}^\infty \|[A^{\top}\Lambda(\tau_m)]^k\|_1^{op} \left(\|\mu(t,c)\|_1 + \|\dot{L}(t)^{\top}\vec{1}\|_1 + \|A^{\top}\dot{L}(t)\vec{1}\|_1 + \|\sigma(t,c)\|_1^{op}\right)\\
&\leq \left(1 + \sum_{k = 1}^\infty (1-\delta)^{k-1}\right) \left(\|\mu(t,c)\|_1 + \|\dot{L}(t)^{\top}\vec{1}\|_1 + \|A^{\top}\|_1^{op} \|\dot{L}(t)\vec{1}\|_1 + \|\sigma(t,c)\|_1^{op}\right)\\
&\leq \left(1 + \frac{1}{\delta}\right) \left(\|\mu(t,c)\|_1 + \|[\dot{L}(t)^{\top}\vec{1}\|_1 + \|\dot{L}(t)\vec{1}\|_1 + \|\sigma(t,c)\|_1^{op}\right)\\
&\leq \frac{1 + \delta}{\delta} \sup_{s \in [\tau_m,\tau_{m+1}]} \left(\|\mu(s,c)\|_1 + \|\dot{L}(s)^{\top}\vec{1}\|_1 + \|\dot{L}(s)\vec{1}\|_1 + \|\sigma(s,c)\|_1^{op}\right)\\
&\leq \theta (1 + \|c\|_1)\end{aligned}$$ The second line follows from the triangle inequality and definition of the operator norm. The third line is a result of Proposition \[prop:Leontief\] and further use of the triangle inequality. The fourth line follows from Proposition \[prop:A\] and noting that, by assumption, $\Lambda_{00} = 0$. The upper bound $\theta \geq 0$ can be determined by Assumption \[ass:society\] and since all terms are continuous and being evaluated on a compact interval of time (since $\tau_{m+1} \leq T$ by definition). Further, we wish to prove $\bar\mu: {\mathbb{T}}\times {\mathbb{R}}^{n+1} \times {\mathbb{A}}\times {\mathbb{R}}^{n+1} \to {\mathbb{R}}^{n+1}$ and $\bar\sigma: {\mathbb{T}}\times {\mathbb{R}}^{n+1} \times {\mathbb{A}}\times {\mathbb{R}}^{n+1} \to {\mathbb{R}}^{(n+1) \times (n+1)}$ are jointly locally Lipschitz in $(c,A,V)$. First $(c,A,V) \in {\mathbb{R}}^{n+1} \times {\mathbb{A}}\times {\mathbb{R}}^{n+1} \mapsto \mu(t,c) - [\dot{L}(t)^{\top}- A^{\top}\dot{L}(t)] \vec{1}$ and $(c,A,V) \in {\mathbb{R}}^{n+1} \times {\mathbb{A}}\times {\mathbb{R}}^{n+1} \mapsto \sigma(t,c)$ are Lipschitz continuous by their linear (or constant) forms with Lipschitz constants that can be taken independently of time (via continuity and the compact time domain) as well as the definitions of $\mu$ and $\sigma$. It remains to show that $(c,A,V) \in {\mathbb{R}}^{n+1} \times {\mathbb{A}}\times {\mathbb{R}}^{n+1} \mapsto (I - A^{\top}\Lambda(\tau_m))^{-1}$ is Lipschitz continuous. Let $A,B \in {\mathbb{A}}$, then by the same argument as above on the bounds of the norm of the matrix inverse, $$\begin{aligned}
\|(I - A^{\top}&\Lambda(\tau_m))^{-1} - (I - B^{\top}\Lambda(\tau_m))^{-1}\|_1^{op}\\
&= \|(I - A^{\top}\Lambda(\tau_m))^{-1}[(I - B^{\top}\Lambda(\tau_m)) - (I - A^{\top}\Lambda(\tau_m))](I - B^{\top}\Lambda(\tau_m))^{-1}\|_1^{op}\\
&= \|(I - A^{\top}\Lambda(\tau_m))^{-1}[A - B]^{\top}\Lambda(\tau_m)(I - B^{\top}\Lambda(\tau_m))^{-1}\|_1^{op}\\
&\leq \|(I - A^{\top}\Lambda(\tau_m))^{-1}\|_1^{op} \|(I - B^{\top}\Lambda(\tau_m))^{-1}\|_1^{op} \|\Lambda(\tau_m)\|_1^{op} \|[A - B]^{\top}\|_1^{op}\\
&\leq \left(\frac{1 + \delta}{\delta}\right)^2 \|\Lambda(\tau_m)\|_1^{op} \|A - B\|_{\infty}^{op}\\
&\leq n\left(\frac{1 + \delta}{\delta}\right)^2 \|\Lambda(\tau_m)\|_1^{op} \|A - B\|_1^{op}.\end{aligned}$$ Thus $\bar\mu$ and $\bar\sigma$ are appropriately locally Lipschitz continuous on $[\tau_m,\tau_{m+1}]$.

Now we wish to consider the differential form for the relative exposures matrix . First, if $\Lambda_{ii}(\tau_m) = 0$ (and in particular, $\Lambda_{00}(\tau_m) = 0$ by assumption of the societal node) then $a_{ij}(t) = \frac{dL_{ij}(t)}{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(t)}1_{\{i \neq 0\}} + \frac{1}{n}1_{\{i = 0, \; j \neq 0\}}$ is the unique solution for any firm $j \in {\mathcal{N}}_0$ over all times $t \in (\tau_m,\tau_{m+1}]$. In particular, this is independent of the evolution of the wealths $V$, so we need only consider the joint differential equation between the wealths $V$ and the relative exposures $a_{ij}$ where bank $i$ is in distress between times $\tau_m$ and $\tau_{m+1}$, i.e., $\Lambda_{ii}(\tau_m) = 1$. Consider bank $i \in {\mathcal{N}}$ with $\Lambda_{ii}(\tau_m) = 1$. Therefore by construction $V_i(t) < 0$ for all $t \in (\tau_m,\tau_{m+1})$. If $V_i(\tau_{m+1}) = 0$ then from Proposition \[prop:A\], it already follows that the unique solution $a_{ij}(\tau_{m+1}) = \frac{dL_{ij}(\tau_{m+1})}{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(\tau_{m+1})}$ must hold, otherwise we can extend $V_i(t) < 0$ for $t \in (\tau_m,\tau_{m+1}]$. The differential form for all relative exposures on the interval $(\tau_m,\tau_{m+1}]$ is provided by $da_{ij}(t) = \frac{dL_{ij}(t) - a_{ij}(t)\sum_{k \in {\mathcal{N}}_0}dL_{ik}(t)}{V_i(t)^-}$. By construction $(a_{ij},V_i) \in [0,1] \times -{\mathbb{R}}_{++} \mapsto \frac{\dot{L}_{ij}(t) - a_{ij} \sum_{k \in {\mathcal{N}}_0} \dot{L}_{ik}(t)}{-V_i}$ is locally Lipschitz and satisfies a local linear growth condition (with constants bounded independent of time as above utilizing continuity of the parameters and the compact time domain).

Combining our results for the joint differential system for the cash flows $c$, clearing wealths $V$ from , and relative exposures $A$ from , we find that this system satisfies a joint local linear growth and local Lipschitz property on the interval $(\tau_m,\tau_{m+1}]$. Therefore, there exists some $\epsilon \in {\mathcal{L}}_{T}^{\infty}({\mathbb{R}}_{++})$ (such that $\tau_m + \epsilon$ is a stopping time) for which a strong solution for $(c,V,A): [\tau_m,\tau_m + \epsilon] \to {\mathbb{R}}^{n+1} \times {\mathbb{R}}^{n+1} \times {\mathbb{A}}$ exists and is unique. Using the same logic with local properties, we can continue our unique strong solution sequentially. This can be continued until the stopping time $\tau_{m+1}$ is reached (found along the path of $(c,V,A)$ as a stopping time) or this process reaches some maximal time $T^* < \tau_{m+1}$ for which a unique strong solution exists on the time interval $[\tau_m,T^*)$. First, as $c(t)$ can be calculated separately from the clearing wealths and relative exposures, we can immediately determine that $c(T^*) = \lim_{t \nearrow T^*} c(t)$ exists. Further, we note that any solution $V(t)$ must, almost surely, exist in the (almost surely) compact space $$\left[V(\tau_m) - \left(I + \frac{1+\delta}{\delta} \mathbbm{1}\right)\left(\int_{\tau_m}^t dc(s)^- + \left(L(t) - L(\tau_m)\right)\vec{1}\right) , V(\tau_m) + c(t) - c(\tau_m)\right] 
\subseteq {\mathcal{L}}_t^2({\mathbb{R}}^{n+1})$$ where $\mathbbm{1} = \{1\}^{(n+1) \times (n+1)}$. The lower bound is determined to be based on the bounding of the Leontief inverse; the upper bound follows from the continuous-time version of , i.e., $$V(t) = V(0) + c(t) - A(t)^{\top}V(t)^-.$$ Additionally, $a_{ij}(t)$ almost surely exists in the compact neighborhood $[0,1]$ by definition. Therefore $(V(T^*),A(T^*)) = \lim_{t \nearrow T^*} (V(t),A(t))$ exists by continuity of the solutions and compactness of the range space. Thus we can continue the differential equation from time $T^*$ with values $(c(T^*),V(T^*),A(T^*))$ which contradicts the nature that $T^*$ is the maximal time. Notably, if $V_i(T^*) = 0$ for some bank $i$ then it is imperative to check if $\tau_{m+1} = T^*$ to update the set of distressed banks $\Lambda$.

Therefore, by induction, there exists a unique strong solution $(V,A)$ to and on the domain $[0,\tau_m]$ for any index $m \in {\mathbb{N}}$ by use of [@Oksendal Theorem 5.2.1]. In particular this holds up to $\tau^* = \sup_{m \in {\mathbb{N}}} \tau_m$. If $\tau^* \geq T$ then the proof is complete. If $\tau^* < T$, then by the same argument as above we can find $(V(\tau^*),A(\tau^*))$ as we can bound both the wealths and relative exposures into an almost surely compact neighborhood (and a subset of ${\mathcal{L}}_{\tau^*}^2({\mathbb{R}}^{n+1})$). Therefore, as before, we can start the process again at time $\tau^*$, which contradicts the terminal nature of $\tau^*$. This concludes the proof.

Consider the dynamic Eisenberg-Noe systems as fixed point problems for processes. That is, consider the fixed point problem $(V,A) = (\Phi_V({{\Delta t}},V,A),\Phi_A({{\Delta t}},V,A))$ for ${{\Delta t}}\geq 0$ where $\Phi_V: {\mathbb{R}}_+ \times ({\mathbb{R}}^{n+1})^{\mathbb{T}}\times {\mathbb{A}}^{\mathbb{T}}\to ({\mathbb{R}}^{n+1})^{\mathbb{T}}$ and $\Phi_A: {\mathbb{R}}_+ \times ({\mathbb{R}}^{n+1})^{\mathbb{T}}\times {\mathbb{A}}^{\mathbb{T}}\to {\mathbb{A}}^{\mathbb{T}}$ defined by $$\begin{aligned}
\Phi_V({{\Delta t}},A,V) &:= \left(V(-{{\Delta t}}) + \int_0^t dc(s) - A(t)^{\top}V(t)^-\right)_{t \in {\mathbb{T}}}\\
\Phi_{a_{ij}}({{\Delta t}},A,V) &:= \begin{cases} \left(\begin{array}{l}\frac{\int_{t-{{\Delta t}}}^t dL_{ij}(s) + a_{ij}(t-{{\Delta t}})V_i(t-{{\Delta t}})^-}{\max\{\sum_{k \in {\mathcal{N}}_0} \int_{t-{{\Delta t}}}^t dL_{ik}(s) + V_i(t-{{\Delta t}})^- \; , \; V_i(t)^-\}}1_{\{i \neq 0\}}\\ \quad + \frac{1}{n}1_{\{i = 0,\; j \neq 0\}}\end{array}\right)_{t \in {\mathbb{T}}} &\text{if } {{\Delta t}}> 0 \\ \left(\begin{array}{l} \left[\frac{dL_{ij}(t)}{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(t)}1_{\{i \neq 0\}} + \frac{1}{n}1_{\{i = 0,\; j \neq 0\}}\right]1_{\{V_i(t) \geq 0\}} \\ + \left[\begin{array}{l}\frac{dL_{ij}(\tau(t))}{\sum_{k \in {\mathcal{N}}_0} dL_{ik}(\tau(t))}\\ \quad + \int_{\tau(t)}^t \frac{dL_{ij}(u) - a_{ij}(u)\sum_{k \in {\mathcal{N}}_0} dL_{ik}(u)}{V_i(u)^-}\end{array}\right]1_{\{V_i(t) < 0\}}\end{array}\right)_{t \in {\mathbb{T}}}&\text{if } {{\Delta t}}= 0\end{cases}\end{aligned}$$ where $\tau(t) := \sup\{s < t \; | \; V_i(s) \geq 0\}$ is the last time that bank $i$ was not in distress before time $t$. Note that $\Phi_V$ follows from the logic of by expanding out the recursive formulation or differential systems . By construction $$(\Phi_V(0,A,V),\Phi_A(0,A,V)) = \lim_{{{\Delta t}}\searrow 0} (\Phi_V({{\Delta t}},A_{{{\Delta t}}},V_{{{\Delta t}}}),\Phi_A({{\Delta t}},A_{{{\Delta t}}},V_{{{\Delta t}}}))$$ for paths of convergent relative exposure matrices $A_{{{\Delta t}}} \to A \in {\mathbb{A}}^{\mathbb{T}}$ and wealths processes $V_{{{\Delta t}}} \to V \in ({\mathbb{R}}^{n+1})^{\mathbb{T}}$ (in the product topologies). Thus by the uniqueness of the discrete-time and continuous-time clearing solutions (Corollary \[cor:discrete-cont\] and Theorem \[thm:continuous\]) and an application of [@feinstein2014measures Proposition A.2], the proof is completed.

\[prop:Leontief\] For any relative exposure matrix $A \in {\mathbb{A}}$ and any distress matrix $\Lambda \in \{0,1\}^{(n+1) \times (n+1)}$ such that $\Lambda_{00} = 0$ and $\Lambda_{ij} = 0$ for $i \neq j$, the matrix $I - A^{\top}\Lambda$ is invertible with Leontief form, i.e., $(I-A^{\top}\Lambda)^{-1}=\sum_{k=0}^\infty(A^{\top}\Lambda)^k$.

By inspection, for any $A \in {\mathbb{A}}$, $(I-A^{\top}\Lambda)(I+A^{\top}(I-\Lambda A^{\top})^{-1} \Lambda)=I$, i.e., the form of the inverse is provided by $I + A^{\top}(I - \Lambda A^{\top})^{-1} \Lambda$. We refer to [@feinstein2017sensitivity Theorem 2.6] for a detailed proof that $(I - \Lambda A^{\top})^{-1}$ is nonsingular and is provided by the Leontief inverse. Therefore, by construction $$\begin{aligned}
(I-A^{\top}\Lambda)^{-1}&=(I+A^{\top}(I-\Lambda A^{\top})^{-1} \Lambda) 
= I+ A^{\top}\left(\sum_{k=0}^\infty (\Lambda A^{\top})^k\right) \Lambda\\ 
&= I + \sum_{k=0}^\infty A^{\top}\Lambda (A^{\top})^k \Lambda^k 
= I + \sum_{k=0}^\infty (A^{\top}\Lambda)^{k+1}
=\sum_{k=0}^\infty(A^{\top}\Lambda)^k.\end{aligned}$$

[^1]: Washington University in St. Louis, Department of Electrical and Systems Engineering, St. Louis, MO 63130, USA.

[^2]: University of California Santa Barbara, Department of Statistics & Applied Probability, Santa Barbara, CA 93106. Part of this work was undertaken while Alex Bernstein was at Washington University.

[^3]: Washington University in St. Louis, Department of Electrical and Systems Engineering, St. Louis, MO 63130, USA.
