Dynamic Clearing and Contagion in Financial Networks
aa r X i v : . [ q -f i n . M F ] M a y Dynamic clearing and contagion in financial networks
Tathagata Banerjee ∗ Alex Bernstein † Zachary Feinstein ‡ May 29, 2018
Abstract
In this paper we will consider a generalized extension of the Eisenberg-Noe model of finan-cial contagion to allow for time dynamics in both discrete and continuous time. Derivation andinterpretation of the financial implications will be provided. Emphasis will be placed on thecontinuous-time framework and its formulation as a differential equation driven by the oper-ating cash flows. Mathematical results on existence and uniqueness of firm wealths under thediscrete and continuous-time models will be provided. Finally, the financial implications of timedynamics will be considered. The focus will be on how the dynamic clearing solutions differfrom those of the static Eisenberg-Noe model.
Keywords:
Systemic risk; financial contagion; financial network; dynamic network
Financial networks and the contagion of bank failures have been widely studied beginning withthe seminal work on financial payment networks by Eisenberg & Noe [14]. The 2007-2009financial crisis and credit crunch showed the severe impacts that systemic crises can have onthe financial sector and the economy as a whole. As the costs of such cascading events istremendous, the modeling of such events is imperative. Recently there have been significantstudies on modeling financial systemic risk and financial contagion. Two major classes of modelsexist for systemic risk, i.e., those based on network models from [14] and those based on a meanfield approach [25, 10]. Notably, the network model approach generally is considered in only astatic, 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 [14] thusclosing the gap between these two streams of literature.Interbank networks were studied first in [14] to model the spread of defaults in the finan-cial system. In the Eisenberg-Noe framework, financial firms must satisfy their liabilities bytransferring assets. One firm being unable to meet its liabilities due to a shortfall of assetscan cause other firms to default on some of their liabilities as well, causing a cascading fail-ure in the financial system. The existence and uniqueness of the clearing payments of thisbaseline model was proven in [14]. That paper additionally provides methods for numericallycomputing the realized interbank payments. This baseline model has been extended in multipledirections, including bankruptcy costs, cross-holdings, and fire sales. We refer to [44, 41] forreviews of the prior literature. In regards to bankruptcy costs in financial networks, we referto [16, 37, 15, 30, 44, 8, 43]. Cross-holdings have been studied in [16, 15, 44, 31]. Fire sales ∗ Washington University in St. Louis, Department of Electrical and Systems Engineering, St. Louis, MO 63130,USA. † University of California Santa Barbara, Department of Statistics & Applied Probability, Santa Barbara, CA93106. Part of this work was undertaken while Alex Bernstein was at Washington University. ‡ Washington University in St. Louis, Department of Electrical and Systems Engineering, St. Louis, MO 63130,USA. [email protected] or a single (representative) illiquid asset have been studied in [12, 35, 27, 2, 11, 44, 1] and formultiple illiquid assets in [18, 20, 19]. These network models have been implemented by centralbanks and regulators for stress testing of and studying cascading failures in the banking systemsunder their jurisdiction, see, e.g., [3, 32, 6, 17, 42, 26].Mean field models have also been considered for studying financial contagion and systemicrisk. [25] provides a model of agents who revert to the ensemble mean to provide understandingof “systemic risk events” in which many firms fail. Similar mean field diffusion models withoutcontrols were studied in, e.g., [24, 28, 29]. In contrast, mean field and stochastic games havebeen proposed for the study of systemic risk in, e.g., [10, 9]. In such models the firms are allowedto borrow from (or lend to) a central bank, the amount of which is optimized to minimize aquadratic cost function. Thus the choice of borrowing and lending provides an optimal controlproblem beyond the simpler mean field model of [25]. [34] proposes a separate particle systemmodel with mean field interactions.The current work will focus on adding the time dynamics, which make the mean field mod-els attractive, to the interbank network approach. In fact, the conclusion of [14] provides adiscussion of future extensions, one of which is the inclusion of multiple clearing dates. Thishas been studied directly in [7, 23]. Additionally, [33] considers a similar approach to modelfinancial networks with multiple maturities. [19] further provides another approach to financialnetworks with multiple maturities by considering each clearing date as a different asset. All ofthese works, however, only consider clearing at discrete times. [40] presents a continuous-timeclearing model that exactly replicates the static Eisenberg-Noe framework. In this work we willpresent both discrete and continuous-time clearing models. However, our emphasis will be onthe derivation and the characterization of the continuous-time model. This in part is motivatedby the prospect of unification with the mean-field models as well as traditional financial mod-els which typically employ continuous-time models. Additionally, as we will demonstrate, thecontinuous-time framework no longer requires monotonicity for existence and uniqueness whichis generally assumed for static and discrete-time systems. This is valuable for future worksthat may model network formation and payments as a non-cooperative game; such games maynot satisfy the strong monotonicity assumptions usually considered in static and discrete-timesystems, but would likely satisfy the sufficient conditions for the continuous-time framework.The organization of this paper is as follows. In Section 2 we will provide a review of thestatic Eisenberg-Noe framework. Of particular interest, in this section, we consider the clearingto be in terms of the equity and losses of the firms, as considered in, e.g., [43, 5] rather thanpayments as originally studied in [14]. In Section 3 we propose a discrete-time formulation forthe Eisenberg-Noe model. In discrete time we provide results on existence and uniqueness, aswell as a numerical algorithm based on the fictitious default algorithm of [14]. We then extendour model to a continuous-time setting in Section 4. For continuous time we consider existenceand uniqueness of the clearing solutions, and a numerical algorithm for finding sample pathsof this clearing solution, under cash flows modeled by Itô processes. We additionally provideconditions for the discrete-time setting to converge to the continuous-time solution as the timestep limits to 0. Section 5 provides discussion on the financial implications of time dynamics ininterbank networks. In particular, we find that the static Eisenberg-Noe clearing solution canbe recovered in the continuous-time setting by choosing the network parameters precisely. Thisallows for a notion of determining the true order of defaults as opposed to the fictitious defaultorder discussed in the static literature based on [14]. However, if the continuous-time networkparameters 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 ofdefaulting and solvent institutions can be altered by rearranging the timing of obligations. Assuch, using the static Eisenberg-Noe framework for stress testing may result in an incorrectassessment of the health of the financial system. The proofs of the main results are provided inthe Appendix. Static clearing systems
We begin with some simple notation that will be consistent for the entirety of this paper. Let x, y ∈ R n for some positive integer n , then x ∧ y = (min( x , y ) , min( x , y ) , . . . , min( x n , y n )) ⊤ ,x − = − ( x ∧ , and x + = ( − x ) − . Further, to ease notation, we will denote [ x, y ] := [ x , y ] × [ x , y ] × . . . × [ x n , y n ] ⊆ R n to be the n -dimensional compact interval for y − x ∈ R n + . Similarly,we will consider x ≤ y if and only if y − x ∈ R n + .Throughout this paper we will consider a network of n financial institutions. We will denotethe set of all banks in the network by N := { , , . . . , n } . Often we will consider an additionalnode , which encompasses the entirety of the financial system outside of the n banks; this node will also be referred to as society or the societal node. The full set of institutions, includingthe societal node, is denoted by N := N ∪ { } . We refer to [22, 30] for further discussion ofthe meaning and concepts behind the societal node.We will be extending the model from [14] in this paper. In that work, any bank i ∈ N mayhave obligations L ij ≥ to any other firm or society j ∈ N . We will assume that no firm hasany obligations to itself, i.e., L ii = 0 for all firms i ∈ N , and the society node has no liabilitiesat all, i.e., L j = 0 for all firms j ∈ N . Thus the total liabilities for bank i ∈ N is givenby ¯ p i := P j ∈N L ij ≥ and relative liabilities π ij := L ij ¯ p i if ¯ p i > and arbitrary otherwise;for simplicity, in the case that ¯ p i = 0 , we will let π ij = n for all j ∈ N \{ i } and π ii = 0 toretain the property that P j ∈N π ij = 1 . On the other side of the balance sheet, all firms areassumed to begin with some amount of external assets x i ≥ for all firms i ∈ N . The resultant clearing payments , under a no priority of payments assumption, satisfy the fixed point problemin payments p ∈ [0 , ¯ p ] p = ¯ p ∧ (cid:0) x + Π ⊤ p (cid:1) . (1)That is, each bank pays the minimum of what it owes ( ¯ p i ) and what it has ( x i + P j ∈N π ji p j ).The resultant vector of wealths for all firms is given by V = x + Π ⊤ p − ¯ p. (2)Noting that payments can be written as a simple function of the wealths ( p = ¯ p − V − ), weprovide the following proposition. We refer also to [43, 5, 4] for similar notions of utilizingclearing wealth instead of clearing payments. Proposition 2.1.
A vector p ∈ [0 , ¯ p ] is a clearing payments in the Eisenberg-Noe setting (1) ifand only if p = [¯ p − V − ] + for some V ∈ R n +1 satisfying the following fixed point problem V = x + Π ⊤ [¯ p − V − ] + − ¯ p. (3) Vice versa, a vector V ∈ R n +1 is a clearing wealths (i.e., satisfying (3) ) if and only if V isdefined as in (2) for some clearing payments p ∈ [0 , ¯ p ] as defined in the fixed point problem (1) .Proof. We will prove the first equivalence only, the second follows similarly.Let p ∈ [0 , ¯ p ] be a clearing payment vector. Define the wealth vector V by (2), then it isclear that V − = ¯ p − p by definition as well, i.e., p = ¯ p − V − ≥ . Thus from (2) we immediatelyrecover that the wealth vector V must satisfy (3).Let p = [¯ p − V − ] + for some wealth vector V ∈ R n +1 satisfying (3). By construction we find p = [¯ p − V − ] + = ¯ p − (cid:0) x + Π ⊤ [¯ p − V − ] + − ¯ p (cid:1) − = ¯ p − (cid:0) x + Π ⊤ p − ¯ p (cid:1) − = ¯ p ∧ (cid:0) x + Π ⊤ p (cid:1) . We note that ¯ p ≥ (cid:0) x + Π ⊤ [¯ p − V − ] + − ¯ p (cid:1) − can be shown trivially. ue to the equivalence of the clearing payments and clearing wealths provided in Proposi-tion 2.1, we are able to consider the Eisenberg-Noe system as a fixed point of equity and lossesrather than payments. In [14] 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 thereexists a unique clearing solution in the Eisenberg-Noe framework so long as L i > for all firms i ∈ N . We will take advantage of this result later in this paper. This is a reasonable assumption(as discussed in, e.g., [30]) as obligations to society include, e.g., deposits to the banks. Consider now a discrete set of clearing times T , e.g., T = { , , . . . , T } for some (finite) terminaltime T < ∞ or T = N . Such a setting is presented in [7]. For processes we will use thenotation from [13] such that the process Z : T → R n has value of Z ( t ) at time t ∈ T and history Z t := ( Z ( s )) ts =0 .In this setting, we will consider the external (incoming) cash flow x : T → R n +1+ and nominalliabilities L : T → R ( n +1) × ( n +1)+ to be functions of the clearing time, i.e., as assets and liabilitieswith different maturities. The external cash in-flows and nominal liabilities can explicitly dependon the clearing results of the prior times (i.e., x ( t, V t − ) and L ( t, V t − ) ) without affecting theexistence and uniqueness results we present, but for simplicity of notation we will focus on thecase where the external assets and nominal liabilities are independent of the health and wealthof the firms. Throughout we are considering the discounted cash flows and liabilities so as tosimplify notation.In contrast to the static Eisenberg-Noe framework, herein we need to consider the resultsof the prior times. In particular, if firm i has positive equity at time t − (i.e., V i ( t − > )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 − (i.e., V i ( t − < ) then the debts thatthe 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 currentunpaid liabilities may be paid at a future time, but before the terminal time. That is, a firmcan be considered in distress at a time if it is unable to satisfy its obligations at that time, butonly defaults if it has negative wealth at the terminal time. Thus in this paper we primarilyfocus on the intra-day dynamics rather than the inter-day dynamics. See Figure 1b for a stylized(snapshot of the) balance sheet example for a firm that has positive wealth at time that rollsforward to time . The full (actualized) balance sheet for this example with only those twotime periods is displayed in Figure 1a; we note that the full balance sheet as depicted considersactualized payments rather than the book value of the obligations. Remark 3.1.
To incorporate the inter-day dynamics in this framework we can “zero out” afirm before the terminal date if it is deemed to default in much the same as in [4]. A broaderframework for dealing with various default mechanisms is discussed in Remark 3.7. We canfurther consider the Nash game in which firms decide if they will allow debts to be rolledforward in time. In such a setting, if we include a delay for payment due to, e.g., bankruptcycourt so that defaulting firms do not pay any obligations until after the terminal time T , thenthe optimal strategy for all firms (up until the terminal time T ) would be to always allow otherfirms to roll all debts forward so as to maximize payments. Assumption 3.2.
Before the time of interest, all firms are solvent and liquid. That is, V i ( − ≥ for all firms i ∈ N . alance SheetAssets Liabilities Cash-Flow @ t = 0 x i (0) Cash-Flow @ t = 1 x i (1) Interbank @ t = 0 P nj =1 π ji (0) p j (0) Interbank @ t = 1 P nj =1 π ji (1) p j (1) Cash-Flow @ t = 0 P nj =1 L ij (0) Cash-Flow @ t = 1 P nj =1 L ij (1) Capital V i (1) (a) Stylized actualized balance sheet for firm i with two time periods. Balance Sheet @ t = 0 Assets Liabilities
Cash-Flow x i (0) Interbank P nj =1 π ji (0) p j (0) Cash-Flow P nj =1 L ij (0) Capital V i (0) Balance Sheet @ t = 1 Assets Liabilities
Cash-Flow x i (1) Carry-Forward V i (0) + Interbank P nj =1 π ji (1) p j (1) Cash-Flow P nj =1 L ij (1) Carry-Forward V i (0) − = 0 Capital V i (1) (b) Stylized “snapshot” of actualized balance sheet for firm i at times and . Figure 1: Comparison of the full balance sheet to the snapshot of maturities utilized for Section 3.5 e can now construct the total liabilities and relative liabilities at time t ∈ T as ¯ p i ( t, V t − ) := X j ∈N L ij ( t ) + V i ( t − − π ij ( t, V t − ) := L ij ( t )+ π ij ( t − ,V t − ) V i ( t − − ¯ p i ( t,V t − ) if ¯ p i ( t, V t − ) > n if ¯ p i ( t, V t − ) = 0 , j = i if ¯ p i ( t, V t − ) = 0 , j = i ∀ i, j ∈ N . In this way, coupled with the accumulation of positive equity over time, the clearing wealthsmust satisfy the following fixed point problem in time t wealths: V ( t ) = V ( t − + + x ( t ) + Π( t, V t − ) ⊤ (cid:2) ¯ p ( t, V t − ) − V ( t ) − (cid:3) + − ¯ p ( t, V t − ) . (4)That is, all firms have a clearing wealth that is the summation of their positive equity at theprior time, the new incoming external cash flow, and the payments made by all other firmsminus the total obligations of the firm (including the prior unpaid liabilities). In this way wecan construct the wealths of firms forward in time. This can be considered a discrete-timeextension of (3).We now wish to consider a reformulation of (4). To accomplish this, we consider a processof cash flows c and functional relative exposures A . These we define by c ( t ) := x ( t ) + L ( t ) ⊤ ~ − L ( t ) ~ a ij ( t, V t ) := ( π ij ( t, V t − ) if ¯ p i ( t, V t − ) ≥ V i ( t ) − L ij ( t )+ π ij ( t − ,V t − ) V i ( t − − V i ( t ) − if ¯ p i ( t, V t − ) < V i ( t ) − ∀ i, j ∈ N . (5)In the above, ~ , , . . . , ⊤ ∈ R n is the vector of ones. Here we consider c ( t ) = x ( t ) + L ( t ) ⊤ ~ − L ( t ) ~ ∈ R n +1 to be the vector of book capital levels at time t , i.e., the new wealth of eachfirm assuming all other firms pay in full. We wish to note that the new total liabilities are givenby L ( t ) ~ and the new incoming interbank obligations are given by L ( t ) ⊤ ~ . 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 : T × R n +1 → [0 , ( n +1) × ( 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 ∈ T . This isin contrast to Π , the relative liabilities, in that it endogenously imposes the limited exposuresconcept. In this work the two notions will generally coincide, but for mathematical simplicitywe introduce this relative exposure matrix. For the equivalence we seek, we define the relativeexposures so that L ( t ) ⊤ ~ A ( t − , V t − ) ⊤ V ( t − − − A ( t, V t ) ⊤ V ( t ) − = Π( t, V t − ) ⊤ [¯ p ( t, V t − ) − V ( t ) − ] + for any V ( t ) ∈ R n +1 . This formulation is such that if the positive part were removed from theright hand side, the relative exposures A would be defined exactly as the relative liabilities Π by construction. In particular, we will define the relative exposures element-wise and pointwiseso as to encompass the limited exposures as in (5). If ¯ p i ( t, V t − ) > then we can simplify thisfurther as a ij ( t, V t ) = L ij ( t )+ a ij ( t − ,V t − ) V i ( t − − max { ¯ p i ( t,V t − ) ,V i ( t ) − } .Using the notation and terms above we can rewrite (4) with respect to the cash flows c andrelative exposures A as V ( t ) = V ( t − + + x ( t ) + Π( t, V t − ) ⊤ [¯ p ( t, V t − ) − V ( t ) − ] + − ¯ p ( t, V t − )= V ( t − + + x ( t ) + L ( t ) ⊤ ~ A ( t − , V t − ) ⊤ V ( t − − − A ( t, V t ) ⊤ V ( t ) − − L ( t ) ~ − V ( t − − = V ( t −
1) + x ( t ) + L ( t ) ⊤ ~ A ( t − , V t − ) ⊤ V ( t − − − A ( t, V t ) ⊤ V ( t ) − − L ( t ) ~ V ( t −
1) + c ( t ) − A ( t, V t ) ⊤ V ( t ) − + A ( t − , V t ) ⊤ V ( t − − . (6) or 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 (6) and (5) for wealths and relative exposures.With this setup we now wish to extend the existence and uniqueness results of [14] to discretetime. Theorem 3.3.
Let ( c, L ) : T → R n +1 × R ( n +1) × ( n +1)+ define a dynamic financial network suchthat every bank has cash flow at least at the level dictated by nominal interbank liabilities, i.e., c i ( t ) ≥ P j ∈N L ji ( t ) − P j ∈N L ij ( t ) , and so that every bank owes to the societal node at alltimes t ∈ T , i.e., L i ( t ) > for all banks i ∈ N and times t ∈ T . Under Assumption 3.2, thereexists a unique solution of clearing wealths V : T → R n +1 to (6) . Remark 3.4.
The assumption that all firms have obligations to the societal node at all times t ∈ T guarantees that the financial system is a “regular network” (see [14, Definition 5]) at alltimes.The analysis of the discrete-time framework can be extended to a probabilistic setting overthe filtered probability space (Ω , F , ( F ( t )) t ∈ T , P ) . That is, we can consider the clearing wealthsin the same manner assuming the cash flow c : T × Ω → R n +1 and nominal liabilities L : T × Ω → R ( n +1) × ( n +1)+ be adapted processes. Let L t ( R m ) be the space of F t -measurablerandom vectors in R m . Let L pt ( R m ) ⊆ L t ( R m ) for p ∈ (0 , ∞ ] be the space of equivalence classesof F t -measurable functions X : Ω → R m such that k X k p := (cid:16)R Ω pP mk =1 X k ( ω ) d P (cid:17) /p < ∞ for p < ∞ and k X k ∞ := ess sup ω ∈ Ω pP mk =1 X k ( ω ) for p = ∞ . The following corollary considersthe boundedness and measurability properties of the discrete-time clearing wealths. Thoughwe will not utilize this discrete-time result in this paper, we consider it important to discussrandom events to more closely match reality. Further, this result will implicitly appear in theconstruction and analysis of the continuous-time Eisenberg-Noe formulation of the next section. Corollary 3.5.
Consider the setting of Theorem 3.3 where the random network parameters ( c, L ) adapted to the filtered probability space (Ω , F , ( F ( t )) t ∈ T , P ) . If c ( s ) ∈ L ps ( R n +1 ) and L ( s ) ∈ L ps ( R ( n +1) × ( n +1)+ ) for all times s ≤ t for some p ∈ [0 , ∞ ] , then the unique clearingsolution at time t has finite p -norm, i.e., V ( t ) ∈ L pt ( R n +1 ) . With the construction of the existence and uniqueness of the solution we now want toemphasize the fictitious default algorithm from [14] to construct this clearing wealths vectorover time. This algorithm is presented for the deterministic setting; if a stochastic setting isdesired then Algorithm 3.6 provides a method for computing a single sample path. We notethat at each time t this algorithm takes at most n iterations. Thus with a terminal time T , thisalgorithm will construct the full clearing solution over T in nT iterations. Algorithm 3.6.
Under the assumptions of Theorem 3.3 in a deterministic setting the clearingwealths process V : T → R n +1 can be found by the following algorithm. Initialize t = − and V ( − ≥ as a given. Repeat until t = max T :(i) Increment t = t + 1 .(ii) Initialize k = 0 , V = V ( t −
1) + c ( t ) , and D = ∅ . Repeat until convergence:(a) Increment k = k + 1 ;(b) Denote the set of insolvent banks by D k := (cid:8) i ∈ { , , ..., n } | V k − i < (cid:9) .(c) If D k = D k − then terminate and set V ( t ) = V k − .(d) Define the matrix Λ k ∈ { , } n × n so that Λ kij = ( if i = j ∈ D k else .(e) Define V k = ( I − Π( t, V t − ) ⊤ Λ k ) − (cid:0) V ( t −
1) + c ( t ) + A ( t − , V t − ) ⊤ V ( t − − (cid:1) . Remark 3.7.
Note that in the construction of V k in step (iie) of the fictitious default algorithmwe utilize the relative liabilities Π( t, V t − ) in the matrix inverse rather than the relative exposures ( t, ( V t − , V k )) . This has the added benefit that this definition of V k is not a fixed pointproblem, which it would be if the relative exposures matrix at time t were considered. Thischange is possible since, as discussed in the proof of Theorem 3.3, any clearing solution must bein the domain so that the relative liabilities and exposures coincide. This additionally providesthe invertibility of this matrix using standard input-output results as discussed in [14, 21].We wish to finish up our discussion of the discrete-time Eisenberg-Noe framework by con-sidering some extensions involving loans. Remark 3.8.
The theoretical framework presented in this paper can be easily extended toincorporate the concepts of loans until some (deterministic) insolvency condition is hit. Inparticular, we will consider loans made from a central bank or lender of last resort who we willassume are part of the societal node . From this perspective we consider three cases that afirm might be in: • solvent and liquid in which case the firm has positive equity and pays off its obligationsin full; • solvent and distressed in which case the firm has negative equity, but receives anovernight loan (with interest rate set at the risk-free rate for simplicity) to cover all obli-gations 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 solvencyfunction. We will assume that once a firm is deemed insolvent it can never recover to solvencyagain. Two possible systems for considering insolvent firms are:(i) Receivership:
In such a system, when a firm is deemed insolvent it is placed in receiver-ship so that obligations are payed out on a first-come first-serve basis.(ii)
Auctions:
In such a system, when a firm is deemed insolvent its future assets are auctionedoff 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 toconsider c ( t, V t − ) and L ( t, V t − ) to truly consider this case. We refer to [7] for a detaileddiscussion of the auction model for insolvency. The auction system can be interpretedas an internal mechanism for determining bankruptcy costs in contrast to the exogenousparameter in, e.g., [37].The existence and uniqueness of the clearing solutions in these scenarios require an additionalmonotonicity property; we can use the notion a speculative system from [4] to get the desiredresults. This condition encodes the notion that a firm does not benefit from any firm’s distress. Consider now a continuous set of clearing times T , e.g., T = [0 , T ] for some (finite) terminaltime T < ∞ or T = R + . As before, for processes we will use the notation from [13] such thatthe process Z : T → R n has value of Z ( t ) at time t ∈ T and history Z t := ( Z ( s )) s ∈ [0 ,t ] . We willnow construct an extension of the continuous-time setting of [40] in that we allow for liabilitiesto 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 networkparameters of cash flows and nominal liabilities. Instead of considering c ( t ) to be the net cashflow at time t ∈ 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 attime t ; we note that by assumption dL ij ( t ) ≥ for all firms i, j ∈ N as, without any paymentsmade, total liabilities should accumulate over time. Our main result in this section (Theorem 4.5)provides existence and uniqueness of the clearing wealths driven by ( dc, dL ) when c ( t ) = R t dc ( s ) is an Itô process and L ( t ) = R t dL ( s ) is deterministic and continuous (e.g., dL does not include ny 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 additionallyfunctions of the wealths V . For simplicity, in this section we will restrict ourselves so that theparameters are independent of the current wealths. In order to construct a continuous-timedifferential system, we will consider again the discrete-time setting with explicit time steps ∆ t . Assumption 4.1.
The cash flows c are defined by the Itô stochastic differential equation dc ( t ) = µ ( t, c ( t )) dt + σ ( t, c ( t )) dW ( t ) for ( n + 1) -vector of Brownian motions W over somefiltered probability space (Ω , F , ( F t ) t ∈ T , P ) . Additionally, the drift and diffusion functions µ : T × R n +1 → R n +1 and σ : T × R n +1 → R ( n +1) × ( n +1) are jointly continuous and satisfy thelinear growth and Lipschitz continuous conditions, i.e., there exist constants C, D > such thatfor all times t ∈ T and cash flows c, d ∈ R n +1 k µ ( t, c ) k + k σ ( t, c ) k op ≤ C (1 + k c k ) k µ ( t, c ) − µ ( t, d ) k + k σ ( t, c ) − σ ( t, d ) k op ≤ D k c − d k where k · k is the 1-norm and k · k op is the corresponding operator norm. The nominal liabilities L : T → R ( n +1) × ( n +1)+ are deterministic and twice differentiable; for notation we will define dL ( t ) = ˙ L ( t ) dt and d L ( t ) = ¨ L ( t ) dt . Further, the relative liabilities to society is bounded frombelow by a level δ > , i.e., inf t ∈ T dL i ( t ) P k ∈N dL ik ( t ) = δ > for all banks i ∈ N . We remark that the assumption on the cash flows can be relaxed so long as the stochasticdifferential equation has a unique strong solution on T and µ, σ satisfy a local linear growthcondition and are locally Lipschitz. This relaxation will be applied in Examples 5.3 and 5.7.In the prior section on a discrete-time model for clearing wealths, we implicitly assumeda constant time-step between each clearing date of ∆ t = 1 throughout. In order to con-struct a continuous-time clearing model we will begin by making a discrete-time model withan explicit ∆ t > term. In fact, this is immediate from the prior construction with a mi-nor alteration to the cash flow term. Herein we construct the net cash flow at time t to begiven by ∆ c ( t, ∆ t ) := R tt − ∆ t dc ( s ) and the nominal liabilities at time t are similarly providedby ∆ L ( t, ∆ t ) := R tt − ∆ 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 < ). The choice of notation for ∆ c and ∆ L are tomake explicit the “change” inherent in the construction.With these parameters we can construct the ∆ t -discrete-time clearing process V ( t, ∆ t ) andexposure matrix A ( t, ∆ t, V t (∆ t )) by: V ( t, ∆ t ) = V ( t − ∆ t, ∆ t ) + ∆ c ( t, ∆ t ) − A ( t, ∆ t, V t (∆ t )) ⊤ V ( t, ∆ t ) − + A ( t − ∆ t, ∆ t, V t − ∆ t (∆ t )) ⊤ V ( t − ∆ t, ∆ t ) − (7) a ij ( t, ∆ t, V t (∆ t )) = ∆ L ij ( t, ∆ t ) + a ij ( t − ∆ t, ∆ t, V t − ∆ t (∆ t )) V i ( t − ∆ t, ∆ t ) − max { P k ∈N ∆ L ik ( t, ∆ t ) + V i ( t − ∆ t, ∆ t ) − , V i ( t, ∆ t ) − } { i =0 } + 1 n { i =0 , j =0 } ∀ i, j ∈ N . (8)Here we assume that V ( t ) = V ( − ≥ for every time t < as in Assumption 3.2. Thisconstruction can be computed either in continuous time t ∈ T with sliding intervals of size ∆ t or at the discrete times t ∈ { , ∆ t, ..., T } . The existence and uniqueness of this system followexactly as in Theorem 3.3 under Assumption 4.1. Corollary 4.2.
Let ( dc, dL ) : T → R n +1 × R ( n +1) × ( n +1)+ define a dynamic financial networksatisfying Assumption 4.1 such that every bank has cash flow at least at the level dictated bynominal interbank liabilities, i.e., ∆ c i ( t, ∆ t ) ≥ P j ∈N ∆ L ji ( t, ∆ t ) − P j ∈N ∆ L ij ( t, ∆ t ) for allbanks i ∈ N , times t ∈ T , and step-sizes ∆ t > . Under Assumption 3.2, there exists a uniquesolution of clearing wealths V : T × R ++ → R n +1 to (7) . Further, the clearing wealths are jointlycontinuous in time and step-size. ow we want to consider the limiting behavior of this discrete-time system as ∆ t tends to0. To do so, first, we will consider the formulation of the relative exposures a ij from bank i to j . From Corollary 4.2 and Assumption 4.1, we know that for any time t ∈ T and bank i ∈ N it must follow that P k ∈N ∆ L ik ( t, ∆ t ) + V i ( t − ∆ t, ∆ t ) − ≥ V i ( t, ∆ t ) − for ∆ t > small enoughdue to the joint continuity of the wealths in time and step-size. Thus in the limiting case, as ∆ t ց , we find that we can consider the relative liabilities rather than the relative exposures,i.e., for ∆ t small enough a ij ( t, ∆ t, V t (∆ t )) = ∆ L ij ( t, ∆ t ) + a ij ( t − ∆ t, ∆ t, V t − ∆ t (∆ t )) V i ( t − ∆ t, ∆ t ) − P k ∈N ∆ L ik ( t, ∆ t ) + V i ( t − ∆ t, ∆ t ) − { i =0 } + 1 n { i =0 , j =0 } ∀ i, j ∈ N . (9)Rearranging these terms we are able to deduce that, for any firm i ∈ N , [ a ij ( t, ∆ t, V t (∆ t )) − a ij ( t − ∆ t, ∆ t, V t − ∆ t (∆ t ))] V i ( t − ∆ t, ∆ t ) − = ∆ L ij ( t, ∆ t ) − a ij ( t, ∆ t, V t (∆ t )) X k ∈N ∆ L ik ( t, ∆ t ) . (10)Coupled with the assumption that the societal node always has positive wealth, we are thusable to consider the limiting behavior of (7) as the step-size ∆ t tends to 0. To do so, consider V ( t, ∆ t ) = V ( t − ∆ t, ∆ t ) + ∆ c ( t, ∆ t ) − A ( t, ∆ t, V t (∆ t )) ⊤ V ( t, ∆ t ) − + A ( t − ∆ t, ∆ t, V t − ∆ t ) ⊤ V ( t − ∆ t, ∆ t ) − = V ( t − ∆ t, ∆ t ) + ∆ c ( t, ∆ t ) − A ( t, ∆ t, V t (∆ t )) ⊤ V ( t, ∆ t ) − + A ( t, ∆ t, V t (∆ t )) ⊤ V ( t − ∆ t, ∆ t ) − − A ( t, ∆ t, V t (∆ t )) ⊤ V ( t − ∆ t, ∆ t ) − + A ( t − ∆ t, ∆ t, V t − ∆ t ) ⊤ V ( t − ∆ t, ∆ t ) − = V ( t − ∆ t, ∆ t ) + ∆ c ( t, ∆ t ) − A ( t, ∆ t, V t (∆ t )) ⊤ [ V ( t, ∆ t ) − − V ( t − ∆ t, ∆ t ) − ] − ∆ L ( t, ∆ t ) ⊤ ~ A ( t, ∆ t, V t (∆ t )) ⊤ ∆ L ( t, ∆ t ) ~ . Consider the notation for the matrix of distressed firms from the fictitious default algorithm(Algorithm 3.6), i.e., Λ( V ) ∈ { , } ( n +1) × ( n +1) is the diagonal matrix of banks in distress Λ ij ( V ) = ( if i = j = 0 and V i < else ∀ i, j ∈ N . We are able to set Λ ( V ) = 0 without loss of generality since, by assumption, the outside node has no obligations into the system. Thus, as with (9), by continuity of the clearing wealthsand ∆ t small enough, we can conclude that except at specific event times (to be consideredlater, see Algorithm 4.7) it follows that Λ( V ( t, ∆ t )) = Λ( V ( t − ∆ t, ∆ t )) . Thus, with this addednotation we can reformulate the clearing wealths equation (7) as V ( t, ∆ t ) = V ( t − ∆ t, ∆ t ) + A ( t, ∆ t, V t (∆ t )) ⊤ Λ( V ( t, ∆ t ))[ V ( t, ∆ t ) − V ( t − ∆ t, ∆ t )] + ∆ c ( t, ∆ t ) − ∆ L ( t, ∆ t ) ⊤ ~ A ( t, ∆ t, V t (∆ t )) ⊤ ∆ L ( t, ∆ t ) ~ . For the construction of a differential form we can consider the equivalent formulation V ( t, ∆ t ) − V ( t − ∆ t, ∆ t ) =[ I − A ( t, ∆ t, V t (∆ t )) ⊤ Λ( V ( t, ∆ t ))] − (cid:18) ∆ c ( t, ∆ t ) − ∆ L ( t, ∆ t ) ⊤ ~ A ( t, ∆ t, V t (∆ t )) ⊤ ∆ L ( t, ∆ t ) ~ (cid:19) . (11)Note that I − A ( t, ∆ t, V t (∆ t )) ⊤ Λ( V ( t, ∆ t )) is invertible by standard input-output results andas proven in Proposition B.1. tilizing (11) and (9) and taking the limit as ∆ t ց , we are thus able to construct the jointdifferential system: dV ( t ) = [ I − A ( t ) ⊤ Λ( V ( t ))] − (cid:16) dc ( t ) − dL ( t ) ⊤ ~ A ( t ) ⊤ dL ( t ) ~ (cid:17) (12) da ij ( t ) = d L ij ( t ) − a ij ( t ) P k ∈N d L ik ( t ) P k ∈N dL ik ( t ) if i ∈ N , V i ( t ) ≥ dL ij ( t ) − a ij ( t ) P k ∈N dL ik ( t ) V i ( t ) − if i ∈ N , V i ( t ) < if i = 0 ∀ i, j ∈ N (13)with initial conditions V (0) ≥ given and a ij (0) = dL ij (0) P k ∈N dL ik (0) { i =0 } + n { i =0 , j =0 } for allfirms i, j ∈ N . As in (11), I − A ( t ) ⊤ Λ( V ( t )) is invertible by standard input-output resultsand as proven in Proposition B.1. The first case in (13) is constructed by noting that a ij ( t ) = dL ij ( t ) P k ∈N dL ik ( t ) if V i ( t ) ≥ and i ∈ N and da j ( t ) = 0 for any firm j ∈ N for all times t ; thesecond case in (13) follows from (10) and taking the limit as ∆ t ց . Note that this differentialsystem is discontinuous, with events at times when firms cross the 0 wealth boundary, i.e.,when Λ( V ( t )) = Λ( V ( t − )) . As such, we will consider the differential system on the inter-eventintervals, then update the differential system between these intervals. This is made more explicitin the proof of Theorem 4.5 and in Algorithm 4.7. As with the discrete-time system (8), therelative 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 relativeobligations between new liabilities and the prior unpaid liabilities. Remark 4.3.
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 bankruptcycourt 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 insolventand restart the differential system with updated parameters from that time point, e.g., using aninstantaneous auction as in [7]; see also Remarks 3.7 and 4.8.We will complete our discussion of the construction of this differential system by providingsome properties on the relative liabilities and exposures matrix A . Notably, these properties arethose 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 ofincoming liabilities, that the relative exposures are bounded from below by 0 (and to societyby δ as provided in Assumption 4.1), and the relative exposure matrix is row stochastic at alltimes. Proposition 4.4.
Let ( dc, dL ) : T → R n +1 × R ( n +1) × ( n +1)+ define a dynamic financial networksatisfying Assumption 4.1. Let ( V, A ) : T → R n +1 × R ( n +1) × ( n +1) be any solution of the dif-ferential system (12) and (13) satisfying Assumption 3.2. The relative exposure matrix A ( t ) satisfies the following properties:(i) For any bank i ∈ N , if V i ( t ) ր as t ր τ then lim t ր τ a ij ( t ) = dL ij ( τ ) P k ∈N dL ik ( τ ) .(ii) For all times t ∈ T and for any bank i ∈ N , the elements a ij ( t ) ≥ for all banks j ∈ N and a i ( t ) ≥ δ ;(iii) For all times t ∈ T and for any bank i ∈ N , the row sums P k ∈N a ik ( t ) = 1 ; With this differential construction (12) and (13), we seek to prove existence and uniquenessof the clearing solutions. For notational simplicity, define the space of relative exposure matrices A := (cid:26) A ∈ [0 , ( n +1) × ( n +1) | A~ ~ , a ii = 0 , a i ≥ δ ∀ i ∈ N , a j = 1 n ∀ j ∈ N (cid:27) . From Proposition 4.4, we have already proven that if ( V, A ) : T → R n +1 × R ( n +1) × ( n +1) is asolution to the continuous-time Eisenberg-Noe system then A ( t ) ∈ A for all times t ∈ T . heorem 4.5. Let T = [0 , T ] be a finite time period and let ( dc, dL ) : T → R n +1 × R ( n +1) × ( n +1)+ define a dynamic financial network satisfying Assumption 4.1. There exists a unique strongsolution to the clearing wealths and relative exposures ( V, A ) satisfying (12) and (13) if V (0) ∈ R n +1++ . Remark 4.6.
The restrictions on the cash flows dc made in Assumption 4.1 can be relaxed todepend explicitly on the wealths and relative exposures, i.e., dc ( t ) = µ ( t, c ( t ) , V ( t ) , A ( t )) dt + σ ( t, c ( t ) , V ( t ) , A ( t )) dW ( t ) . This would still guarantee a unique strong solution of the clearing wealths and relative exposuresas in Theorem 4.5 so long as µ, σ satisfy a local linear growth condition, local Lipschitz condition,and c ( t ) can be bounded above and below by elements of L t ( R n +1 ) for all time t .We now present an algorithm for numerically computing an approximation of a single samplepath for the continuous-time Eisenberg-Noe clearing system. To do so we consider Euler’smethod for differential equations with an event finding algorithm. Algorithm 4.7.
Under the assumptions of Theorem 4.5 for a fixed event ω ∈ Ω the clear-ing wealths process V : T → R n +1 and relative exposures A : T → A can be found by thefollowing algorithm. Fix a step-size ∆ t > . Initialize t = 0 , V (0) ≥ given, a ij (0) = dL ij (0) P k ∈N dL ik (0) { i =0 } + n { i =0 , j =0 } , and Λ = { } ( n +1) × ( n +1) . Repeat until t ≥ T :(i) Initialize Λ = Λ and ∆ t = ∆ t .(ii) Sample Z ∼ N (0 , I ) .(iii) Repeat until Λ = Λ :(a) Set Λ = Λ .(b) Compute ¯ µ ( t ) = ( I − A ( t ) ⊤ Λ) − (cid:16) µ ( t, c ( t )) − ˙ L ( t ) ⊤ ~ A ( t ) ⊤ ˙ L ( t ) ~ (cid:17) ¯ σ ( t ) = ( I − A ( t ) ⊤ Λ) − σ ( t, c ( t )) Z. (c) Loop through each bank i ∈ N :i. If V i ( t ) > , ¯ µ i ( t ) < , and ¯ σ i ( t ) − µ i ( t ) V i ( t ) ≥ then ∆ t = min ∆ t , − ¯ σ i ( t ) − p ¯ σ i ( t ) − µ i ( t ) V i ( t )2¯ µ i ( t ) ! . ii. If V i ( t ) < , ¯ µ i ( t ) = 0 , and ¯ σ i ( t ) − µ i ( t ) V i ( t ) ≥ then ∆ t = min ∆ t , − ¯ σ i ( t ) + p ¯ σ i ( t ) − µ i ( t ) V i ( t )2¯ µ i ( t ) ! . iii. If ¯ µ i ( t ) = 0 and V i ( t )¯ σ i ( t ) < then ∆ t = min (cid:8) ∆ t , V i ( t ) / ¯ σ i ( t ) (cid:9) .iv. If ¯ µ i ( t )¯ σ i ( t ) < then ∆ t = min (cid:8) ∆ t , ¯ σ i ( t ) / ¯ µ i ( t ) (cid:9) .(d) Compute ∆ V ( t ) = ¯ µ ( t )∆ t + ¯ σ ( t ) √ ∆ t .(e) Define the matrix Λ ∈ { , } ( n +1) × ( n +1) such that Λ ij = if i = j = 0 , V i ( t ) > or [ V i ( t ) = 0 , ∆ V i ( t ) ≥ if i = j = 0 , V i ( t ) < or [ V i ( t ) = 0 , ∆ V i ( t ) < else iv) Define the matrix ¯Λ ∈ { , } ( n +1) × ( n +1) so that ¯Λ = ( if i = j = 0 , V i ( t ) < else .(v) Set c ( t + ∆ t ) = c ( t ) + µ ( t, c ( t ))∆ t + σ ( t, c ( t )) √ ∆ tZV ( t + ∆ t ) = V ( t ) + ∆ V ( t ) A ( t + ∆ t ) = ¯Λ h A ( t ) + diag( V ( t ) − ) − [ ˙ L ( t ) − A ( t ) ∗ ( ˙ L ( t ) )]∆ t i + ( I − ¯Λ) diag( ˙ L ( t ) ~ − ˙ L ( t ) . where = { } ( n +1) × ( n +1) and ∗ denotes the element-wise multiplication operator.(vi) Increment t = t + ∆ t .If t > T then set c ( T ) = c ( t − ∆ t ) + c ( t ) − c ( t − ∆ t )∆ t ( T − [ t − ∆ t ]) V ( T ) = V ( t − ∆ t ) + V ( t ) − V ( t − ∆ t )∆ t ( T − [ t − ∆ t ]) A ( T ) = A ( t − ∆ t ) + A ( t ) − A ( t − ∆ t )∆ t ( T − [ t − ∆ t ]) . In the above event-finding algorithm for the continuous-time Eisenberg-Noe system, themain 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 describedin step (iiic). In particular, (iii(c)i)-(iii(c)iii) guarantee that V i ( t ) + ¯ µ i ( t )∆ t + ¯ σ i ( t ) √ ∆ t isnonnegative if V i ( t ) > and nonpositive if V i ( t ) < . The additional condition in (iii(c)iv)guarantees that the direction of ¯ µ i ( t )∆ t + ¯ σ i ( t ) √ ∆ t is maintained as ∆ t shrinks, i.e., if ∆ t istoo 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 (iii(c)iv) as itimproves the accuracy of the algorithm. Remark 4.8.
As with the discrete-time setting discussed in Remark 3.7, we can introducethe concept of loans from a central bank to the continuous-time Eisenberg-Noe system. To doso 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 needfor 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 updatethe cash flow parameters for each firm going forward, but no dynamic contagion would need tobe modeled.We wish to conclude our discussion of the continuous-time Eisenberg-Noe system by provid-ing a result on how the unique solution to the discrete-time solution converges to the continuous-time solution as ∆ t ց . That is, we wish to consider how the unique clearing wealths andrelative exposures solving the discrete-time systems (7) and (8) converge to those in continuous-time Eisenberg-Noe system (12) and (13) as the step-size decreases to 0. Lemma 4.9.
Consider the setting of Corollary 4.2 and Theorem 4.5. Then the continuous-timeclearing solutions at any time t ∈ T is the limit of the discrete-time solution as the step-sizetends to 0, i.e., ( V ( t ) , A ( t )) = lim ∆ t ց ( V ( t, ∆ t ) , A ( t, ∆ t, V t (∆ t ))) where ( V ( · , ∆ t ) , A ( · , ∆ t )) satisfy (7) and (8) and ( V, A ) satisfy (12) and (13) . 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 conclu- ions hold in the discrete-time setting as well. Notably, we deduce rules so as to recreate thestatic Eisenberg-Noe clearing solution via our continuous-time differential system, which (in-dependently) replicates the results from [40]. Further, we consider the implications of timedynamics on the health of the financial system by determining bounds on how different thestatic clearing solution and a dynamic solution might be. This demonstrates the importance oftime dynamics on accurately assessing the health and wealth of the financial system. 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 ) / P k ∈N dL ik ( s ) = dL ij ( t ) / P k ∈N dL ik ( t ) for all times s, t ∈ T and firms i, j ∈ N so long as P k ∈N dL ik ( s ) , P k ∈N dL ik ( t ) > . Thekey implication of this assumption is that the relative exposures matrix in (13) can be foundexplicitly to equal the relative liabilities a ij ( t ) = π ij := dL ij ( s i ) P k ∈N dL ik ( s i ) if s i < sup T n if s i = sup T , j = i if s i = sup T , j = i for all times t and banks i, j ∈ N where s i ∈ { t ∈ T | P k ∈N dL ik ( t ) > } chosen arbitrarilystrictly less than sup T (and s i = sup T if the supremum is taken over the empty set).Further, expanding and solving the differential system (12), we deduce that the continuous-time clearing wealths must satisfy the fixed point problem V ( t ) = V (0) + Z t dc ( s ) − Π ⊤ V ( t ) − (14)at all time t ∈ T . Therefore, if R t dc ( s ) ≥ R t dL ( s ) ⊤ ~ − R t dL ( s ) ~ at some time t , it followsthat V ( t ) are the static clearing wealths to the Eisenberg-Noe system with aggregated datawith nominal liabilities matrix defined by R t dL ( s ) and (incoming) external cash flow given by R t dc ( s ) − (cid:16)R t dL ( s ) ⊤ ~ − R t dL ( s ) ~ (cid:17) . Importantly, this means that, if the relative liabilitiesare kept constant over time, taking aggregated data and considering the static Eisenberg-Noeframework will produce the same final clearing wealths as the dynamic Eisenberg-Noe settingpresented in this paper. However, though the set of defaulting banks is the same as in the staticsetting, the order of defaults need not strictly follow the order given in the fictitious defaultalgorithm of [14]. Definition 5.1.
A bank is called a k th-order default in the static Eisenberg-Noe setting if itis determined to be in default in the k th iteration of the fictitious default algorithm (see, e.g.,[14, Section 3.1] or the inner loop of Algorithm 3.6). We note that the first-order defaults are exactly those firms that have negative wealth evenif it has no negative exposure to other firms (i.e., all other firms satisfy their obligations in full).
Proposition 5.2.
Let ( x, ¯ L ) ∈ R n +1+ × R ( n +1) × ( n +1)+ denote the static incoming external cashflow and nominal liabilities. Define a dynamic system over the time period T = [0 , T ] such that V (0) ∈ [0 , x ] , dL ( t ) = T ¯ Ldt , and dc ( t ) = T (cid:16) x − V (0) + ¯ L ⊤ ~ − ¯ L~ (cid:17) dt . The clearing wealthsat the terminal time V ( T ) are equal to those given in the static setting. Additionally, no firmwill ever recover from distress in the dynamic setting. Finally, the first k th-order default willoccur only after the first ( k − th-order default in the static fictitious default algorithm; inparticular, the first firm to become distressed will be a first-order default in the static fictitiousdefault algorithm.Proof. The fact that the clearing wealths V ( T ) are equal to the static Eisenberg-Noe clearingwealths (as defined in Proposition 2.1) follows from (14) and the logic given in the proof of emma 4.9. Additionally, since dc ( t ) is constant in time and firms are beginning in a solventstate, over time the unpaid liabilities may accumulate as a negative factor on bank balancesheets, 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 − th-order defaults(and not solely by the ( k − th-order defaults). Therefore, by way of contradiction, if a k th-orderdefault were to occur before any ( k − th-order default then such a firm must default withoutregard to what happens to the ( k − th-order defaults, i.e., this firm must be a ( k − th-orderdefault. 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 fictitiousdefault algorithm of [14] is unsurprising. Consider a financial system with two subgraphs thatare only connected through their obligations to the societal node. By construction, the defaultof a firm in one subgraph will have no impact on the firms in the other subgraph. Thus wecan construct a network so that all defaults in one subgraph (including higher order defaults asdefined in Definition 5.1) occur before any first-order defaults in the other subgraph.Notably, Proposition 5.2 states that, provided the aggregate data (until the terminal time) iskept 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 4bank (plus societal node) system. In particular, we will consider the cash flows c to be definedas a Brownian bridge so as to provide the appropriate aggregate data at the terminal time. Example 5.3.
Consider a financial system with four banks, each with an additional obligationto an external societal node. Consider the time interval T = [0 , with aggregated data such thatthe initial wealths are given by V (0) = (100 , , , , ⊤ , cash flows dc are such that R dc ( s ) =¯ L ⊤ ~ − ¯ L~ , and where the nominal liabilities matrix dL = ¯ Ldt is defined by ¯ L = . The static
Eisenberg-Noe clearing wealths, with nominal liabilities ¯ L and external assets V (0) ,are found to be V (1) ≈ (109 . , − . , − . , − . , . ⊤ . Further, from the static fictitiousdefault algorithm, we can determine that bank 1 is a first-order default, bank 2 is a second-orderdefault, and bank 3 is a third-order default. Consider now three dynamic settings which aredifferentiated only by the choice of the cash flows dc :(i) Consider the deterministic setting introduced in Proposition 5.2, i.e., dc ( t ) = [ ¯ L ⊤ ~ − ¯ L~ dt for all times t ∈ T .(ii) Consider a Brownian bridge with low volatility, i.e., dc ( t ) = ¯ L ⊤ ~ − ¯ L~ − c ( t )1 − t dt + dW ( t ) forvector of independent Brownian motions W and with c (0) = 0 .(iii) Consider a Brownian bridge with high volatility, i.e., dc ( t ) = ¯ L ⊤ ~ − ¯ L~ − c ( t )1 − t dt + 5 dW ( t ) forvector of independent Brownian motions W and with c (0) = 0 .A single sample path for each dynamic setting is provided. In each plot we reduce the equityof the societal node by 100 so that it begins with an initial wealth of 0, but more importantlyso that it can easily be displayed on the same plot as the other 4 institutions. First, we pointout that, as indicated by the circles at the terminal time in each plot, the terminal wealths ofthe continuous-time setting match up with the clearing wealths in the static model. We furthernote that in the deterministic setting (Figure 2a) and the low volatility setting (Figure 2b) theorder of defaults is maintained. However, in the high volatility setting (Figure 2c) the order ofdefaults given by the fictitious default algorithm no longer holds. Time -10-8-6-4-20246810 W ea l t h Society: V (t)-100Bank 1: V (t)Bank 2: V (t)Bank 3: V (t)Bank 4: V (t) (a) Example 5.3: Clearing wealths over time un-der deterministic and constant cash flows. Time -10-8-6-4-20246810 W ea l t h Society: V (t)-100Bank 1: V (t)Bank 2: V (t)Bank 3: V (t)Bank 4: V (t) (b) Example 5.3: Clearing wealths over time un-der low volatility Brownian bridge cash flows. Time -10-8-6-4-20246810 W ea l t h Society: V (t)-100Bank 1: V (t)Bank 2: V (t)Bank 3: V (t)Bank 4: V (t) (c) Example 5.3: Clearing wealths over time un-der high volatility Brownian bridge cash flows. Figure 2: Example 5.3: Comparison of clearing wealths under deterministic and random cash flowsthat aggregate to the same terminal values.
Now we will consider the case in which the relative liabilities change over time. As in the priordiscussion, we will focus on the setting in which the aggregate cash flows and interbank liabilitiescorrespond to a static Eisenberg-Noe model. As the liabilities are now changing over time thereis an inherent prioritization in the obligations due to the rolling forward of unpaid debts. Anyearlier obligations are more likely to be paid, and accumulate to be paid proportionally with anynew obligations. As such, by altering only the rate at which the liabilities are due, the terminalwealths and also the set of defaulting firms can be modified. Proposition 5.5 provides analysison which banks will always be solvent and which will always be in default at the terminal time.In particular, the results of Proposition 5.5 show that the static
Eisenberg-Noe model appliedto aggregate data can produce a viewpoint on the health of the financial system that is eitherincorrectly optimistic or pessimistic; without explicitly knowing the dynamics of the cash flowsand liabilities, only rough estimates can be considered. This is in contrast to, e.g., [30] in whichdata from the European Banking Authority’s 2011 stress test was utilized to assess the healthof the European financial system without time dynamics.
Definition 5.4.
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 Time -10-8-6-4-2024681012 W ea l t h Society: V (t)-100Bank 1: V (t)Bank 2: V (t)Bank 3: V (t)Bank 4: V (t) (a) Example 5.6: Clearing wealths over time un-der setting to have all but the first order-defaultssolvent at the terminal time. Time -10-8-6-4-2024681012 W ea l t h Society: V (t)-100Bank 1: V (t)Bank 2: V (t)Bank 3: V (t)Bank 4: V (t) (b) Example 5.6: Clearing wealths over time un-der setting to have all but the first order-solvencydefaulting at the terminal time. Figure 3: Example 5.6: Comparison of clearing wealths under different ordering of the nominalliabilities in time that aggregate to the same terminal values. all).
Note that, by assumption, the societal node will always be a first-order solvent institution. Proposition 5.5.
Let ( x, ¯ L ) ∈ R n +1+ × R ( n +1) × ( n +1)+ denote the static incoming external cashflow and nominal liabilities. Define a dynamic system over the time period T = [0 , T ] such that V (0) ∈ [0 , x ] , R T dL ( t ) = ¯ L , and R T dc ( t ) = x − V (0) + ¯ L ⊤ ~ − ¯ L~ . At time T , those banks thatare 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 dynamicsetting at the terminal time.Proof. This result follows from the definition of a first-order default or solvency as such firmsallow us to disregard all interbank dynamics.To conclude this discussion, we will consider two examples with the same aggregate values asgiven in Example 5.3. The first example considers the case in which the nominal liabilities areshifted 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 considersa fixed structure for the nominal liabilities in time (but non-constant relative liabilities), thusdemonstrating the path-dependence of the clearing wealths on the cash flows.
Example 5.6.
Consider the financial system described in Example 5.3 over the time interval T = [0 , with aggregated data such that the initial wealths V (0) = (100 , , , , ⊤ and wherethe aggregate nominal liabilities matrix is defined by ¯ L . Further, consider the cash flows dc ( t ) = dL ( t ) ⊤ ~ − dL ( t ) ~ for all times t ∈ T where dL is either:(i) prioritizing the defaulting firms: dL ( t ) = 5 ¯ L (cid:0) E { t ∈ (0 . , } + P i ∈N E i { t ∈ (0 . i − , . i ] } (cid:1) ,or(ii) prioritizing society: dL ( t ) = 5 ¯ L (cid:0) E { t ∈ [0 , . } + P i ∈N E i { t ∈ (0 . i, . i +1)] } (cid:1) where the collection of matrices E i ∈ { , } ( n +1) × ( n +1) are such that ( E i ) ii = 1 and all otherelements are set to . As in Figure 2, the circles at the terminal time in both plots denote theclearing wealths under the static Eisenberg-Noe setting. It is clear in both examples that theterminal dynamic clearing wealths now are not equal to the static wealths. Further, by choosingthe liabilities to be introduced in the order provided we provide the settings so that only the × marks the societal wealth under the static Eisenberg-Noe framework with aggregateddata. first-order defaults, Bank 1, have negative terminal wealth (Figure 3a) or so that only the first-order solvencies, the societal firm, have positive terminal wealth (Figure 3b). In Figure 3a,we notice that firms 2 and 3 have a terminal wealth of , so although they are not defaulting,they do not have any positive equity either. Further, it is clear that though all financial firmshave improved their wealth given this ordering of the nominal liabilities, the societal wealthis decreased (though to a lesser amount than the aggregate improvement for the banks) incomparison to the static results. In contrast, in the second scenario in which obligations tosociety are first (Figure 3b), the societal wealths are greater than those provided in the staticsetting but all banks have less wealth. Notice further that, even after the obligations to societyhave “ended” at time 0.2 the societal wealth still increases. This occurs as the banks in distressreceive money as their incoming liabilities come due and thus they have cash to immediatelytransfer to cover the prior unpaid obligations to, e.g., society. Finally, this numerically verifiesthe results of Proposition 5.5 and demonstrates the importance of understanding the order ofobligations for an accurate measure of the health of the financial system. Example 5.7.
Consider the financial system described in Example 5.3 over the time interval T = [0 , with aggregated data such that the initial wealths V (0) = (100 , , , , ⊤ and wherethe aggregate nominal liabilities matrix is defined by ¯ L . Further, consider the nominal liabilitiesdetermined by dL ( t ) = ¯ L (cid:18) E + . E { t ∈ [0 . , . } + . E { t ∈ [0 . , . } + . E { t ∈ [0 . , . } + . E { t ∈ [0 . , . } (cid:19) where the collection of matrices E i ∈ { , } ( n +1) × ( n +1) are such that ( E i ) ii = 1 and all otherelements are set to . Finally, consider the cash flows determined by a Brownian bridge withvolatility of 2, i.e., dc ( t ) = ¯ L ⊤ ~ − ¯ L~ − c ( t )1 − t dt + 2 dW ( t ) for vector of independent Brownian motions W and with c (0) = 0 . Figure 4 depicts the empirical distribution of the terminal societalwealths under 10,000 samples of the Brownian bridge cash flows. The black curve depicts thekernel density for this empirical distribution. The × illustrates the societal wealth under thestatic Eisenberg-Noe framework considering the aggregated data (as provided in Example 5.3).The key takeaway of this figure is the payments to society range from 8.12 to 10.20 out ofan obligated 12, i.e., society can experience anywhere from 16% to 32% shortfall in paymentsdepending on the sample path. This also implies that society can experience anywhere froma 13.4% decrease to an 8.8% increase over the payments found under the static Eisenberg-Noe odel. 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 inwhich they are defaulting on their obligations. Recall none of these three firms are first-orderdefaults or first-order solvencies. Empirically, firm 2 (a second-order default) is found to defaultin approximately 98% of the simulations; firm 3 (a third-order default) is found to default inapproximately 3.6% of simulations; firm 4 (which does not default in the static setting) is foundto 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 havea significant impact on the health of the system. In this paper we considered an extension of the financial contagion model of [14] to allow forcash flows and obligations to be dynamic in time. We presented this model in both discreteand continuous time, thus extending the frameworks of [7, 23, 33] which consider only discrete-time clearing. Notably, we determine conditions for existence and uniqueness of the clearingsolutions under deterministic and Itô settings. In this way, we have written a dynamical systemfor 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 dynamicEisenberg-Noe model presented herein will reproduce the static system at the terminal timein a path-independent manner. Notably, in such a setting, we are able to determine the truedefaulting order rather than the fictitious order found in the fictitious default algorithm thatis widely used in computing static clearing models. If, however, the relative liabilities are notconstant over time, then we determine that the static Eisenberg-Noe model may report anincorrectly optimistic or pessimistic picture of the financial system.Three clear extensions of this model are apparent to us, and which we foresee creating furtherdivergence between static and dynamic models. The first extension is the inclusion of illiquidassets and fire sales. In the static models, e.g., [12, 1, 18], there is no first mover advantage toliquidating assets as all firms receive the same price. However, in a dynamic model there maybe advantage to liquidating early in order to receive a higher price, but which may precipitatea larger fire sale amongst the other firms. The second extension is the inclusion of contingentpayments and credit default swaps. In the static setting this has recently been considered by[4, 38, 39]. By considering the network dynamics to be dependent on the history of clearingwealths, many of the difficulties reported in the static works are likely to be resolved naturally;we refer to [4] which provides an initial discussion of this extension. The final extension, forwhich we believe the proposed dynamic model will be especially useful, is in considering strate-gic or dynamic actions by the market participants, e.g., incorporating bankruptcy costs andstrategic decisions on rolling forward of debt. We feel that the continuous-time framework willbe particularly suitable for these extensions as it allows us to construct unique clearing solutionswithout requiring strong monotonicity assumption.
A Proof of results in Section 3
Proof of Theorem 3.3.
We will prove this result inductively. First consider time t = 0 . Recallfrom Assumption 3.2 that V ( − ≥ . The clearing wealths at time follow the fixed pointequation V (0) = Φ(0 , V (0)) := V ( −
1) + c (0) − A (0 , V ) ⊤ V (0) − . Note that, by construction, A (0 , V ) ⊤ V (0) − ≤ L (0) ⊤ ~ . Therefore any clearing solution mustfall within the compact range [ V ( −
1) + c (0) − L (0) ⊤ ~ , V ( −
1) + c (0)] ⊆ R n +1 . It is clear from thedefinition that Φ(0 , · ) is a monotonic operator, and thus there exists a greatest and least clearingsolution V ↑ (0) ≥ V ↓ (0) by Tarski’s fixed point theorem [45, Theorem 11.E], both of which mustfall within this domain. Further, a ij (0 , V ) = L ij P k ∈N L ik (for i ∈ N and j ∈ N ) for any wealth (0) in this domain since V ( −
1) + c (0) − L (0) ⊤ ~ ≥ − L (0) ~ − ¯ p (0 , V − ) . We will proveuniqueness as it is done in [14] by noting additionally that we can assume that the societal nodewill always have positive equity (i.e., V ↓ (0) ≥ ). First, we will show that the positive equitiesare the same for every firm no matter which clearing solution is chosen, i.e., V ↑ i (0) + = V ↓ i (0) + for every firm i ∈ N . By definition V ↑ (0) ≥ V ↓ (0) and using P j ∈N a ij (0) = 1 for every firm i ∈ N we recover X i ∈N V ↑ i (0) + = X i ∈N h V ↑ i (0) + V ↑ i (0) − i = X i ∈N V i ( −
1) + c i (0) − X j ∈N a ji (0 , V ↑ ) V ↑ j (0) − + V ↑ i (0) − = X i ∈N [ V i ( −
1) + c i (0)] − X j ∈N V ↑ j (0) − X i ∈N a ji (0 , V ↑ ) + X i ∈N V ↑ i (0) − = X i ∈N [ V i ( −
1) + c i (0)] = X i ∈N V ↓ i (0) + . Therefore it must be the case that V ↑ i (0) + = V ↓ i (0) + for all firms i ∈ N . Since we assume thatthe societal node will always have positive equity, it must be the case that V ↑ (0) = V ↓ (0) . Nowsince we assume that each node i ∈ N owes to the societal node, if any firm i ∈ N is such that ≥ V ↑ i (0) > V ↓ i (0) then it must be that V ↑ (0) > V ↓ (0) , which is a contraction.Continuing with the inductive argument, assume that the history of clearing wealths V t − upto time t − is fixed and known. The clearing wealths at time t follow the fixed point equation V ( t ) = Φ( t, V ( t )) := V ( t −
1) + c ( t ) − A ( t, V t ) ⊤ V ( t ) − + A ( t − , V t − ) ⊤ V ( t − − . Note that, by construction, A ( t, V t ) ⊤ V ( t ) − ≤ L ( t ) ⊤ ~ A ( t − , V t − ) ⊤ V ( t − − . Therefore anyclearing solution must fall within the compact range [ V ( t −
1) + c ( t ) − L ( t ) ⊤ ~ , V ( t −
1) + c ( t ) + A ( t − , V t − ) ⊤ V ( t − − ] ⊆ R n +1 . Further, a ij ( t, V t ) = L ij + a ij ( t − ,V t − ) V i ( t − − P k ∈N L ik + V i ( t − − (for i ∈ N and j ∈ N ) for any wealth V ( t ) in this domain since V ( t − c ( t ) − L ( t ) ⊤ ~ ≥ − V ( t − − − L ( t ) ~ − ¯ p ( t, V t − ) . Thus we can apply the same logic as in the time case to recover existence anduniqueness of the clearing wealths V ( t ) at time t . Proof of Corollary 3.5.
This follows immediately from the proof of Theorem 3.3 using inductionand noting that the lattice upper and lower bounds for the domain and range spaces of Φ( s, · ) are subsets of L ps ( R n +1 ) . Therefore any clearing solution V ( t ) is bounded above and below byan element of L pt ( R n +1 ) and the result is proven. B Proof of results in Section 4
Proof of Corollary 4.2.
Existence and uniqueness of the clearing solutions follows from Theo-rem 3.3. To prove continuity we will employ an induction argument. To do so, we will considerthe reduced domain V : T × [ ǫ, ∞ ) → R n +1 for some ǫ > . That is, we restrict the step-size ∆ t ≥ ǫ . As we will demonstrate that the continuity argument holds for any ǫ > then the de-sired result must hold as well. Before continuing, consider an expanded version of the recursiveformulation of (7), i.e., V ( t, ∆ t ) = V ( −
1) + Z t dc ( s ) − A ( t, ∆ t, V t (∆ t )) ⊤ V ( t, ∆ t ) − (15)for all times t ∈ T . Fix the minimal step-size ǫ > . Note that the relative exposures satisfy a ij ( t, ∆ t, V t (∆ t )) := R t dL ij ( s ) P k ∈N R t dL ik ( s ) for any time t ∈ [0 , ǫ ) by the assumption that V ( − ≥ . Thus we can conclude V : [0 , ǫ ) × [ ǫ, ∞ ) → R n +1 is continuous by an application of [22,Proposition A.2]. Now, by way of induction, assume that V : [0 , s ) × [ ǫ, ∞ ) → R n +1 is continuousfor some s > . Again, by [22, Proposition A.2], we are able to immediately conclude that V : [0 , s + ǫ ) ∩ T × [ ǫ, ∞ ) → R n +1 is continuous. As we are able to always extend the continuityresult by ǫ > in time, the result is proven. Proof of Proposition 4.4. (i) Consider firm i ∈ N . By assumption we have that a ij ( t ) for t ր τ solves the first order differential equation: da ij ( t ) dt + P k ∈N dL ik ( t ) /dtV i ( t ) − a ij ( t ) = dL ij ( t ) /dtV i ( t ) − . For sake of simplicity, let this differential equation start at time with V i (0) < and someinitial value a ij (0) . Then this differential equation can be solved via the integrating factor ν ( t ) := R t P k ∈N dL ik ( s ) V i ( s ) − ds . Thus for t ր τ it follow that a ij ( t ) = e − ν ( t ) (cid:20)Z t e ν ( s ) dL ij ( s ) V i ( s ) − + a ij (0) (cid:21) . Therefore, utilizing L’Hôspital’s rule, lim t ր τ a ij ( t ) = lim t ր τ e − ν ( t ) (cid:20)Z t e ν ( s ) dL ij ( s ) V i ( s ) − + a ij (0) (cid:21) = lim t ր τ e ν ( t ) dL ij ( t ) V i ( t ) − e ν ( t ) ddt ν ( t ) = lim t ր τ dL ij ( t ) /V i ( t ) − P k ∈N dL ik ( t ) /V i ( t ) − = dL ij ( τ ) P k ∈N dL ik ( τ ) . (ii) First, if V i ( t ) ≥ then by construction (and the above result) it follows that a ij ( t ) = dL ij ( t ) P k ∈N dL ik ( t ) ≥ for any i, j ∈ N and a i ( t ) ≥ δ by this construction. Consider nowthe case for V i ( t ) < and assume a ij ( t ) < . Let τ = sup { s ≤ t | V i ( s ) = 0 } . Since a ij ( τ ) ∈ [0 , by construction and the relative exposures are continuous, this implies thereexists some time s ∈ [ τ, t ) such that a ij ( s ) = 0 . By the definition of the relative exposures,this must follow that da ij ( s ) ≥ for any time a ij ( s ) ≤ (with da ij ( s ) > if a ij ( s ) < ),thus a ij ( t ) < can never be reached. Further, assume a i ( t ) < δ . By Assumption 4.1, if a i ( s ) ≤ dL i ( s ) P k ∈N dL ik ( s ) then da i ( s ) ≥ . In particular, if a i ( s ) ≤ δ then da i ( s ) ≥ (with da i ( s ) > if a i ( s ) < δ ). Thus, by the same contradiction found in the case for j ∈ N ,we are able to bound a i ( t ) ≥ δ .(iii) First, if i = 0 then P j ∈N a j ( t ) = 1 by property that a j ( t ) = n { j =0 } for all times t .Now consider i ∈ N , if V i ( t ) ≥ then by construction (and the above result) it followsthat P j ∈N a ij ( t ) = P j ∈N dL ij ( t ) P k ∈N dL ik ( t ) = 1 . Consider now the case for V i ( t ) < andlet τ = sup { s ≤ t | V i ( s ) = 0 } . Since P j ∈N a ij ( τ ) = 1 by prior results, we will assumethat P j ∈N a ij ( t ) = 1 to deduce X j ∈N da ij ( t ) = X j ∈N dL ij ( t ) − a ij ( t ) P k ∈N dL ik ( t ) V i ( t ) − = P j ∈N dL ij ( t ) V i ( t ) − − (cid:16)P j ∈N a ij ( t ) (cid:17) (cid:0)P k ∈N dL ik ( t ) (cid:1) V i ( t ) − = 0 . Therefore based on the initial conditions, a ij ( t ) must evolve so that it maintains theconstant row sum of . roof of Theorem 4.5. Recall that the initial values to the Eisenberg-Noe differential system are V i (0) > and a ij (0) = dL ij (0) P k ∈N dL ik (0) { i =0 } + n { i =0 , j =0 } for all banks i, j ∈ N . For ease ofnotation, consider τ := 0 and recursively define the stopping times τ m +1 := inf { t ∈ ( τ m , T ] | V i ( τ m ) V i ( t ) < or [ V i ( τ m ) = 0 , dV i ( τ m ) V i ( t ) < } . That is, τ m ∈ T is the time of the m th change in Λ( V ) . Without loss of generality, we willassume that τ m = T if the infimum is taken over an empty set. We note that the times τ m areall stopping times with respect to the natural filtration.With these times, note that in particular, on the interval ( τ m , τ m +1 ] we can consider the setof distressed banks to be constant; to simplify, and slightly abuse, notation we can thus considera constant matrix of distressed firms Λ( τ m ) in the interval ( τ m , τ m +1 ] . We will now constructthe unique strong solution forward in time over these time intervals, noting that we update Λ and τ m +1 once the next event is found.First, by construction, on [0 , τ ] there exists a unique solution to the differential systemprovided by V ( t ) = V (0) + c ( t ) and a ij ( t ) = dL ij ( t ) P k ∈N dL ik ( t ) { i =0 } + n { i =0 , j =0 } for all banks i, j ∈ N . Assume there exists a strong solution in the time interval [0 , τ m ] for τ m < T . Nowwe want to prove the existence and uniqueness for the clearing wealths and relative exposureson the interval ( τ m , τ m +1 ] . Expanding dc ( t ) based on its differential form allows us to consider(12) as dV ( t ) = [ I − A ( t ) ⊤ Λ( τ m )] − ( µ ( t, c ( t )) − [ ˙ L ( t ) ⊤ − A ( t ) ⊤ ˙ L ( t )] ~ dt + [ I − A ( t ) ⊤ Λ( τ m )] − σ ( t, c ( t )) dW ( t )= ¯ µ ( t, c ( t ) , A ( t ) , V ( t )) dt + ¯ σ ( t, c ( t ) , A ( t ) , V ( t )) dW ( t ) . Let us first consider the linear growth condition for dV . Utilizing the -norm and where k · k op denotes the corresponding operator norm, let A ∈ A and V ∈ R n +1 , then k ¯ µ ( t, c, A, V ) k + k ¯ σ ( t, c, A, V ) k op ≤ k ( I − A ⊤ Λ( τ m )) − k op (cid:16) k µ ( t, c ) k + k [ ˙ L ( t ) ⊤ − A ⊤ ˙ L ( t )] ~ k + k σ ( t, c ) k op (cid:17) ≤ ∞ X k =0 k [ A ⊤ Λ( τ m )] k k op (cid:16) k µ ( t, c ) k + k ˙ L ( t ) ⊤ ~ k + k A ⊤ ˙ L ( t ) ~ k + k σ ( t, c ) k op (cid:17) ≤ ∞ X k =1 (1 − δ ) k − ! (cid:16) k µ ( t, c ) k + k ˙ L ( t ) ⊤ ~ k + k A ⊤ k op k ˙ L ( t ) ~ k + k σ ( t, c ) k op (cid:17) ≤ (cid:18) δ (cid:19) (cid:16) k µ ( t, c ) k + k [ ˙ L ( t ) ⊤ ~ k + k ˙ L ( t ) ~ k + k σ ( t, c ) k op (cid:17) ≤ δδ sup s ∈ [ τ m ,τ m +1 ] (cid:16) k µ ( s, c ) k + k ˙ L ( s ) ⊤ ~ k + k ˙ L ( s ) ~ k + k σ ( s, c ) k op (cid:17) ≤ θ (1 + k c k ) The second line follows from the triangle inequality and definition of the operator norm. Thethird line is a result of Proposition B.1 and further use of the triangle inequality. The fourthline follows from Proposition 4.4 and noting that, by assumption, Λ = 0 . The upper bound θ ≥ can be determined by Assumption 4.1 and since all terms are continuous and beingevaluated on a compact interval of time (since τ m +1 ≤ T by definition). Further, we wish toprove ¯ µ : T × R n +1 × A × R n +1 → R n +1 and ¯ σ : T × R n +1 × A × R n +1 → R ( n +1) × ( n +1) are jointlylocally Lipschitz in ( c, A, V ) . First ( c, A, V ) ∈ R n +1 × A × R n +1 µ ( t, c ) − [ ˙ L ( t ) ⊤ − A ⊤ ˙ L ( t )] ~ and ( c, A, V ) ∈ R n +1 × A × R n +1 σ ( t, c ) are Lipschitz continuous by their linear (or constant)forms with Lipschitz constants that can be taken independently of time (via continuity andthe compact time domain) as well as the definitions of µ and σ . It remains to show that c, A, V ) ∈ R n +1 × A × R n +1 ( I − A ⊤ Λ( τ m )) − is Lipschitz continuous. Let A, B ∈ A , thenby the same argument as above on the bounds of the norm of the matrix inverse, k ( I − A ⊤ Λ( τ m )) − − ( I − B ⊤ Λ( τ m )) − k op = k ( I − A ⊤ Λ( τ m )) − [( I − B ⊤ Λ( τ m )) − ( I − A ⊤ Λ( τ m ))]( I − B ⊤ Λ( τ m )) − k op = k ( I − A ⊤ Λ( τ m )) − [ A − B ] ⊤ Λ( τ m )( I − B ⊤ Λ( τ m )) − k op ≤ k ( I − A ⊤ Λ( τ m )) − k op k ( I − B ⊤ Λ( τ m )) − k op k Λ( τ m ) k op k [ A − B ] ⊤ k op ≤ (cid:18) δδ (cid:19) k Λ( τ m ) k op k A − B k op ∞ ≤ n (cid:18) δδ (cid:19) k Λ( τ m ) k op k A − B k op . Thus ¯ µ and ¯ σ are appropriately locally Lipschitz continuous on [ τ m , τ m +1 ] .Now we wish to consider the differential form for the relative exposures matrix (13). First,if Λ ii ( τ m ) = 0 (and in particular, Λ ( τ m ) = 0 by assumption of the societal node) then a ij ( t ) = dL ij ( t ) P k ∈N dL ik ( t ) { i =0 } + n { i =0 , j =0 } is the unique solution for any firm j ∈ N overall times t ∈ ( τ m , τ 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 relativeexposures a ij where bank i is in distress between times τ m and τ m +1 , i.e., Λ ii ( τ m ) = 1 . Considerbank i ∈ N with Λ ii ( τ m ) = 1 . Therefore by construction V i ( t ) < for all t ∈ ( τ m , τ m +1 ) . If V i ( τ m +1 ) = 0 then from Proposition 4.4, it already follows that the unique solution a ij ( τ m +1 ) = dL ij ( τ m +1 ) P k ∈N dL ik ( τ m +1 ) must hold, otherwise we can extend V i ( t ) < for t ∈ ( τ m , τ m +1 ] . The differ-ential form for all relative exposures (13) on the interval ( τ m , τ m +1 ] is provided by da ij ( t ) = dL ij ( t ) − a ij ( t ) P k ∈N dL ik ( t ) V i ( t ) − . By construction ( a ij , V i ) ∈ [0 , × − R ++ ˙ L ij ( t ) − a ij P k ∈N ˙ L ik ( t ) − V i islocally Lipschitz and satisfies a local linear growth condition (with constants bounded indepen-dent 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 (12), and relative exposures A from (13), we find that this system satisfies a jointlocal linear growth and local Lipschitz property on the interval ( τ m , τ m +1 ] . Therefore, thereexists some ǫ ∈ L ∞ T ( R ++ ) (such that τ m + ǫ is a stopping time) for which a strong solution for ( c, V, A ) : [ τ m , τ m + ǫ ] → R n +1 × R n +1 × A exists and is unique. Using the same logic with localproperties, we can continue our unique strong solution sequentially. This can be continued untilthe stopping time τ m +1 is reached (found along the path of ( c, V, A ) as a stopping time) or thisprocess reaches some maximal time T ∗ < τ m +1 for which a unique strong solution exists on thetime interval [ τ m , T ∗ ) . First, as c ( t ) can be calculated separately from the clearing wealths andrelative exposures, we can immediately determine that c ( T ∗ ) = lim t ր T ∗ c ( t ) exists. Further, wenote that any solution V ( t ) must, almost surely, exist in the (almost surely) compact space (cid:20) V ( τ m ) − (cid:18) I + 1 + δδ (cid:19) (cid:18)Z tτ m dc ( s ) − + ( L ( t ) − L ( τ m )) ~ (cid:19) , V ( τ m ) + c ( t ) − c ( τ m ) (cid:21) ⊆ L t ( R n +1 ) where = { } ( n +1) × ( n +1) . The lower bound is determined to be based on the bounding of theLeontief inverse; the upper bound follows from the continuous-time version of (15), i.e., V ( t ) = V (0) + c ( t ) − A ( t ) ⊤ V ( t ) − . Additionally, a ij ( t ) almost surely exists in the compact neighborhood [0 , by definition. There-fore ( V ( T ∗ ) , A ( T ∗ )) = lim t ր T ∗ ( V ( t ) , A ( t )) exists by continuity of the solutions and compactnessof 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 τ m +1 = T ∗ to update the set ofdistressed banks Λ . herefore, by induction, there exists a unique strong solution ( V, A ) to (12) and (13) on thedomain [0 , τ m ] for any index m ∈ N by use of [36, Theorem 5.2.1]. In particular this holds up to τ ∗ = sup m ∈ N τ m . If τ ∗ ≥ T then the proof is complete. If τ ∗ < T , then by the same argumentas above we can find ( V ( τ ∗ ) , A ( τ ∗ )) as we can bound both the wealths and relative exposuresinto an almost surely compact neighborhood (and a subset of L τ ∗ ( R n +1 ) ). Therefore, as before,we can start the process again at time τ ∗ , which contradicts the terminal nature of τ ∗ . Thisconcludes the proof. Proof of Lemma 4.9.
Consider the dynamic Eisenberg-Noe systems as fixed point problems forprocesses. That is, consider the fixed point problem ( V, A ) = (Φ V (∆ t, V, A ) , Φ A (∆ t, V, A )) for ∆ t ≥ where Φ V : R + × ( R n +1 ) T × A T → ( R n +1 ) T and Φ A : R + × ( R n +1 ) T × A T → A T definedby Φ V (∆ t, A, V ) := (cid:18) V ( − ∆ t ) + Z t dc ( s ) − A ( t ) ⊤ V ( t ) − (cid:19) t ∈ T Φ a ij (∆ t, A, V ) := R tt − ∆ t dL ij ( s )+ a ij ( t − ∆ t ) V i ( t − ∆ t ) − max { P k ∈N R tt − ∆ t dL ik ( s )+ V i ( t − ∆ t ) − , V i ( t ) − } { i =0 } + n { i =0 , j =0 } t ∈ T if ∆ t > h dL ij ( t ) P k ∈N dL ik ( t ) { i =0 } + n { i =0 , j =0 } i { V i ( t ) ≥ } + dL ij ( τ ( t )) P k ∈N dL ik ( τ ( t )) + R tτ ( t ) dL ij ( u ) − a ij ( u ) P k ∈N dL ik ( u ) V i ( u ) − { V i ( t ) < } t ∈ T if ∆ t = 0 where τ ( t ) := sup { s < t | V i ( s ) ≥ } is the last time that bank i was not in distress before time t . Note that Φ V follows from the logic of (15) by expanding out the recursive formulation (7)or differential systems (12). By construction (Φ V (0 , A, V ) , Φ A (0 , A, V )) = lim ∆ t ց (Φ V (∆ t, A ∆ t , V ∆ t ) , Φ A (∆ t, A ∆ t , V ∆ t )) for paths of convergent relative exposure matrices A ∆ t → A ∈ A T and wealths processes V ∆ t → V ∈ ( R n +1 ) T (in the product topologies). Thus by the uniqueness of the discrete-time andcontinuous-time clearing solutions (Corollary 4.2 and Theorem 4.5) and an application of [22,Proposition A.2], the proof is completed. Proposition B.1.
For any relative exposure matrix A ∈ A and any distress matrix Λ ∈{ , } ( n +1) × ( n +1) such that Λ = 0 and Λ ij = 0 for i = j , the matrix I − A ⊤ Λ is invert-ible with Leontief form, i.e., ( I − A ⊤ Λ) − = P ∞ k =0 ( A ⊤ Λ) k .Proof. By inspection, for any A ∈ A , ( I − A ⊤ Λ)( I + A ⊤ ( I − Λ A ⊤ ) − Λ) = I , i.e., the form ofthe inverse is provided by I + A ⊤ ( I − Λ A ⊤ ) − Λ . We refer to [21, Theorem 2.6] for a detailedproof that ( I − Λ A ⊤ ) − is nonsingular and is provided by the Leontief inverse. Therefore, byconstruction ( I − A ⊤ Λ) − = ( I + A ⊤ ( I − Λ A ⊤ ) − Λ) = I + A ⊤ ∞ X k =0 (Λ A ⊤ ) k ! Λ= I + ∞ X k =0 A ⊤ Λ( A ⊤ ) k Λ k = I + ∞ X k =0 ( A ⊤ Λ) k +1 = ∞ X k =0 ( A ⊤ Λ) k . References [1] Hamed Amini, Damir Filipović, and Andreea Minca. Uniqueness of equilibrium in a pay-ment system with liquidation costs.
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