A Dynamic Default Contagion Model: From Eisenberg-Noe to the Mean Field
AA Dynamic Default Contagion Model:From Eisenberg–Noe to the Mean Field
Zachary Feinstein ∗ Andreas Sojmark † December 19, 2019
Abstract
In this work we introduce a model of default contagion that combines the approaches ofEisenberg–Noe interbank networks and dynamic mean field interactions. The proposed conta-gion mechanism provides an endogenous rule for early defaults in a network of financial institu-tions. The main result is to demonstrate a mean field interaction that can be found as the limitof the finite bank system generated from a finite Eisenberg–Noe style network. In this way, weconnect two previously disparate frameworks for systemic risk, and in turn we provide a bridgefor exploiting recent advances in mean field analysis when modelling systemic risk. The meanfield limit is shown to be well-posed and is identified as a certain conditional McKean–Vlasovtype problem that respects the original network topology under suitable assumptions.
Keywords: systemic risk; financial networks; mean field limit; default contagion; cascades;heterogeneous interactions; core-periphery;
More than a decade after the collapse of Lehman Brothers and the threat of contagious defaultsthroughout the global financial system in 2008, systemic risk is still of vital importance to study.Systemic risk is the risk of financial contagion, i.e., when the failure of one institution spreads toothers due to interlinkages in balance sheets both direct (e.g., via obligations) or indirect (e.g.,via overlapping portfolios). The 2008 financial crisis demonstrated the magnitude of the coststhat systemic crises produce; this necessitates the design of models to consider stress testing offinancial institutions to improve regulation and mitigate the worst effects of a crisis.In this work, we aim to bridge the divide between two, currently unrelated, modeling tech-niques for financial contagion. That is, we will connect the Eisenberg–Noe network approachpopularized by Eisenberg and Noe [2001] to the more recent mean field approaches of systemicrisk. As the goal of this paper is primarily to highlight the overlapping notions between thoseworks, and demonstrate that the network models in fact converge to the mean field limit, wewill focus on simple, but realistic, financial settings that illustrate this point.Briefly, there are two main contagion channels for systemic risk: (i) default contagion and(ii) liquidity contagion .(i) Default contagion occurs if the failure of one bank or institution to repay its debts in fullcauses other banks to default triggering a chain reaction of failing banks. This occursthrough, e.g., a network of interbank obligations as studied in the seminal works of Eisen-berg and Noe [2001], Rogers and Veraart [2013] in a static, network-based setting. Morespecifically, in those works, the default of a bank causes direct impacts to the balancesheets of other banks in the financial system. This loss of capital (potentially) causes ∗ Stevens Institute of Technology, School of Business, Hoboken, NJ 07030, USA. [email protected] † Imperial College London, Department of Mathematics, London, SW7 2AZ, UK. [email protected] a r X i v : . [ q -f i n . M F ] D ec ther banks to default, thus spreading the original shock further throughout the financialsystem. Such an event is denoted “default contagion” as the contagion is via default events.(ii) Liquidity contagion occurs if the illiquidity of one bank or institution (as measured by,e.g., the leverage ratio) causes other banks to also become illiquid. This occurs through,e.g., a fire sale of assets; the liquidation of assets causes the prices fall, this harms theleverage ratio of all other institutions via mark-to-market accounting thus causing furtherliquidations. This has been studied in Cifuentes et al. [2005], Braouezec and Wagalath[2019], Feinstein [2020].In this work we will focus solely on default contagion. Utilizing just this notion of contagionallows us to focus on the two main streams of literature mentioned above—namely balancesheet constructed network models and dynamic mean field models—in order to compare themand, ultimately, show that these at first sight divergent areas are, in fact, studying the samephenomena.The specific modeling assumptions undertaken in this work are chosen in order to capturerealistic financial networks that incorporate dynamic defaults that come as a random shock tothe system. This dynamic and stochastic default contagion is due to all banks holding assetsthat evolve stochastically over time (in this work, often assumed to be a generalised geometricBrownian motion) and having interbank contractual obligations with fixed repayment schedules.Key to this construction is the realistic determination of default considered herein; as in realfinancial systems, we impose that a bank enters default once it has negative capital (on itsbalance sheet due to appropriate accounting techniques). We are primarily interested in howthese default shocks propagate through the system over time as a default contagion event.Moreover, we are interested in how these events depend on the common exposures of the banks,which we model through a common noise component of the external assets. Passing to the meanfield limit, the direct effects of the idiosyncratic noise are averaged out, while the propagationof feedback effects from default contagion remains conditional on the common noise.The remainder of this paper will be organized as follows. Section 2 provides a more detailed,but succinct, overview of both the network-based models and mean field models for systemic risk.It seems that such a combined literature review has not previously been attempted, so we hope itcan spark more interaction between these two areas of research in systemic risk; however, thosecomfortable with this background can safely begin this work in Section 3. From here, Section 3provides an extension of the dynamic network model of Banerjee et al. [2018] to include earlydefaults based on the realised capital of the institutions. This section is studied with a finitenetwork of banks to provide understanding of the system dynamics and is novel on its own.To illustrate the workings of our model, simple numerical examples are provided. Focusing ona particular core-periphery setting, this model is extended in Section 4 to consider the limitas the number of institutions becomes large. In so doing, we introduce a mean field model ofsystemic risk directly built from the balance sheet approach, and we provide a simple numericalsimulation based on a concrete core-periphery example from Section 3. Finally, Section 5 isdedicated to a more general mean field analysis of which the model in Section 4 is a special case.This model and the associated mathematical results are the main contributions of this work. Bypresenting both a finite network and mean field limit for the same systems, we are able to takeadvantage of the positive aspects of both modelling frameworks, including the balance sheetstructure from the (finite) Eisenberg–Noe setting combined with the lower parameter space andmore concise mathematical results of the mean field approach. Network approaches to systemic risk consider a directed and weighted graph of interbank obli-gations to determine the resultant clearing payments made between financial institutions. These etworks determine the defaulting set of banks by finding exactly those institutions who do notpay off their obligations in full when the network clears. A fundamental concept for such modelsis the stylized banking balance sheet. By and large, these models provide a static snapshot ofthe health of the financial system, though dynamics have begun to be included in select networkmodels; more details on these models will be provided herein. The Eisenberg–Noe model.
To fix the concepts, we will present first the Eisenberg–Noeclearing payment system from the seminal work Eisenberg and Noe [2001]. In short, consider n banks that make up the financial system, each with some initial endowment x i ≥ for firm i .Each firm additionally has liabilities to other institutions denoted by the total liabilities ¯ p i ≥ for firm i and the relative obligations between two firms π ij is given by the proportion of thetotal liabilities of firm i owed to j . The realised payments p ∈ R n that each firm makes isobtained by the fixed point equation p = ¯ p ∧ (cid:0) x + Π (cid:62) p (cid:1) (2.1)where a ∧ b := (min( a , b ) , . . . , min( a n , b n )) denotes the lattice minimum in the usual way.One of the key strengths of this model is in its relation to a simple balance sheet that can becalibrated to data, as undertaken in, e.g., Upper and Worms [2004], Mastromatteo et al. [2012],Anand et al. [2015], Gandy and Veraart [2016] using, e.g., data from the European BankingAuthority. Analysis of (2.1) . The Eisenberg–Noe model for clearing payments inherently codifies threekey financial constructs:(i) priority of debt over equity : a firm must first pay off its debts in full before it accumulatesany equity;(ii) limited liabilities : no firm pays more than their contractual obligations; and(iii) pro-rata repayment : there is no seniority structure of debt.This last assumption, on pro-rata repayment, has been weakened to allow for varying senioritystructures and prioritized repayment in, e.g., Elsinger [2009], Feinstein [2019]. A fourth assump-tion is also considered in the Eisenberg–Noe model, though it is relaxed in many subsequentworks. That is,(iv) full recovery in default : a firm has no costs associated with defaulting on its obligations.This was studied in Rogers and Veraart [2013], Glasserman and Young [2015], Weber and Weske[2017] as a strict extension of the Eisenberg–Noe model. The model of Gai and Kapadia [2010]can also be viewed as an extension (2.1) with 0 recovery in default, i.e., no payment is made incase of default. Given that these financial constructs are all rules that define the value of assetsand liabilities, the Eisenberg–Noe model and its extensions are often described as a balancesheet description of the financial system.Mathematically, as the clearing payment problem inherent to the Eisenberg–Noe model isa fixed point equation (2.1), the question of existence and uniqueness is of the paramountimportance. Under the first three financial constructs, and thus allowing for bankruptcy costs,there exists a lattice of clearing payments via the application of Tarski’s fixed point theorem.In particular, this implies that there exists a greatest clearing payment vector; such a paymentscheme would always be chosen as all institutions in the system have the greatest possibleequity under this scheme, i.e., it is the Nash equilibrium of all clearing payments. In addition,if we introduce the fourth financial construct (full recovery of assets) then, under very simpleassumptions (e.g., all banks in the system hold some initial endowment), the clearing paymentis unique. In addition, in the Eisenberg–Noe setting, the sensitivity of the clearing payments tochanges in the system parameters has been studied in Liu and Staum [2010] for the endowmentsand Feinstein et al. [2018] for consideration of the relative liabilities. he fictitious default algorithm , first presented in Eisenberg and Noe [2001], is used toefficiently find the greatest clearing solution. Briefly, this algorithm initially assumes no banksare in default and determines the clearing payments under such an assumption. If any banksdefault in that scenario then in the greatest clearing solution they must also be in default.Fixing only those banks as defaulting, a new clearing payment is computed under such a setting.This process of checking for defaults and determining new clearing payments under fixed setof insolvent banks is repeated until there are no new defaults. As the Eisenberg–Noe modelconsists of finite number n of banks, this process is guaranteed to converge in at most n iterations.Though this algorithm is efficient, and the underlying problem is mathematically well-structured,analytical results are typically not feasible to provide. Random graph approach.
In order to obtain some analytical results in this networkframework, prior works consider passing to the n → ∞ limit of banks in the system. Thesenetwork asymptotics are considered in settings in which the interbank liabilities are describedby a random graph. The simplest random graph model is the Erdös–Rényi network in which afixed valued connection is made between any two banks based on a fixed probability. In manyof these asymptotic studies the model of Gai and Kapadia [2010] is utilised, i.e., there is norecovery in case of default. Such an all-or-nothing payment setting leads to tractable formulaefor the probability of defaults in the graph. We refer to Hurd [2016], and references therein, fora detailed survey of this random graph approach.The analytical results from this random graph approach allow for further considerationsof system stability as well. For the finite network setting, systemic risk measures to determineacceptable capital requirements for the entire financial system have been proposed in, e.g., Chenet al. [2013], Kromer et al. [2016], Feinstein et al. [2017], Biagini et al. [2019b], but these objectsrequire Monte Carlo simulation for any computation. In contrast, Amini et al. [2016, 2012],Amini and Minca [2016], Detering et al. [2019] are able to define a resiliency metric and determinecapital requirements for banks to make the system acceptable to regulators. This asymptoticframework thus provides for simple comparative statics. Dynamic network models of default contagion.
Considerations given thus far aresolely in a static setting. However, banking balance sheets are highly dynamic and subject tofluctuations due to, e.g., market movements. Indeed the conclusion of Eisenberg and Noe [2001]gives a discussion of how to include multiple clearing dates and time dynamics, which has beenstudied in Capponi and Chen [2015], Ferrara et al. [2016]. Additionally, Kusnetsov and Veraart[2018] considers a similar approach to a financial model with multiple maturities.As Capponi and Chen [2015] presented in a discrete time setting: A firm is liquid andsolvent at some time t if it has positive equity and has not previously been insolvent. Due tothe assumptions inherent in (2.1), if a firm is liquid then it must pay in full. A firm is illiquidand solvent at some time t if it has negative cash account, but is able to obtain a loan to coverits deficits, and has not previously been insolvent. As will be described in this work, and asundertaken in Banerjee et al. [2018], Sonin and Sonin [2017], these loans will take the form ofrolling forward unpaid debts from solvent firms to their obligees. This is the key distinction thatcannot exist in the pure Eisenberg-Noe framework as there is no future time point to repay theloan. Finally a firm is insolvent at time t if either it has negative equity or it was previouslydeemed insolvent.Most prior works on dynamic network models consider a discrete time setting Capponi andChen [2015], Ferrara et al. [2016], Kusnetsov and Veraart [2018]. As far as the authors are aware,the only two extensions of the Eisenberg–Noe framework to continuous time are Banerjee et al.[2018], Sonin and Sonin [2017]. Neither of those works considers the mark-to-market equity ofa bank in order to determine insolvency. That is the key innovation being provided in Section 3of this work. .2 Mean Field Approaches In the dynamic mean field approaches to systemic risk, the starting point is to identify each bankin a large financial system with some notion of its financial robustness or distance-to-default atany given time. Next, the financial system is then modelled as a system of interacting stochasticprocesses, whose values represent the current robustness. Typically, these models start fromsome form of Brownian dynamics, but colloquially one could say that they are built on thefollowing premise: in contrast to a risk-neutral Black–Scholes world, the modelling of systemicrisk calls for room to play with the drift and other aspects of the coefficients, in a way thattakes into account the system as a whole.
A concrete particle system.
To fix ideas, let us consider a particular system of n banks,described by their distances-to-default X i and corresponding default times τ i := inf { t ≥ X i ( t ) < } for i = 1 , . . . , n. Letting X τi ( t ) := X i ( t ∧ τ i ) , a simple ‘structural’ model of systemic risk (inspired by Carmonaet al. [2015]) could then be based on dynamics of the form dX τi ( t ) = (cid:16) θn n (cid:88) j =1 (cid:0) X τj ( t ) − X τi ( t ) (cid:1) + µ ( t ) (cid:17) dt + σ ( t ) dW i ( t ) , t ≤ τ i , i = 1 , . . . , n, (2.2)where W i ( t ) = (cid:112) − ρ B i ( t )+ ρB ( t ) , for a family of independent Brownian motions B , . . . , B n .Here ρ > models the presence of a ‘common’ noise, namely B , that captures exposure tocommon risk factors, while θ > incorporates an element of ‘herding’ in the drift that could, forexample, be the result of banks engaging in similar strategies and other interbank connections.To capture systemic risk, the natural quantities in this model are the average distance-to-defaultand the proportion of defaults given, respectively, by M n ( t ) := 1 n n (cid:88) i =1 X τi ( t ) and L n ( t ) := 1 n n (cid:88) i =1 t ≥ τ i . Of course, an obvious weakness is the inherent symmetry in this model, and this is indeed acentral point to be addressed later in this work when we return to the Eisenberg–Noe approach,as discussed in Section 2.1 above. For the purposes of this overview, however, we remain in thesymmetric setting.
Mean field analysis of (2.2) . Sending n → ∞ in (2.2), one can hope to simplify theanalysis and simulation of the system, provided there is a law of large numbers effect. To seethat there is indeed such an effect, note that M nt = (cid:104) ν nt , Id (cid:105) and L nt = 1 − ν nt (0 , ∞ ) , where theempirical measures ν nt := n (cid:80) ni =1 t<τ i δ X ni ( t ) are tracking the surviving banks. From Hamblyand Søjmark [2019] it is then known that ( ν n , M n , L n ) converges to a unique limit ( ν, M , L ) ,where M t = (cid:104) ν t , Id (cid:105) , L t = 1 − ν t (0 , ∞ ) , and ν t has a density V t which solves the nonlinear SPDE dV t ( x ) = σ ∂ xx V t ( x ) dt − ∂ x (cid:0) [ θ ( M ( t ) − x ) + µ ] V t ( x ) (cid:1) dt − ρσ∂ x V t ( x ) dB ( t ) , (2.3)for x ∈ (0 , ∞ ) , with an absorbing boundary condition at the origin, i.e. V t (0) = 0 . The distri-bution of the limiting processes M and L can be used to give measures of systemic risk. Forexample, one can track the probability of seeing changes in M and L above some threshold overa short period of time. A simple observation is that, for larger θ > and ρ > , the distributionof the change in L over a given period becomes more concentrated at the extremes with banksmore likely to either survive or default together.Translating (2.3) to the language of McKean–Vlasov SDEs, we have M ( t ) = E [ X ( t ) t<τ | B ] and L ( t ) = P ( t ≥ τ | B ) , where τ = inf { t ≥ X ( t ) ≤ } and dX ( t ) = (cid:0) θ ( M ( t ) − X ( t )) + µ (cid:1) dt + σd ( (cid:112) − ρ B + ρB )( t ) . (2.4) his follows by an application of Itô’s formula, which shows that (2.3) is the (nonlinear) stochas-tic Fokker–Planck equation for (2.4) absorbed at the origin, conditional on B . That is, we have E [ φ ( X t ) t<τ | B ] = (cid:82) ∞ φ ( x ) V t ( x ) dx for all φ ∈ C b ( R ) .Building on the above, one could consider strategic interactions in (2.2) with costs andcontrols depending on M n and L n . This would then yield a mean field game involving the limitprocesses M and L . Without the common noise, a framework for this type of mean field gamehas recently been developed in the two consecutive papers Campi and Fischer [2018], Campiet al.. Mean field models of contagion.
Recently, a new line of mean field modelling hasbeen proposed in Hambly et al. [2019], Hambly and Søjmark [2019], Nadtochiy and Shkolnikov[2019a] aimed at studying default contagion in large financial systems. Based on a ‘structural’approach, these models introduce an endogenous notion of contagion in systems such as (2.2), byimposing that bankruptcies should cause a drop in the distances-to-default of the other banks.Mathematically, this amounts in one way or another to incorporating the proportion of defaults L n into the dynamics, thus leading to positive feedback loops whereby defaults can shift otherbanks into default. Variations of this approach and further theoretical results can be foundin Ledger and Søjmark [2018a,b], Nadtochiy and Shkolnikov [2019b]. Moreover, we note thatclosely related approaches to contagion (in a dynamic but finite-dimensional setting) have alsobeen considered in Battiston et al., Lipton [2016].In terms of numerical implementation, Kaushansky et al. [2018c], Kaushansky and Reisinger[2019] have proposed and analysed numerical schemes for the mean field model of Hambly et al.[2019], and it is noted in Kaushansky et al. [2018c] that a modified version of Lipton [2016] fallswithin this framework. These developments can be seen as following on from Itkin and Lipton[2017], Kaushansky et al. [2018a,b], where similar models are studied for systems of two or threebanks, and we note that passing to the mean field limit yields a way of alleviating the curseof dimensionality arising from the couplings due (in particular) to mutual obligations in largefinancial systems.In this paper we will show how a variant of these ‘structural’ approaches to contagion isintrinsically connected to a dynamic Eisenberg–Noe model with early defaults (as developed inSection 3). Moreover, we will show (in Sections 4 and 5) that the associated mean field limitcan be derived and analysed rigorously by extending the techniques from Hambly et al. [2019],Ledger and Søjmark [2018a,b]. The broader mean field literature on systemic risk.
If, for simplicity, the con-straints on the state space in particle system (2.2) are dropped, then the dynamics are preciselythose of the early papers Carmona et al. [2015], Fouque and Sun [2013], where X i now denotesthe (logarithmic) cash-reserves of bank i and the mean-reversion models borrowing and lendingin the interbank market. In Carmona et al. [2015] these dynamics emerge as a Nash equilibriumof a stochastic game (where drifts are controlled and it is costly to diverge from the mean) witha variant of (2.4) without absorption arising as the equilibrium dynamics for the limiting meanfield game.Starting from Fouque and Ichiba [2013], several other papers on systemic risk have studieddifferent versions of this mean-reverting setup (mostly without the common noise). These con-tributions can be loosely grouped into: systems with stabilisation by a central agent Garnieret al. [2013, 2017], games with delay Carmona et al. [2018], Fouque and Zhang [2018], gameswith model uncertainty Huang and Jaimungal [2017], utility optimisation by the individualbanks and a central bank Maheshwari and Sarantsev [2017], methods for introducing hetero-geneity Chong and Kluppelberg, Fang et al. [2017], jump-diffusion dynamics Bo and Capponi[2018], Borovykh et al. [2018], Benazzoli et al. and connections to the theory of risk measuresBiagini et al. [2019a]. Still focusing on mean-reversion, constraints on the state space havebeen considered via Feller type square root diffusions in Bo and Capponi [2018], Fouque andIchiba [2013], Shkolnikov and Ichiba [2013], Sun [2018] (with various additional features) and, insuch a framework, Capponi et al. [2019] has recently proposed a network structure with finitely any clusters of banks, where each cluster mean-reverts around different predetermined levelsmodelling the presence of target leverage ratios.In addition to the ‘structural’ approaches to contagion discussed earlier, there is a separateliterature on contagion in large financial systems, wherein defaults are dictated by exponentialclocks as in the ‘reduced-form’ approach to credit risk. This leads to more implicit notions ofcontagion occurring at the level of the intensities. Firstly, Giesecke et al. [2013, 2015] proposea system of interacting intensities that are self-exciting via dependence on the proportion ofdefaults. Secondly, somewhat closer in spirit to (2.2), Ichiba et al. [2019] identifies the financialhealth of each bank in a large system with a geometric Brownian motion, but with defaultdictated by an exponential clock whose intensity can depend on the banks own health andthe average healthiness of the system. Contagion amounts to each default causing a drop inthe healthiness of the other banks by a random fraction, which in turn increases the defaultintensities. n Banks
As mentioned above, the primary goal of this section is to introduce a dynamic network modelto study default contagion in a finite system of banks. To do so, we seek to extend the dynamicnetwork model of Banerjee et al. [2018] to incorporate early defaults due to negative (accounting)capital. This extension to include early defaults is novel and important in its own right. Thiswill be utilised in Section 4 to consider a comparison with a mean field limit. More details onthe reasons for undertaking that analysis are provided in Section 4, and we also refer to thebrief discussion in the introduction above.This section is broken into two subsections. First, in Section 3.1, we describe the stylizedbalance sheet of all banks in the system. This is used to define a general dynamic model fordefault contagion in the vein of Eisenberg and Noe [2001], Rogers and Veraart [2013]. Second,we simplify the parameters so as to study a specialized network setup that facilitates the lateranalyses of this work. As the primary goal of this work is to merge the network and meanfield approaches in the literature, we find this specialized network setup is instructive. For theanalysis of this section, none of these additional assumptions are required for the theoreticalresults.Briefly, before undertaking the analysis, we wish to consider some notation utilised through-out this work. Consider a financial system with n ∈ N financial institutions. This system doesnot include the central bank or other financial entities not included within this system; we willconsider such an entity, called the “societal node” and denote it by node . Notationally, let N = { , , ..., n } be the set of banks and N = N ∪ { } include the societal node. As we areconsidering a dynamic network model, consider a continuous set of clearing times T = [0 , T ] for some (finite) terminal time T < ∞ . For simplicity, assume throughout this work that therisk-free rate is 0 ( r = 0 ). Finally, we will use the notation Z ( t ) for the value at time t ∈ T ofa process Z : T → R n . We will now consider a model akin to the continuous-time setting ofBanerjee et al. [2018] in that we allow for liabilities to change over time and for firms to havestochastic cash flows. In order to construct a continuous-time model we will begin by considering the stylized balancesheet for a generic bank i ∈ N in our system. This balance sheet comes from a dynamic versionof Eisenberg and Noe [2001]. Throughout time, all assets are of only two types: interbank assetsand external assets. All liabilities are either interbank (and thus assets for another bank j ∈ N in the system) or external and owed to the societal node .In order to construct a continuous-time model we will begin by considering our networkparameters of cash flows and nominal liabilities. We will now consider a banking system withstylized balance sheet as depicted in Figure 1. alance Sheet @ t Assets Liabilities
External (Mark-to-Market) E [ x i ( T ) | F t ] Interbank (Solvent) (cid:80) j ∈A t L ji ( T ) Interbank (Insolvent) (cid:80) j ∈N \A t (cid:18) (1 − R ) L ji ( τ j )+ R L ji ( T ) (cid:19) Total (cid:80) j ∈N L ij ( T ) Capital K i ( t ) Figure 1:
Stylized balance sheet for firm i ∈ N at time t ∈ T . Let x i ( T ) be the value of the external assets for firm i ∈ N at the terminal time T .This will often be denoted in vector notation as x ( T ) . In mark-to-market accounting, at time t ∈ T , these external assets should be valued in (risk-neutral) expectation, i.e., E [ x ( T ) | F t ] = x (0)+ (cid:82) t dx ( s )+ E [ (cid:82) Tt dx ( s ) | F t ] . The value of the external assets can, equivalently be describedby the (marginal) cash flows external to the system, e.g., from depositors at the banks, as utilisedin Banerjee et al. [2018] for a dynamic version of Eisenberg and Noe [2001]. In this context, wedescribe dx ( t ) to be the marginal change in the external assets at time t ∈ T , i.e., firm i ∈ N has incoming external cash flows (cid:82) t t dx i ( t ) between times t < t . Throughout this work wewill take the external assets to follow a non-negative (Itô) process.In contrast, we will assume the total nominal liabilities matrix L is a deterministic processof time as these obligations are contractually generated and have fixed repayment schedule.In other words, by looking at all outstanding contracts at time , the total amount that isowed between any two institutions (and externally) up to any time can be determined exactly.Generally we will consider dL ( t ) to be the marginal change in nominal liabilities matrix attime t , i.e., the liabilities owed from time t to t are defined by (cid:82) t t dL ( t ) ∈ R ( n +1) × ( n +1)+ . Byassumption dL ij ( t ) ≥ for all firms i, j ∈ N as, without any payments made, total liabilitiesshould accumulate over time. Additionally, dL ii ( t ) = 0 for all firms i ∈ N to remove thepossibility of self-dealing. This nominal liabilities matrix appears on both the asset and liabilitiesside firms. The liabilities for firm i is the total amount owed over T , i.e., (cid:80) j ∈N L ij ( T ) = (cid:80) j ∈N (cid:82) T dL ij ( s ) . To simplify notation, we will define d ¯ p ( t ) := dL ( t ) (cid:126) for any time t ∈ T to denote the marginal change in the total liabilities vector where (cid:126) , . . . , (cid:62) ∈ R n +1 ;correspondingly, the total liabilities owed by firm i over T are given by ¯ p i ( T ) . The interbankassets require consideration of historical price accounting since the interbank assets are generallynonmarketable. As such, firm i ∈ N will give full value to all obligations (both past and future) (cid:80) j ∈A t L ji ( T ) = (cid:80) j ∈A t (cid:82) T dL ji ( s ) from solvent institutions A t at time t ; for insolvent firms ∈ N \A t , firm i will give full value up to the insolvency time τ j ∈ T (discussed furtherbelow), but only a fixed recovery rate R ∈ [0 , on obligations from j after insolvency, i.e., (cid:82) τ j dL ji ( s ) + R (cid:82) Tτ j dL ji ( s ) = (1 − R ) L ji ( τ j ) + R L ji ( T ) . Assumption 3.1.
The modeling assumptions expressed above can be summarized thusly:(i) the external assets of each bank follow a stochastic process (which can be correlated to eachother) and (being marketable) are marked-to-market with risk-neutral measure P ;(ii) the interbank assets and liabilities are solely based on contracts written prior to time andhave fixed repayment schedule; and(iii) interbank assets (being nonmarketable) are valued using historical price accounting, i.e.,priced at face value prior to a default event and reevaluated with the true recovery rateafter default.These three key modeling assumptions lead to a contagion mechanism in which defaults come asa shock to the system and cause a jump in the capital of any connected institution. The shocks due to default outlined above are realistic since the interbank assets are non-marketable. If, however, banks attempted a counterparty or network valuation adjustment (see,e.g., Barucca et al. [2016], Banerjee and Feinstein [2019]) default shocks would still be expecteddue to the assymetric and incomplete information available to the different banks. We also wishto note that the historical price accounting rule undertaken herein provides the greatest possiblevalue for interbank assets and thus provides a bound on any other valuation system.The balance sheet capital for firm i at time t ∈ T is exactly the difference on its balancesheet between assets and liabilities, i.e., K i ( t ) = E [ x i ( T ) | F t ] + (cid:88) j ∈A t L ji ( T ) + (cid:88) j ∈N \A t [(1 − R ) L ji ( τ j ) + R L ji ( T )] − ¯ p i ( T ) . (3.1)Insolvency for bank i occurs at the first time that it has negative capital, i.e., τ i = inf { t ∈ T | K i ( t ) < } and the set of solvent firms at time t is given by A t := { i ∈ N | τ i > t } . Remark 3.2.
The stochastic structure introduced herein is necessary for consideration of earlydefaults. Without it, the capital of banks would be deterministic and all defaults would beknown at the initial time . Though Banerjee et al. [2018] introduces a stochastic system forfinancial networks, it does not consider endogenous early defaults. That is an innovation of thiswork. Remark 3.3.
In the balance sheet approach considered herein, due to the full recovery ofinterbank assets prior to default and a fixed recovery after default, the details of the Eisenberg–Noe Eisenberg and Noe [2001] are only subtly utilised in the background. That is, the constantrecovery implies a pro-rata repayment scheme as in the Eisenberg–Noe framework; the differencebetween this repayment scheme and that of Eisenberg–Noe and Rogers-Veraart Rogers andVeraart [2013] is that recovery is on the liability side rather than the asset side. We take thisas a simplification to ease the discussion and mathematics to focus primarily on the stylizedcontagion in this work.Additionally, we can consider this balance sheet framework as akin to the dynamic networkmodels of Banerjee et al. [2018], Sonin and Sonin [2017], but adding in notions from the discrete-time model of Capponi and Chen [2015] in which firms can default before the terminal time. Inthat work there is a detailed discussion on an auction model for determining the recovered assetsin case of default from which the remaining debts are paid; this is in contrast to the simplifiedexogenous recovery rates. If we take the approach from Banerjee et al. [2018] in which firms payoff debts as they arrive and may have unpaid prior liabilities, the construction of the systemdynamics requires further considerations. Briefly, let V i ( t ) denote the cash holdings of firm i Time t C ap i t a l K Bank 1Bank 2Bank 3
Figure 2:
A realisation of the 3 bank system described in Example 3.8 with a contagious default at t ≈ . . at time t ∈ T . Let π ij ( t ) be the relative liabilities at time t . This is constructed in detail in acontinuous-time setting in Banerjee et al. [2018]; we refer to that paper for a detailed discussionof the construction of the relative liabilities in this general setting. It is possible that a firm haspositive capital K i ( t ) > but insufficient funds to cover short term liabilities; in such a settingwe assume that the debts roll-forward when they go unpaid by a solvent firm as in Banerjee et al.[2018], Sonin and Sonin [2017]. When a firm defaults, we consider a recovery rate R ∈ [0 , on the unpaid previous debts and R ∈ [0 , on future obligations. As such we have the cashholdings and (modified) capital equations at time t ∈ T as: V ( t ) = x i ( t ) + (cid:88) j ∈N (cid:2)(cid:0) L ji ( t ) − π ji ( t ) V j ( t ) − (cid:1) t<τ j + (cid:0) (1 − R ) L ji ( τ j ) + R L ji ( T ) − (1 − R ) π ji ( τ j ) V j ( τ j ) − (cid:1) t ≥ τ j (cid:3) − ¯ p i ( t ) K i ( t ) = V i ( t ) + E [ x i ( T ) | F t ] − x i ( t ) + (cid:88) j ∈N (cid:0) π ji ( t ) V j ( t ) − + L ji ( T ) − L ji ( t ) (cid:1) t<τ j − [¯ p i ( T ) − ¯ p i ( t )] . Much of the results of this work can be undertaken in this setting with a liability structuredefined in Assumption 3.5. For simplification and from financial interpretation: we will beinterested in the setting where R = 1 . We assume this from the idea that, prior to the defaulteven though a firm may be illiquid, it is solvent and thus some lender of last resort will guaranteethese obligations that rolled forward. We will make the following assumptions for the remainder of this paper. These can be relaxedas in Banerjee et al. [2018] and discussed briefly in Remark 3.3, but as the goal of this workis to demonstrate a simple and clear comparison between a dynamic Eisenberg–Noe model anddefault contagion in the mean field limit we will consider this simplification.
Assumption 3.4.
Throughout this work we consider a short time horizon T model wherein theliability repayment schedule is constant over time, i.e., dL ij ( t ) = λ ij dt for i, j ∈ N with λ ii = 0 and λ j ( t ) = 0 , and L (0) = 0 . The constant nature of the network, as defined in Assumption 3.4, is valid for a shorttime frame. Since financial crises occur over short time horizons, this fixed nature is therefore ppropriate. Further, this allows us to study how the initial network topology can cause defaultcontagion and, ultimately, a systemic crisis.For the remainder of this paper we will introduce the notation ¯ L i ( T ) := (cid:88) j ∈N [ L ij ( T ) − L ji ( T )] = T λ i + T (cid:88) j ∈N [ λ ij − λ ji ] to denote the difference between obligations (external and interbank) and interbank assets.Typically we will assume ¯ L i ( T ) > for all firms i , i.e., bank i has liabilities that cannot beoffset solely by interbank assets. Under such an assumption, every bank is a net borrower overall(in the sense that total obligations are larger than interbank assets); this is true even if a bankis not a net borrower in the interbank system. Assumption 3.5.
For simplification and ease of use, we will assume for the remainder of thispaper that the external cash flows follow (possibly time-dependent) correlated geometric Brownianmotions, i.e., dx i ( t ) = x i ( t )[ µ i ( t ) dt + σ i ( t ) dW i ( t )] for vector of correlated Brownian motions W . Under the setting of Assumptions 3.4 and 3.5, we can compute the capital process K ( t ) from (3.1) as: K i ( t ) = x i ( t ) e (cid:82) Tt µ i ( s ) ds − ¯ L i ( T ) − (1 − R ) (cid:88) j ∈N \A t ( T − τ j ) λ ji . (3.2)In order to determine (3.2), we take advantage of the external assets following a geometricBrownian motion to find E [ x i ( T ) | F t ] = x i ( t ) e (cid:82) Tt µ i ( s ) ds . Assumption 3.6.
We wish to assume that no banks are in default at time t = 0 . This isequivalent to bounding the initial external assets from below, i.e., x i (0) > T [ λ i + (cid:80) j ∈N [ λ ij − λ ji ]] e − (cid:82) T µ i ( s ) ds almost surely for every bank i ∈ N . Lemma 3.7.
If Assumptions 3.4-3.6 are satisfied, there exists a greatest and least clearingcapital K ↑ ≥ K ↓ (component-wise and for every time t ).Proof. First, recall that the default times are defined by τ i = inf { t ∈ T | K i ( t ) < } for bank i . The fixed point problem for the capital process K only depends on itself through the defaulttimes τ . Now, consider the fixed point in the capital process. Note that as the capital process K decreases the default times τ all decrease. Further, as banks default, the entire system’swealth drops as well since R < . With this, we are able to complete this proof through a useof Tarski’s fixed point theorem.Throughout the remainder of this work, we will focus on the greatest clearing solution K ↑ .In the Eisenberg–Noe framework, this is computed using a fictitious default algorithm . Briefly,such an algorithm assumes that at time t ∈ T , any bank that was solvent prior to t ( A t − ) isassumed to still be solvent; this is the best case scenario for all banks due to the downwardstresses from a default. Solvency ( K i ( t ) ≥ ) of all banks is then checked under this scenario; ifno banks default we can move forward in time, otherwise any new defaults may cause a dominoeffect of further defaults. In the case of defaults, we update the balance sheet of all solventfirms to determine if this shock causes a cascade of failures. This sequential testing for newdefaults and updating the balance sheets continue until no new defaults occur. In practice thisalgorithm is run using an event finding algorithm to determine the time of the initial default,at that time the cascading defaults are determined until the system re-stabilises at a new set ofsolvent institutions, and the stochastic processes evolve normally until the next default event.This is demonstrated in Figure 2 where the insolvency of one bank causes another bank todefault as well. If desired, the least clearing solution K ↓ could be found analogously with afictitious solvency algorithm instead. e wish to preview a consideration of the cascade condition of Section 4.3.1 and Section 5.3.1which is a rule for determining the size of default cascades (tailored to our later reformulationof the Eisenberg–Noe banking system as a stochastic interacting particle system). We do thisto highlight the similarity between this condition and the fictitious default algorithm widelyused in the (finite) network setting. Further, we wish to emphasise that, in the mean fieldframework, a cascade needs to be defined more carefully, as liabilities between institutions areinfinitesimals and it is no longer meaningful to talk about defaults of individual banks. Thedetails underlying the definition of the cascade condition are provided in Section 5.2, and weemphasise already here that the iterations of this cascade condition correspond analogously withthe fictitious default algorithm (see, in particular, Section 5.2.2 for the precise description ofthe resolution of default cascades). As far as the authors are aware, a connection between thefictitious default algorithm and the cascade condition presented herein, or its predecessors inthe mean field literature, has not previously been investigated.We conclude this section with consideration of two numerical examples. The first is thedescription of the small 3 bank system shown in Figure 2. We then consider a larger systemwith a core-periphery structure. As noted in Craig and Von Peter [2014], Fricke and Lux [2015],many real world financial systems exhibit a core-periphery structure. Example 3.8.
This example will be constructed to demonstrate the primary features of themodel, namely early defaults and the contagion thereof. Consider a n = 3 bank system overa time horizon T = [0 , for simplicity. These banks are connected to each other throughinterbank obligations λ = where the last column denotes the obligations λ · to the societal node. These banks holdidentical, correlated external assets following the geometric Brownian motion dx i ( t ) = x i ( t )[ dt + dW i ( t )] with correlations dW i ( t ) dW j ( t ) = dt for i (cid:54) = j ∈ { , , } . The initial value of theseexternal assets are chosen so that the initial capital of all banks is 1, i.e., set x i (0) = 2 e − (whichresults in K i (0) = 1 ) for all banks i ∈ { , , } . By setting the parameters in this way we satisfyAssumptions 3.4-3.6. Finally, we fix the recovery rate for defaulted assets to be R = 0 . . Onerealisation of this system is shown in Figure 2. In that realisation, bank defaults due to itsown investments at time τ ≈ . . This causes bank to default at that time τ ≈ . as welldue to the shock to its own capital from re-marking its interbank assets. Bank remains solventfor the studied time frame T but a large negative shock is exhibited on its capital due to thedefault of both banks and .The following example is a 10 bank system exhibiting the core-periphery structure. This net-work topology is discussed in more detail in Borgatti and Everett [2000], Fricke and Lux [2015].Of particular note, empirical studies (see, e.g., Craig and Von Peter [2014], Fricke and Lux [2015])have demonstrated that real-world financial systems exhibit this structure. In undertaking thisstudy, two network structures will be considered: first, a highly connected networks with smallobligations from peripheral to peripheral institutions; second, we consider the same network butwith no obligations between peripheral institutions. The second, approximate, network has alimit to its rank, i.e., the rank of the second network obligation matrix is bounded by twice thenumber of core institutions. This approximating network, with a much sparser network, is moretractable computationally and, as such, will be used to motivate a rank decomposition structurein the mean field limit of the dynamic network model discussed within this section. For moredetails, we refer to the next section. Example 3.9.
Consider a n = 10 bank core-periphery system with core banks and peripheral ank Table 1:
Default times for institutions in Example 3.9 under a single realisation of the external assets. Normeddifference between these default times is . . ones. These banks are interconnected with the (randomized) interbank liabilities λ = .
01 0 0 0 0 3 .
43 2 .
87 2 .
87 2 . .
35 0 3 .
08 2 .
36 2 .
78 2 .
80 1 .
13 0 .
94 0 .
94 0 . .
54 2 .
23 0 0 .
04 0 0 .
05 0 .
06 0 .
05 0 .
05 05 .
90 2 .
90 0 0 0 0 .
06 0 .
07 0 0 04 .
67 2 .
29 0 .
05 0 .
04 0 0 0 0 .
05 0 0 . .
40 2 .
16 0 .
05 0 .
04 0 0 0 .
05 0 .
05 0 .
05 03 .
64 4 .
47 0 0 .
04 0 0 .
05 0 0 0 0 . .
41 4 .
18 0 0 .
04 0 .
04 0 .
04 0 .
05 0 0 0 . .
25 3 .
99 0 0 0 0 0 .
05 0 .
04 0 04 .
31 5 .
29 0 0 .
05 0 0 0 0 0 0 and all banks owe $1 to the societal node. As in real financial systems, there are sparse, andsmall, obligations between peripheral firms. For a comparison, consider a reduced system ofobligations so that the obligations between peripheral firms are zeroed out, i.e., ˆ λ = .
01 0 0 0 0 3 .
43 2 .
87 2 .
87 2 . .
35 0 3 .
08 2 .
36 2 .
78 2 .
80 1 .
13 0 .
94 0 .
94 0 . .
54 2 .
23 0 0 0 0 0 0 0 05 .
90 2 .
90 0 0 0 0 0 0 0 04 .
67 2 .
29 0 0 0 0 0 0 0 04 .
40 2 .
16 0 0 0 0 0 0 0 03 .
64 4 .
47 0 0 0 0 0 0 0 03 .
41 4 .
18 0 0 0 0 0 0 0 03 .
25 3 .
99 0 0 0 0 0 0 0 04 .
31 5 .
29 0 0 0 0 0 0 0 0 . This reduced system ˆ λ has rank instead of full rank for the original network. In Table 1, thedefault time for each bank is reported under the original (full) and reduced networks. Notably,though these default times are not identical, they capture the general behavior quite accurately.We will take advantage of this notion of the reduced system in the following sections.In the next section, we start from the above example and discuss the mean field limit thatresults from sending the number of banks to infinity in a suitable way. As already mentioned in the introduction of this paper, the main motivation for the presentsection is theoretical in nature: the aim being to close a gap between the network literatureon systemic risk and recent mean field contagion models. Specifically, we will relate the finiteinterbank system from Section 3 to a mean field limit described by a conditional McKean–Vlasov problem akin to the problems studied in Hambly et al. [2019], Hambly and Søjmark[2019], Ledger and Søjmark [2018a], Nadtochiy and Shkolnikov [2019b,a].Aside from this theoretical perspective, there are several good practical reasons for studyingthe mean field limit of the model proposed in Section 3. First of all, the mean field limitrigorously facilitates a low parameter space, which can allow for a clearer identification of the ain mechanisms at work, and which may serve as a vehicle for defining macroscopic events.Secondly, the mean field limit can allow for more efficient numerical simulations by replacinga large system of coupled SDEs with a single limiting object. Thirdly, one is unlikely to haveprecise data for the liabilities matrix, but the mean field limit makes a rigorous case for workingwith an approximate distribution. Finally, the lower parameter space can facilitate calibrationto the average of a large sample of banks, as opposed to the unfeasible task of fitting fullyheterogeneous parameters in the finite dynamic system.In the present section, we focus on the financial motivation and thus restrict attention tothe core-periphery structure discussed at the end of Section 3. This leads us to introduce aparticular intuitive and tractable mean field point of view on the Eisenberg–Noe style interbanksystem from Section 3. While we give a careful presentation of the mathematical results for themean field limit, along with a numerical example, the theoretical details are left to Section 5,which treats a more general framework. Before addressing the mean field setup, consider a finite financial system of size n = m con-sisting of m c core banks and m p := m − m c peripheral banks, where the peripheral banks aredefined by not having any liabilities towards each other. In other words, the liabilities matrixfor the system can be written in the block form λ m × m = (cid:18) A BC (cid:19) = (cid:18) A m c × m c B m c × m p C m p × m c (cid:19) . (4.1)We could also work with sparse connections between the peripheral banks, but the idea here isto keep the model simple and focus on the core-periphery interactions, so we simply zero outthe periphery-to-periphery interactions in line with the discussion in Section 3.2 above.In general, the λ ij ’s can be completely different for each pair of banks ( i, j ) , but it is naturalto suppose that they are nonetheless representative of some underlying structure in terms ofhow the core and peripheral banks interact. One tractable way of capturing this is to declarethat λ ij = (1 + (cid:15) i )(1 + δ j )ˆ λ ij , (4.2)where the ( (cid:15) i , δ i ) ’s are random samples from P ⊗ P , for some distribution P with mean zeroand support in [ − , (or similar), and the ˆ λ ij ’s are the fixed entries of a nicer matrix ˆ λ m × m ,which defines the underlying structure of the network. For concreteness, let us consider thespecific example ˆ λ m × m := (cid:18) ˆ A ˆ B ˆ C (cid:19) := (cid:18) (cid:19) (cid:18) · · · · · · (cid:19) × m p , (cid:18) · · · · · · (cid:19) × m p , ... ... m p , × ... ... m p , × . (4.3)In this case, there are two core banks (i.e., m c = 2 ) and the peripheral banks can be dividedinto two groups (of size m p , and m p , with m p = m p , + m p , ) in terms of how they interactwith the core. Nevertheless, the real connections are subject to noise—modelled by (4.2)—andhence λ m × m can feature much more asymmetry in the core-periphery interactions. .1.1 Growing the number of banks to infinity Starting from the above system of size m , we now introduce a natural way of growing it toinfinity. Based on the ‘initializing’ system of size m , for each m ≥ , the idea is to construct asystem of size n = mm according to the following procedure: • multiply each of the m c core banks into m analogous entities (that can be seen as sub-entities comprising a core bank of m times the size of the original), for a total of mm c coreentities • multiply each of the m p peripheral banks into m analogous peripheral entities, for a totalof mm p peripheral entities • let the external assets of each entity have an i.i.d. copy of the same initial condition aswell as the same drift and volatility as the original bank up to an i.i.d. noise. • impose that the m sub-entities of a given core bank do not have liabilities towards eachother (which is enforcing no self-dealing within the core bank) • impose that, up to noise, the liability positions between a given core and peripheral entityare the same as those between the original core and peripheral bank only scaled by m − (meaning that the underlying network structure is preserved and, the noise aside, eachentity has the same total liabilities as the original bank of which it is a copy)To be precise, starting from an underlying matrix ˆ λ m × m as in the example (4.3), weconstruct the n = mm ’th system by first fixing the underlying network structure through themapping ˆ λ m × m = (cid:18) ˆ A ˆ B ˆ C (cid:19) (cid:55)−→ ˆ λ mm × mm := 1 m ˆ A · · · ˆ A ˆ B · · · ˆ B ... . . . ... ... . . . ... ˆ A · · · ˆ A ˆ B · · · ˆ B ˆ C · · · ˆ C · · · ... . . . ... ... . . . ... ˆ C · · · ˆ C · · · , (4.4)and then the liabilities matrix λ mm × mm is defined by setting λ ij := (1 + (cid:15) i )(1 + δ j )ˆ λ ij (4.5)as in (4.2). Here the ˆ λ ij ’s are now the entries of ˆ λ mm × mm given in (4.4), and the ( (cid:15) i , δ i ) ’s arerandom samples drawn from the distribution P ⊗ P , for a given probability measure P .We stress that, due to the noise, the entries of λ mm × mm can be entirely heterogeneousboth across and within groups. In particular, a given sub-entity of the first core bank mayinteract differently with all entities representing the second core bank, and any given core entitymay interact differently with all peripheral entities across the two groupings. Nevertheless, bypassing to the mean field limit we may hope to discover the underlying structure as definedby the ‘initializing’ matrix ˆ λ m × m . The remaining subsections illustrate this for the specificexample provided by (4.3). Returning to the Eisenberg–Noe model from Section 3, consider a finite system of size n , andnote that the capital (3.2) of each bank i = 1 , . . . , n can be written as a coupled system K i ( t ) = x i ( t ) e (cid:82) Tt µ i ( s ) ds − ¯ L i ( T ) − T (1 − R ) (cid:90) t (1 − sT ) d L ni ( s ) , (4.6) here L ni ( t ) := n (cid:88) j =1 λ ji t ≥ τ j with τ j = inf { t ≥ K i ( t ) < } . (4.7)For simplicity, we will assume that ¯ L i ( T ) = T Λ i for some constants Λ , . . . , Λ n > , meaningthat, for each bank i , its total liabilities net of interbank assets over the period [0 , T ] is givenby the positive amount T Λ i . In particular, if a bank is a net lender in the interbank market,then the surplus is more than offset by external liabilities, which is in line with what is observedin practice. Recalling that each x i ( t ) is a geometric Brownian motion, it is convenient to workwith the following logarithmic ‘distances-to-default’ defined by X i ( t ) := log (cid:26) x i ( t ) exp { (cid:82) Tt µ i ( s ) ds } Λ i T + T (1 − R ) (cid:82) t (1 − sT ) d L ni ( s ) (cid:27) , (4.8)for i = 1 , . . . , n . This transforms the system (4.6)-(4.7) into the equivalent system L ni ( s ) = n (cid:88) j =1 λ ji t ≥ τ j , τ j = inf { t ≥ X j ( t ) < } dX i ( t ) = − σ i ( t ) dt + σ i ( t ) dW i ( t ) − d log (cid:110) − R )Λ i (cid:90) t (1 − sT ) d L ni ( s ) (cid:111) X i (0) = log { x i (0) } − log { Λ i T } + (cid:90) T µ i ( t ) dt, (4.9)where we recall that x i (0) e (cid:82) T µ i ( t ) dt > ¯ L i ( T ) = Λ i T , by Assumption 3.6, which guarantees X i (0) > . Moreover, we will assume that the Brownian motions are only correlated througha common noise, meaning that we can write W i ( t ) = ρB ( t ) + (cid:112) − ρ B i ( t ) for independentBrownian motions B , . . . B n .Since the default times τ i are part of the equations for the distances-to-default X i , onehas to be careful that there can be several solutions to (4.9) depending on how one decidesif a bank is in default at time t . For example, even if X i ( t (cid:57) ) > for all the banks, onemay succeed in defaulting a few—or even all—of them at time t , provided the correspondingincrease of the L ni ( t ) ’s make X i ( t ) drop below zero for precisely the banks we decided to default,where X i ( t ) := X i ( t (cid:57) ) − { jump from increase in L ni ( t ) } . Moreover, if it is indeed the case that X i ( t (cid:57) ) ≤ for some bank i , then we need to decide (in a way that is consistent with theequations) how this propagates as it may start a cascade of defaults at the same time t , forwhich there can again be multiple possible choices (much in line with the previous example).The solution we choose to work with here amounts to picking the solution that gives thegreatest clearing capital in the Eisenberg–Noe framework (see Lemma 3.7) with any instanta-neous default cascades resolved by an analogue of the Eisenberg–Noe fictitious default algorithm.In Section 5.2 we show how this corresponds to amending the particle system (4.9) with whatwe call the cascade condition —see (5.12) for its precise definition and derivation, albeit in amore general setting than the specific example considered here. This condition is intrinsic tothe particle system formulation of our interbank model, and it uniquely determines the lossprocesses L ni at ‘time t ’ given the state of the system immediately before, namely at ‘time t (cid:57) ’in the sense of taking a left limit. In particular, this ensures that (4.9) has a unique strongcàdlàg solution, as argued in the proof of Proposition 5.5. We will not discuss this conditionany further here, but we briefly present its mean field analogue in Section 4.3.2 below. For any given m ≥ , and a fixed initial size m , we will now consider the interbank system ofsize n = mm modelled by (4.9), where the liabilities matrix and the other parameters are noisyrealisations of the underlying core-periphery network structure defined by the concrete example(4.3). Specifically, we impose that: the liabilities matrix λ mm × mm is constructed from (4.3) via random samples ( (cid:15), δ ) fromthe distribution P ⊗ P as outlined in the previous subsection. • the i ’th set of parameters ( x i (0) , σ i , µ i , Λ i ) is given as a function of the i ’th random sample δ i , where the function is the same for all entities of the same type (out of the two coretypes and two peripheral types defined by (4.3)).Based on the analysis in Section 5, it follows that the system we just described has a well-defined mean field limit as n → ∞ (see, in particular, Section 5.3.5). This limit captures thecoupled evolution of the four underlying types of banks (two core and two peripheral) afteraveraging over the infinitely many entities within each type. Let I ⊂ [ − , denote the supportof the distribution P , and let θ (cid:55)→ ( σ l,θ , µ l,θ , Λ l,θ ) denote the parameter function for each of thefour types l = 1 , . . . , . Let us say that l = 1 , are the two core types and l = 3 , are thetwo peripheral types (in correspondence with m c = 1 + 1 and m p = m p , + m p , in (4.3)). Tosimplify the presentation of the mean field limit below, we write Y l,θ ( t ) := − (cid:90) t σ l,θ ( s ) ds + (cid:90) t σ l,θ ( s ) d ( ρB ( s ) + (cid:112) − ρ B l ( s )) , and C l,θ := (1 + θ ) 1 − R Λ l,θ , for l = 1 , . . . , , and θ ∈ I , where B and B , . . . , B are independent Brownian motions.As the number of banks grows to infinity (in accordance with Section 4.1.1), the results ofSection 5 below show that mean field limit of the finite interbank system is given by the coupledMcKean–Vlasov problem (cid:101) L l ( t ) = (cid:90) I (cid:90) ∞ P ( t ≥ τ xl,θ | B ) V l ( x | θ ) dxdP ( θ ) , τ xl,θ = inf { t ≥ X xl ( t ) ≤ } ,X xl,θ ( t ) = x + Y l,θ ( t ) − log (cid:16) C l,θ (cid:88) i =1 ˜ λ il (cid:90) t (1 − sT ) d (cid:101) L i ( s ) (cid:17) , (4.10)for l = 1 , . . . , , where the strength of the core-core and core-periphery interactions are fullycaptured by the simplified liabilities matrix ˜ λ × = m p , m p ,
45 0 3 m p , m p , m p , m p , m p , m p , , (4.11)and V l ( · | θ ) are the initial densities for the distances-to-default of the four types l = 1 , . . . , conditional on θ ∈ I . See (4.17) below for how these initial conditions relate to the parametersand the initial laws of the external assets.Note that the contagion in (4.10) is no longer felt as the result of a single default event.Instead, there are now four ‘infinite collections’ of entities (corresponding to the four underlyingtypes) who feel the contagion through the mutual exposures ˜ λ ij in relation to the proportion ofdefaults within each infinite collection (given by the loss processes (cid:101) L l , for l = 1 , . . . , ). In theMcKean–Vlasov formulation (4.10), these proportions of default are really ‘average’ probabilities of default for the entities of each type, but see also the SPDE formulation (4.12)-(4.13) belowwhich makes the interpretation in terms of proportions more explicit. Remark 4.1.
The relative number of core and peripheral entities are specified by m = m c + m p ,where m c = 2 and m p = m p , + m p , . For a finite system of any size, the m c = 2 collections ofcore sub-entities each comprise a fraction m of the system, while a fraction m p , m makes up thefirst collection of peripheral banks, and the final fraction m p , m makes up the second collectionof peripheral banks. As a result, in the mean field limit we have that: (i) the core feels the ontagion from a given proportion of defaults within the two peripheral groups at a strengthmultiplied by m p , and m p , , respectively, and, similarly, (ii) the peripheral groups feel contagionfrom the core at a strength multiplied by m p , and m p , . Consider, for simplicity of presentation, the case where P is a Dirac mass at zero (so θ dropsfrom the equations), meaning that there is no additional heterogeneity within each of the fourtypes (for practical purposes, one can think of having replaced the parameters by their meanvalues averaged over P ). By applying Itô’s formula, and taking expectations conditional on B , we can reformulate (4.10) as a system of four coupled (nonlinear and nonlocal) stochasticpartial differential euqations (SPDEs). These SPDEs govern the (stochastic) densities of thedistances-to-default for the four infinite collections of banks of a given type (conditional on thecommon noise B ). This is arguably the more natural point of view for the dynamics of themean field limit. Specifically, we have (cid:101) L l ( t ) = 1 − (cid:90) ∞ V lt ( x ) dx, for l = 1 , . . . , , (4.12)where V = ( V , . . . , V ) solves a coupled system of SPDEs on the positive half-line of the form dV lt ( x ) = σ l (cid:0) ∂ xx V lt ( x ) − ∂ x V lt ( x ) (cid:1) dt − (cid:88) i =1 ˜ λ il f l ( t ) ∂ x V lt ( x ) d (cid:101) L i ( t ) + ρσ l ∂ x V lt ( x ) dB ( t ) , (4.13)with the Dirichlet boundary condition V lt (0) = 0 , for each l = 1 , . . . , . Note that this pointof view makes clear the precise nature of the contagion: namely a nonlinear transportation ofmass towards the origin, at a rate that is proportional to the current rates of default within eachinfinite collection of banks (as mediated by the mutual exposures ˜ λ ij between the four infinitecollections). Indeed, in dt amounts of time, the proportion of defaults within the l ’th collectionof banks is precisely d (cid:101) L l ( t ) , since (cid:101) L l ( t ) gives the total loss of mass for the l ’th collection of banksup to and including time t (i.e., the accumulated proportion of defaults).We note that, due to the irregularity in time of the common noise B , the time derivativeof (cid:101) L l ( t ) does not exist if ρ > , but the process is increasing, so the integrals against it arewell-defined. Still, in order for the SPDE formulation (4.13) to make sense globally, as it is, weare implicitly relying on each (cid:101) L l being continuous. As we already discussed above, this may beviolated, meaning that one (or more) of the loss processes (cid:101) L l can undergo a jump discontinuity,corresponding to an instantaneous macroscopic default cascade within the infinite collection ofthe l ’th type (or types). Nevertheless, one can still attach a rigorous meaning to the SPDE,as long as it is understood to only hold on the random intervals between jump times in thefollowing sense: at a jump time t , the densities are shifted according to the jump size, and thusthe system of SPDE is restarted from the new set of initial conditions V lt ( x ) := V lt (cid:57) (cid:0) x + Θ l ( t, ∆ L l ( t )) (cid:1) , x ∈ R + , (4.14)where V lt (cid:57) is the pointwise left-limit of V ls as s ↑ t , Θ l ( t, z ) := log (cid:16) C l (cid:90) t (cid:57) (1 − sT ) d L l ( s ) + C l (1 − tT ) z (cid:17) − log (cid:16) C l (cid:90) t (cid:57) (1 − sT ) d L l ( s ) (cid:17) , and L l ( t ) := (cid:88) i =1 ˜ λ il (cid:101) L i ( t ) , for l = 1 , . . . , . Note that we must allow the Dirichlet boundary condition to be violated when restarting at ajump time (and, as Remark 4.2 below points out, it is also a loss of the Dirichlet condition that eads to a jump). As concerns the timing and the sizes of the jumps, these are defined (in acàdlàg fashion) by what we call the mean field cascade condition , namely ∆ L l ( t ) = lim ε ↓ lim m ↑∞ ∆ ( m,ε ) t,l , ∆ ( m,ε ) t,l = Ξ l (cid:0) t, ε + ∆ ( m − ,ε ) t, · (cid:1) , m ≥ , ∆ (0 ,ε ) t,l = Ξ l ( t, ε ) , (4.15)where Ξ l ( t, z ) = Ξ l ( t, z , . . . , z ) := (cid:88) i =1 ˜ λ il (cid:90) Θ i ( t,z i )0 V it (cid:57) ( x ) dx. This condition for the jumps is the mean field analogue of the cascade condition for the finitesystem discussed at the end of Section 4.2. Intuitively, it amounts to subjecting the systemto an arbitrarily small shock that ignites a fictitious default cascade and then keeping trackof how it propagates in relation to the size of the initial shock: as we send the size of theinitial shock to zero, either the size of the fictitious cascade goes to zero, and there is then nojump, or it converges to something positive, and this positive value is then the size of the jumpcorresponding to a bona fide instantaneous default cascade. The mean field cascade conditionis carefully developed and motivated in Section 5. It is a special case of the condition (5.16) inSection 5.3, which addresses a more general framework than the one considered in this section.
Remark 4.2.
As we note in Section 4.3.2 below, the cascade condition gives ∆ L l ( t ) = 0 forevery l = 1 , . . . , , whenever each left-limit density V lt (cid:57) ( x ) vanishes as x ↓ . More generally,there is no jump at time t provided Ξ l ( t, (cid:15) ) < (cid:15) for small enough (cid:15) > , for each l = 1 , . . . , , asfollows by the same arguments as in Section 5.3.2. To see how this condition being violated canlead to a jump, consider the case where, at some time t , we have (cid:90) Θ i ( t,(cid:15) )0 V it (cid:57) ( x ) dx ≥ ˜ λ − ij (cid:15) and (cid:90) Θ j ( t,(cid:15) )0 V jt (cid:57) ( x ) dx ≥ ˜ λ − ji (cid:15), (4.16)for small enough (cid:15) > , for some pair of banks i, j ∈ { , . . . , } with ˜ λ ij > and ˜ λ ji > ,meaning that banks in the i ’th and j ’th groups are exposed to each other (with i = j being apossibility, provided banks within the same group are exposed to each other in a way that issignificant in the mean field limit). Now suppose for a contradiction that ∆ L ( t ) = 0 . Then thecascade condition implies that we can make lim m ↑∞ ∆ ( m,ε ) t, · as small as we like (since it vanishesas (cid:15) ↓ ). Thus, (4.16) together with the dominated convergence theorem gives lim m ↑∞ ∆ ( m,ε ) t,i = Ξ i (cid:0) t, ε + lim m ↑∞ ∆ ( m,ε ) t, · (cid:1) ≥ (cid:15) + lim m ↑∞ ∆ ( m,ε ) t,j , for small enough (cid:15) > , and the same conclusion holds with i and j interchanged. Together,these two inequalities yield a contradiction, and hence we conclude that there must indeed bea jump. With ( i, j ) = (1 , this corresponds to the situation at the jump time in Figure 3.Of course, there is nothing sacred about the size n = 4 , and, unlike the particular interactionsin (4.11), we could in general have a nonzero diagonal, so i = j is perfectly valid if the coresub-entities within a given collection are exposed to contagion from each other.In order to better illustrate the dynamics of the mean field limit, we present a numericalsimulation of the system of SPDEs (4.12)-(4.13) with jumps governed by the mean field cascadecondition (4.15) via (4.14). The outcome is plotted in Figure 3, which shows a heat plot foreach of the four solutions ( t, x ) (cid:55)→ V lt ( x ) to the coupled system of SPDEs. The simulationis performed using an adaptation of the numerical scheme proposed in [Ledger and Søjmark,2018a, Sect. 4.2]. igure 3: The figure shows four heat plots for the distances-to-default of the four infinite collections (for a givenrealisation of the common noise B ), where the horizontal axis is time, and the vertical axis is the distance-to-default.The interactions are given by ˜ λ × in (4.11) with m p , = m p , = 4 , as in Example 3.9. The parameters are constant(with ‘Periphery 1’ and ‘Core 2’ having a more positive drift), and the initial conditions can be read off the heat plotsat time t = 0 . The common noise starts out on a slight negative trend, which instigates a default cascade betweenthe low performing fractions of ‘Core 1’ and ‘Periphery 2’, resulting in both fractions defaulting in their entirety.Moreover, these defaults spill into a severe downgrading of the financial health of ‘Core 2’. However, ‘Core 2’ wasotherwise performing well, so only a very small proportion of it defaults, and since ‘Periphery 1’ is only exposed todefaults in ‘Core 2’, this means that these events have no significant impact on ‘Periphery 1’. In terms of the related mathematical literature, we stress that the papers Delarue et al.[2019], Hambly et al. [2019], Ledger and Søjmark [2018a,b], Nadtochiy and Shkolnikov [2019a] arefocused on ‘one-dimensional’ variations of McKean–Vlasov problems akin to (4.10), whereas therecent paper Nadtochiy and Shkolnikov [2019b] studies a coupled system analogous to (4.10) withonly minor differences. In particular, Nadtochiy and Shkolnikov [2019b] provides an existenceresult based on a Schauder fixed point argument (but no results on uniqueness) and studiescriteria under which any solution to the system must incur a blow-up. However, unlike thepresent paper, the results in Nadtochiy and Shkolnikov [2019b] neither address the relation toa finite particle system, nor do they consider a condition for uniquely specifying the jump sizes(in contrast to our cascade condition).
Due the averaging effect of passing to the mean field limit, one could reasonably expect thelimiting loss processes (cid:101) L l to evolve continuously, and anything else would be somewhat surprising iven that the McKean–Problem is driven by continuous Brownian dynamics. In many cases,it will indeed be true that the system evolves continuously. However, as we have just seen inFigure 3, depending on the parameters, one or more of the loss processes may see their speedof increase diverge to infinity in a way that results in a jump discontinuity (see also [Ledgerand Søjmark, 2018a, Thm. 2.7] in a simplified setting). Naturally, such an event can be seen asdefining an instantaneous ‘macroscopic’ default cascade that survived the passage to the meanfield limit.In order to decide whether the solution is continuous or not, and in order to specify the size ofa potential jump, we must amend (4.10) with the mean field cascade condition introduced above.As already mentioned, the details of this are reserved for Section 5, however, it is worth takinga few moments to preview a simple result on when jumps can be ruled out, which illustrates theworkings of the cascade condition.Section 5.3.2 presents a simple criterion for the initial densities that rules out a jump immedi-ately after initializing the system. As above, we consider the case where there is no dependenceon θ , and note that the initial densities are then of the form V l ( x ) = v l (cid:0) Λ l T e − (cid:82) T µ l ( s ) ds e x (cid:1) Λ l T e − (cid:82) T µ l ( s ) ds e x , for x > , (4.17)where v l is the initial density for the external asset process of banks of type l (which is supportedon x > Λ l T e − µ l T ). If, for every l = 1 , . . . , , there is a small (cid:15) l > such that (1 − R ) T (cid:88) j =1 ˜ λ jl v l ( x ) e − µ l T < , for all x ∈ (Λ l T e − µ l T , Λ l T e − µ l T + (cid:15) l ) , (4.18)then there is not an instant jump at time t = 0 and the solution remains continuous for a smallamount of time after initialization. Remark 4.3.
If each x (cid:55)→ v l ( x ) is continuous near the boundary x = Λ l T e − µ l T and v l ( x ) vanishes as x ↓ Λ l T e − T µ l T , then clearly (4.18) is satisfied. However, if v l ( x ) converges tosomething strictly positive as x ↓ Λ l T e − µ l T , then the values of the parameters become decisive.At any given time t ≥ , the mean field cascade condition (5.16) gives the precise criterionfor whether or not there is a jump, and what the size of the jump is, if there is one. However,here we only note that there is a simple (non-optimal) time- t analogue of (4.18) for ruling outjumps at any given time t and in some short time interval thereafter. To see what this lookslike, let V l be given by (4.13); that is, V ls ( x ) denotes the density of solvent banks of type l with distance-to-default x at time s , for a fixed realisation of the common noise B . If, for each l = 1 , . . . , , there is a small (cid:15) l such that (1 − R )( T − t ) (cid:88) j =1 ˜ λ jl V lt (cid:57) ( x ) < , for all x ∈ (0 , (cid:15) v ) , then there is no jump at time t and the solution is guaranteed to remain continuous for a shorttime thereafter. Note that the criterion involves the left limit V lt (cid:57) ( x ) = lim s ↑ t V ls ( x ) , meaningthat it is based on the state of the system strictly before time t (where the state of the systemis given by the distance-to-default densities for the solvent banks of the four types). The readeris referred to Section 5.3 for further details. Recall that we transformed the capital (3.2) of each bank into an interacting particle system(4.9) based on the notion (4.8) of their logarithmic distances-to-default. The remaining part ofthe paper is dedicated to a careful analysis of this particle system and its mean field limit. Inrelation to the previous section, we carry out the analysis under a more general assumption onthe coefficients and the structure of the liabilities matrix (as n → ∞ ). We then show in Section5.3.5 how to obtain the core-periphery model of Section 4 as a special case of this framework. .1 The finite interbank system To streamline the presentation, we will work with a general version of the system of interactingdistances-to-default (4.9). That is, we will focus on general particle systems of the form X i ( t ) = X i (0) − (cid:90) t b i ( s ) ds + (cid:90) t σ i ( s ) dW i ( s ) − F (cid:16)(cid:90) t g ( s ) d L ni ( s ) (cid:17) L ni ( s ) = n (cid:88) j =1 λ ji t ≥ τ j , τ j = inf { t ≥ X j ( t ) ≤ } , (5.1)where each X i denotes the distance-to-default of ‘bank i ’ as derived from the expression forbank i ’s capital (3.1) in the dynamic Eisenberg–Noe framework of Section 3. We recall thatthe transformation from (3.1) to an interacting system of distances-to-default was carried outin Section 4.2. The precise assumptions for the particle system are outlined in what follows.First of all, we will assume that, for large n , the rank of the liabilities matrices λ n × n isbounded by some value k (uniformly in large n > k ). Then, for large n > k , we have afactorization of the form nλ n × n = U n × k V k × n . (5.2)Here the natural choice of factorization comprises the matrices U n × k := (˜ u ij ) and V k × n := ( ς i ˜ v ij ) built from the singular value decomposition nλ n × n = ˜ U diag ( ς ) ˜ V , where diag ( ς ) is the n × n diagonal matrix with the singular values ς , . . . , ς n on the diagonal (out of which no more thanthe first k values are nonzero, since the rank is bounded by k ).Spectral decompositions and low rank structures are omnipresent in statistical analysis andthe applied sciences more generally. In relation to financial networks and systemic risk, simpleaspects of this has, e.g., been utilised in contagion models Amini and Minca [2016], Cont andSchaanning [2019] and statistical methods for detecting core-periphery network structures Cu-curingu et al. [2016]. More recently, the preprint Spiliopoulos and Yang [2019] studies a reducedform model for default clustering (based on interacting default intensities), using a singularvalue decomposition of the adjacency matrix in a way that is completely analogous to whatwe do here; namely to study the large population limit of the system under a bounded rankassumption which allows for a more tractable reformulation of the interactions. Example 5.1.
Suppose the liabilities matrix λ n × n is constructed from an underlying matrix ˆ λ m × m , as in (4.4)–(4.5), where ˆ λ m × m is of the block form (4.4). Then the rank of ˆ λ m × m is at most m , and one easily verifies that the rank of λ n × n also stays bounded by m for anysystem of size n = mm , for all multiples m ≥ . This yields a particular example where therank remains bouned as n → ∞ . We return to this in Section 5.3.5, where we detail how themodel in Section 4 appears as a special case of the analysis presented here.Note that (5.2) amounts to nλ ij = k (cid:88) l =1 u il v lj , for every i, j = 1 , . . . , n, where u il is ( i, l ) -entry of U n × k and v lj is the ( l, j ) -entry of V k × n . Based on this, the utility of(5.2) lies in the simple fact that we can now decompose the processes L ni from (4.7) as L ni ( t ) = k (cid:88) l =1 v li L nl ( t ) , where L nl ( t ) := 1 n n (cid:88) j =1 u jl t ≥ τ j , for i = 1 , . . . , n. (5.3)Crucially, these new loss processes L nl , for l = 1 , . . . , k , do not depend on i and, equally impor-tant, the number of them, namely k , is fixed as n → ∞ .In order to make precise the financial meaning of (5.3), we interpret the entries of U n × k and V k × n as latent factors identifying k underlying channels of contagion in the network structure(independently of the size n ): u jl captures how strongly bank j contributes to the contagion of channel l , and • v li captures how exposed bank i is contagion from channel l Let u , . . . , u n ∈ R k denote the n row vectors of U n × k and v , . . . , v n ∈ R k denote the n column vectors of V k × n . That is, u i := ( u i , . . . , u ik ) and v i := ( v i , . . . , v ki ) , for i = 1 , . . . , n. (5.4)Then bank i is characterized by the pair of k -dimensional vectors u i and v i , detailing, respec-tively, how it contributes to each of the k (latent) channels of contagion and how it is impactedby them. Nevertheless, once we have identified the k channels of contagion, the vector v i alonecan be seen as identifying bank i in terms of how it is hit by contagion: if two banks have similar v i ’s, they are similar in this crucial sense (although they may of course be dissimilar in termsof how strongly they contribute to contagion overall and to each of the various channels). Remark 5.2.
In Section 4 we considered a specific core-periphery structure where the peripheralgroups could be identified strictly by how they interact with the core (via the underlying matrix ˆ λ m × m ). In practice, the interbank liabilities may comprise a perturbation of this structurewhich is more heterogeneous (in addition to the noisiness) but nonetheless still of low rank(e.g. due to asymmetric but sparse periphery-to-periphery connections). Thus, we may not havea small number of clear-cut groups as in Section 4, but the low rank (uniformly in n ) would stillallow the system to be decomposed into a small number of latent channels of contagion.Relying on the decomposition (5.3), the particle system (5.1) is transformed to take the form X i ( t ) = X i (0) + (cid:90) t b i ( s ) ds + (cid:90) t σ i ( s ) dW i ( t ) − F (cid:16) k (cid:88) l =1 v li (cid:90) t g ( s ) d L nl ( s ) (cid:17) L nl ( t ) = 1 n n (cid:88) j =1 u jl t ≥ τ j , τ j = inf { t ≥ X j ( t ) ≤ } , l = 1 , . . . , k. (5.5)Here, and it what follows, we assume that the Brownian motions W i are correlated througha single common Brownian motion. That is, for each i = 1 , . . . , n , we have W i ( t ) = ρB ( t ) + (cid:112) − ρ B i ( t ) for a family of independent Brownian motions B , . . . , B n . In terms of the coeffi-cients in (5.5) we impose the following structural conditions which are motivated by the desireto include the original system (4.9) and keep the analysis as simple as possible. Assumption 5.3. F is Lipschitz continuous and increasing with F (0) = 0 , while g is continu-ous, non-negative, and decreasing. Furthermore, the asymmetry of the drifts and the volatilitiesis of the form b i ( s ) = b u i , v i ( s ) and σ i ( s ) = σ u i , v i ( s ) . Finally, b u,v and σ u,v are deterministicfunctions of time, and we ask that | ρ | < and (cid:15) ≤ σ u,v ≤ (cid:15) − , for a uniform constant (cid:15) > , aswell as ρσ u,v ∈ C κ ( R ) , for some κ > / . Recall that the pair of k -dimensional vectors u i and v i from (5.4) characterize bank i inrelation to interbank contagion. Together with the (random) initial conditions X i (0) , for i =1 , . . . , n , this describes the asymmetry in the interbank market. In order to obtain somethingmeaningful as the number of banks goes to infinity, we need to impose some structure throughthe convergence of their joint empirical measures defined by (cid:36) n := 1 n n (cid:88) i =1 δ u i ⊗ δ v i ⊗ δ X i (0) , for n ≥ . (5.6) Assumption 5.4.
First of all, we assume | u i | + | v i | ≤ C , for some C > , uniformly in i = 1 , . . . , n and n ≥ . Secondly, we ask that each ( u i , v i , X i (0)) is independent of the drivingBrownian motions. Thirdly, we ask that (cid:36) n converges weakly to a probability measure (cid:36) ∈P ( R k × R k × R + ) , which we write as a joint law (cid:36) = Law( u , v , X (0)) with d(cid:36) ( u, v, x ) = dν ( x | u, v ) d ˆ (cid:36) ( u, v ) , where ˆ (cid:36) = Law( u , v ) and ν ( ·| u, v ) is the regular conditional law of X (0) iven ( u , v ) = ( u, v ) . Finally, letting S ( u ) := supp(Law( u )) and likewise for v , we assume S ( u ) and S ( v ) are compact, and that for any v ∈ S ( v ) we have (cid:80) kl =1 u l v l ≥ for all u ∈ S ( u ) , as inthe finite system of size n where (cid:80) kl =1 u l v l = nλ ij ≥ for every u = u i and v = v j . As we already pointed out in Section 4.2, the particle system (5.5) needs to be amended witha condition for how to resolve defaults (that is consistent with the equations and correspondsto the greatest clearing capital solution in Lemma 3.7). This is achieved by insisting on the cascade condition (5.12) which is the subject of the next subsection (Section 5.2).
Proposition 5.5 (Well-posedness of the particle system) . Let Assumption 5.3 be in place.Equipped with the cascade condition (5.12) , as defined in Section 5.2 below, the system (5.5) hasa unique strong càdlàg solution.Proof.
Up until the first default time, the system trivially has a unique strong solution that iscontinuous in time. Since we insist on the cascade condition (5.12), the number of defaultingbanks at the first default time is uniquely specified, and this then uniquely determines how torestart the system. Defining the solution recursively, for each of at most n stopping times, weobtain a unique strong solution with càdlàg paths. In this section we make precise when the loss processes L nl ( t ) should jump and what the sizeof each jump should be given the possibility of a default cascade—that is, when the defaultof one bank immediately forces more banks into default at the same instance of time. Thistakes some care in order to ensure the consistency with the system (5.5), but ultimately thesituation is resolved by identifying the correct fixed point of an iterated mapping (as presented inSections 5.2.2 and 5.2.1). At first sight, the notation we introduce may appear a little abstract,but it leads to the convenient formulation (5.12) of what we call the cascade condition , whichcharacterizes the jump sizes of the particle system in an intrinsic way, and which guides theidentification of the analogous condition for the mean field limit. As discussed in Section 3.2,this cascade condition is conceptually related to the fictitious default algorithm of Eisenbergand Noe [2001].Fix n ≥ and consider the mapping from the type vector v i to the total losses felt by bank i given by v i (cid:55)→ L n v i ( t ) := k (cid:88) l =1 v li L nl ( t ) , for i = 1 , . . . , n. Recalling the decomposition (5.3), we simply have L n v i ( t ) = L ni ( t ) , but the point is to isolate howthe asymmetric i -dependence arises strictly as a function of the vector v i (whose k componentscapture how significant banks of ‘type’ l = 1 , . . . , k are to bank i ). Notice that, while each t (cid:55)→ L nl ( t ) in principle need not be increasing (depending on the rank factorization), the fullprocess t (cid:55)→ L n v i ( t ) = L ni ( t ) is by definition increasing.Given a càdlàg path t (cid:55)→ η ( t ) , we write ∆ η ( t ) := η ( t ) − η ( t (cid:57) ) . Then we can observe that, atany time t ≥ , the jump sizes of the loss processes must satisfy ∆ L nl ( t ) = 1 n n (cid:88) i =1 u il t = τ i = 1 n n (cid:88) i =1 u il { X i ( t (cid:57) ) ∈ [0 , ∆ F ni ( t )] } t ≤ τ i , (5.7)for each l = 1 . . . , k , where ∆ F ni ( t ) is the amount by which the i ’th distance-to-default (orparticle) is shifted down at time t (due to losses from defaults at time t ), namely ∆ F ni ( t ) := F (cid:18) k (cid:88) j =1 v ji (cid:90) t (cid:57) g ( s ) d L nj ( s ) + g ( t )∆ L n v i ( t ) (cid:19) − F (cid:18) k (cid:88) j =1 v ji (cid:90) t (cid:57) g ( s ) d L nj ( s ) (cid:19) . Consequently, once we have identified the correct sizes of the jumps ∆ L n v i ( t ) , for i = 1 , . . . , n , allthe jump sizes ∆ L nl ( t ) , for l = 1 , . . . , k , are automatically uniquely specified by (5.7). Indeed, the vents { t ≤ τ i } and the values (cid:82) t (cid:57) g ( s ) d L ni ( s ) , for i = 1 . . . , n , are all fixed at time t , since theyare given as left-limits of the evolution of the system strictly before time t . On the other hand,the values ∆ L n v i ( t ) are to be determined at time t , and they will involve a choice, amounting tohow we choose to resolve default cascades. As for L n v i ( t ) above, it is important to realise that the i -dependence of ∆ F ni ( t ) is again a functionof v i alone. In order to make this clear (and to streamline the mathematical presentation), weintroduce the random map Θ : R + × ( R + ) R k × R k → R + given by Θ( t ; f, y ) := F (cid:18) k (cid:88) j =1 y j (cid:90) t (cid:57) g ( s ) d L nj ( s ) + g ( t ) f ( y ) (cid:19) − F (cid:18) k (cid:88) j =1 y j (cid:90) t (cid:57) g ( s ) d L nj ( s ) (cid:19) . (5.8)Clearly, we then have ∆ F ni ( t ) = Θ( t ; ∆ L n , v i ) , so we can rewrite (5.7) as ∆ L nl ( t ) = 1 n n (cid:88) i =1 u il { X i ( t (cid:57) ) ∈ [0 , Θ n ( t ;∆ L n , ˆ v i )] , t ≤ τ i } , l = 1 . . . , k. (5.9)As already alluded to above, this shows that: once we pin down the mapping v (cid:55)→ ∆ L n v , thenthe correct jumps of L n , . . . , L nk are automatically specified by the constraint (5.9).Looking at (5.9), we immediately obtain a constraint for v (cid:55)→ ∆ L n v by simply summing over l = 1 , . . . , k weighted by the v il ’s, which yields the identity ∆ L n v i ( t ) = k (cid:88) l =1 v li (cid:18) n n (cid:88) j =1 u jl { X j ( t (cid:57) ) ∈ [0 , Θ n ( t, ∆ L n , v j )] , t ≤ τ j } (cid:19) , i = 1 , . . . , n. (5.10)Note that this is precisely saying that v (cid:55)→ ∆ L n v arises as a fixed point Ξ( t ; ∆ L n , · ) = ∆ L n ( · ) ( t ) ,where the random map Ξ : R + × ( R + ) R k × R k → R + is defined by Ξ( t ; f, x ) := k (cid:88) l =1 x l (cid:18) n n (cid:88) j =1 u jl { X j ( t (cid:57) ) ∈ [0 , Θ( t ; f,x )] , t ≤ τ j } (cid:19) . (5.11)However, the mapping f (cid:55)→ Ξ( t ; f, · ) can have multiple fixed points, so the above fixed pointconstraint alone is not enough to determine the jump sizes ∆ L n . That is, the system (5.5) is apriori ill-posed without a selection rule.Based on the natural step-by-step resolution of default cascades (explained in detail in Sec-tion 5.2.2 below), the correct selection rule simply amounts to a (suitably initialized) iterativeapplication of the mapping Ξ . This can be formulated succinctly as ∆ L n v ( t ) := lim m → n ∆ mt, v for v = v . . . , v n , ∆ mt, v := Ξ( t, ∆ m − t, v , v ) and ∆ t, v := Ξ( t ; 0 , v ) , (5.12)which we will refer to as the cascade condition for the jump sizes. As concerns the notion of astep-by-step resolution, the number of ‘steps’ or ‘rounds’ in the cascade is given by the smallest ¯ m such that lim m → n ∆ mt, · = ∆ ¯ mt, · . For clarity, further details on this are presented in the separateSection 5.2.2 below, where we give a precise mathematical definition of instantaneous defaultcascades leading to this condition. .2.2 Detailed description of the resolution of cascades Let A t (cid:57) denote the (random) set of indices i ≤ n such that t ≤ τ i for each i ∈ A t (cid:57) , meaning thatbank i was (and may still be) solvent—or more colloquially, ‘alive’—strictly before time t . Notethat there is at least one default at time t precisely when X i ( t (cid:57) ) = 0 for some bank indexedby i ∈ A t (cid:57) . Therefore, we define the (random) set of indices D t as precisely those i ∈ A t (cid:57) for which X i ( t (cid:57) ) = 0 , corresponding to the initial set of defaults at time t (which came aboutwithout any role played by contagion).Supposing that D t (cid:54) = ∅ , we now need to make it mathematically precise how to decide if a cascade is initiated by the contagious effects from these initial defaults—and then we need tomake precise how to resolve the total size of the cascade if it occurs.Recalling the definition of Ξ in (5.11), the isolated effect of the initial defaults (which werecall are indexed by D t ) is to increase each L n v i ( t (cid:57) ) by an initial jump of size ∆ t, v i := Ξ( t ; 0 , v i ) = 1 n k (cid:88) l =1 v li (cid:88) j ∈D (0) t u jl , for i = 1 , . . . , n. Next, we define the (random) set of indices D t ⊂ A t (cid:57) / D t as precisely those i ∈ A t (cid:57) / D t forwhich X i ( t (cid:57) ) − Θ( t ; ∆ t, ( · ) , v i ) ≤ . In other words, the members of D t are precisely those banks that enter into default at time t on account of the contagion from the initiating set of defaults D t . Supposing that D t (cid:54) = ∅ , wemust update the losses to include this first round of contagion, and check if this induces anotherround of contagion. This amounts to considering a jump in L n v i ( t (cid:57) ) of size ∆ t, v i := Ξ( t ; ∆ t, ( · ) , v i ) = 1 n k (cid:88) l =1 v li (cid:88) j ∈D t ∪D t u jl , for i = 1 , . . . , n, and thus defining D t ⊂ A t (cid:57) / D t as precisely those i ∈ A t (cid:57) / D t for which X i ( t (cid:57) ) − Θ( t ; ∆ t, ( · ) , v i ) ≤ . For any m ≤ n , ∆ mt, ( · ) and D mt are defined analogously. Recalling that n − (cid:80) kl =1 v li u jl = λ ji ≥ ,it is always the case that ∆ mt, ( · ) = Ξ( t ; ∆ m − t, ( · ) , · ) ≥ ∆ m − t, ( · ) in the pointwise sense. In particular, each (∆ mt, v ) is an increasing sequence in m = 0 , , . . . , n ,and it is immediate that D ¯ mt = ∅ implies Ξ( t ; ∆ ¯ mt, ( · ) , v ) = ∆ ¯ mt, v , for all v = v . . . , v n , so the sequence eventually reaches a fixed point after the ¯ m ’th round of contagion-induceddefaults (since there are only n banks in total, note that we must have D nt = ∅ ). Once a fixedpoint is reached at the ¯ m ’th iteration, there are no further defaults, and hence the defaultcascade is fully resolved with ∆ L n v ( t ) := ∆ ¯ mt, v = k (cid:88) l =1 v li (cid:88) j ∈D t ∪···∪D ¯ mt u jl , for v = v , . . . , v n . By construction, each sequence (∆ mt, v ) stays fixed after this ¯ m ’th iteration, so the value ∆ L n v ( t ) is indeed the limit of ∆ mt, v as m → n , in agreement with the cascade condition defined in (5.12).This formulation of the jump size as a limit of iterated steps in the default cascade is instructivefor the formulation of the mean field analogue, which follows in the next subsection. Remark 5.6.
If all banks have interbank liabilities, then, for any j , there is an i such that λ ji > . Hence, we get D mt (cid:54) = ∅ if and only if Ξ( t ; ∆ mt, · , v ) ≥ ∆ mt, v for all v ∈ { v . . . , v n } and Ξ( t ; ∆ mt, · , v ) > ∆ mt, v for at least one v ∈ { v . . . , v n } . .3 The mean field limit Provided there is a suitable averaging effect, we can send n → ∞ in the particle system (5.5)and thereby capture the ‘systemic’ or macro-level properties of the finite system through itsmean field limit. As we show in Appendix A.3, the insistence on Assumptions 5.3 and 5.4 issufficient to ensure such a law of large numbers, and it then follows that the resulting mean fieldlimit is given by a McKean–Vlasov problem of the form L l ( t ) = (cid:82) R + × R k × R k u l P ( t ≥ τ xu,v | B ) d(cid:36) ( x, u, v ) , l = 1 , . . . , k,τ xu,v = inf { t ≥ X xu,v ( t ) ≤ } ,X xu,v ( t ) = x + (cid:82) t b u,v ( s ) ds + (cid:82) t σ u,v ( s ) dW s − F (cid:16)(cid:80) kl =1 v l (cid:82) t g ( s ) d L l ( s ) (cid:17) , (5.13)for t ∈ [0 , T ] , where W = ρB + (cid:112) − ρ B , for two independent Brownian motions B ⊥ B .In general, one would expect the above problem to be continuous in time, due to the ‘av-eraging’ effect of passing to the mean field limit, and, as long as this is the case, the system isfully specified by (5.13). However, as we saw already in Section 4, the loss processes t (cid:55)→ L l ( t ) may in fact undergo jump discontinuities (in particular, [Hambly et al., 2019, Thm 1.1] can beadapted to show that such jumps must occur for some parameters). When accounting for this,one needs to be careful that the jump sizes are not pinned down uniquely by the formulation(5.13). Similarly to the cascade condition for the finite system, a concrete choice must be madethat allows the system to be càdlàg and uniquely determines the jumps. This is the topic of thenext subsection. Our first task is to show that the jump sizes must obey certain fixed point constraints. Byanalogy with the analysis of cascades in the finite system, we therefore define the mapping L : S ( v ) → L ∞ (0 , T ) by L v ( t ) := k (cid:88) l =1 v l L l ( t ) , where L ∞ (0 , T ) = L ∞ ([0 , T ] , R ) is the space of bounded real-valued functions on [0 , T ] under theequivalence relation of being equal almost everywhere. By Assumption 5.4 and the definitionof each L l , the process t (cid:55)→ L v ( t ) is indeed bounded, for each v ∈ S ( v ) , and crucially theassumptions also imply that t (cid:55)→ L v ( t ) is increasing. Similarly to the constraint (5.9) for thefinite system, we can infer directly from (5.13) that the jumps of each L l must satisfy ∆ L l ( t ) = (cid:90) R + × R k × R k u l P (cid:0) X xu,v ( t (cid:57) ) ∈ [0 , Θ( t ; ∆ L , v )] , t ≤ τ | B (cid:1) d(cid:36) ( x, u, v ) , (5.14)where Θ( t ; f, v ) := F (cid:18)(cid:90) t (cid:57) g ( s ) d L v ( s ) + g ( t ) f ( v ) (cid:19) − F (cid:18)(cid:90) t (cid:57) g ( s ) d L v ( s ) (cid:19) . Once the jumps of L are pinned down, the constraint (5.14) uniquely determines the jumpsof each loss process L l . Furthermore, from (5.14) and the definition of L , it follows that the(random) value of ∆ L must be a fixed point of the (random) mapping f (cid:55)→ Ξ( t ; f, · ) , where Ξ( t ; f, v ) := k (cid:88) l =1 v l (cid:90) R + × R k × R k ˆ u l P (cid:0) X x ˆ u, ˆ v ( t (cid:57) ) ∈ [0 , Θ( t ; f, ˆ v )] , t ≤ τ | B (cid:1) d(cid:36) ( x, ˆ u, ˆ v ) . (5.15)In general, the map (5.15) can have several, even infinitely many, fixed points (for example, f ≡ is always a fixed point, but this is not compatible with jumps), so ∆ L is not uniquelyspecified a priori. This situation is resolved by the selection rule (5.16) introduced below, imicking the cascade condition (5.12) for the jumps in the finite system. However, unlike thefinite system, the mean field limit always satisfies Ξ( t ; 0 , · ) = 0 , so the occurrence and size of apotential default cascade must be identified by artificially exposing the system to an arbitrarilysmall shock. Specifically, we shift the system down by a small amount Θ( t ; (cid:15), v ) and then keeptrack of how the resulting losses propagate as ε ↓ , meaning that the size of the initial shift Θ( t ; (cid:15), v ) vanishes. Mathematically, this means that, for v ∈ S ( v ) , the jump size ∆ L v ( t ) is givenby the mean field cascade condition ∆ L v ( t ) = lim ε ↓ lim m ↑∞ ∆ ( m,ε ) t,v , ∆ ( m,ε ) t,v = Ξ( t, ε + ∆ ( m − ,ε ) t, · , v ) , m ≥ , ∆ (0 ,ε ) t,v = Ξ( t, ε, v ) , (5.16)where the equalities hold almost surely (recall that Ξ is conditional on the common noise B ).We stress that the limit is well-defined, since (∆ ( m,ε ) t,v ) forms a bounded sequence that increasesas m ↑ ∞ and decreases as ε ↓ . Moreover, dominated convergence shows that the (random)map v (cid:55)→ ∆ L v ( t ) given by (5.16) is a fixed point of f (cid:55)→ Ξ( t, f, · ) , so this choice for the jumpsizes is indeed consistent with the McKean–Vlasov problem (5.13). Remark 5.7.
We emphasise that the iterative structure of (5.16) lends itself easily to numericalimplementation, and this is indeed the starting point for the algorithm behind the simulationsin Figure 3. Moreover, we note that, in the case of a symmetric network of obligations, [Hamblyet al., 2019, Proposition 2.4] gives that the mean field cascade condition (5.16) agrees with thecorresponding notion of a ‘physical’ jump condition considered in Hambly et al. [2019].
We now present a simple criterion, namely (5.18), that rules out a jump discontinuity at agiven time t > and guarantees the solution stays continuous in some small amount of timethereafter. Of course, the mean field cascade condition (5.16) already gives the precise criterionfor whether or not the system undergoes a jump at time t , but our aim here is to provide someintuition for the workings of this condition.At any given time t > , and for any pair ( u, v ) , we let V t ( ·| u, v ) denote the random densityof the random sub-probability measure A (cid:55)→ ν t ( A | u, v ) := (cid:90) R + P (cid:0) X xu,v ( t ) ∈ A, t < τ xu,v | B (cid:1) V ( x | u, v ) dx. (5.17)For the purposes of this subsection, we think of having fixed a realisation of B , so the belowcriteria (5.18) should be understood as holding for this particular realisation: thus, the conclu-sion is that there is no jump for this particular realisation of B . Of course, if the criteria holdsfor all realisations of B , then it is an almost sure conclusion.Recalling the definition of Θ , we have Θ( t ; f, v ) ≤ (cid:107) F (cid:107) Lip g ( t ) f ( v ) , so Ξ( t ; f, v ) ≤ k (cid:88) l =1 v l (cid:90) R k × R k (cid:90) (cid:107) F (cid:107) Lip g ( t ) f (ˆ v )0 ˆ u l V t (cid:57) ( y | ˆ u, ˆ v ) dyd ˆ (cid:36) (ˆ u, ˆ v ) , where V t (cid:57) is the pointwise left limit of V s as s ↑ t . Now fix a time t > , and suppose V t (cid:57) andthe joint distribution ˆ (cid:36) of ( u , v ) satisfy the following criterion: there is a small δ > such that,for each v ∈ S ( v ) , (cid:107) F (cid:107) Lip g ( t ) k (cid:88) l =1 v l (cid:90) R k × R k ˆ u l V t (cid:57) ( y | ˆ u, ˆ v ) d ˆ (cid:36) (ˆ u, ˆ v ) ≤ − δ for all y ∈ (0 , (cid:15) v ) , (5.18)for some small enough (cid:15) v > . Then we get ∆ (0 ,ε ) t,v = Ξ( t, ε, v ) ≤ (1 − δ ) ε, or all ε > sufficiently small such that (cid:107) F (cid:107) Lip g ( t ) ε < (cid:15) v . In turn, for all ε > such that (cid:107) F (cid:107) Lip g ( t )(1 + (1 − δ )) ε < (cid:15) v , we have ∆ (1 ,ε ) t,v = Ξ( t, ε + ∆ (0 ,ε ) t, · , v ) ≤ (1 − δ ) ε + (1 − δ ) ε, and, by recursion, for any given m ≥ we thus have ∆ ( m,ε ) t,v = Ξ( t, ε + ∆ ( m − ,ε ) t, · , v ) ≤ ε m (cid:88) l =0 (1 − δ ) l +1 , for all ε > such that (cid:107) F (cid:107) Lip g ( t )( (cid:80) ml =0 (1 − δ ) l ) ε < (cid:15) v . Since − δ < , the sum forms ageometric series that converges as m → ∞ , so provided (5.18) is satisfied we conclude that ∆ L v ( t ) = 0 for all v ∈ S ( v ) , since ∆ L v ( t ) ≤ lim ε ↓ lim m ↑∞ ε m (cid:88) l =0 (1 − δ ) l +1 = lim ε ↓ ε − δδ = 0 . In particular, each s (cid:55)→ L l ( s ) must indeed continuous at time t . In other words, after the systemtakes an artificial hit of order ε , the induced rounds of contagion quickly become negligible withthe total effect being at most of order ε (i.e., of the same order as the initial ‘artificial’ shock),and so they disappear as the size of the initial shock is sent to zero. Furthermore, now that weknow there is not a jump, a straightforward adaptation of [Søjmark, 2019, Prop. 6.4.3] showsthat a bound of the form (5.18) holds for some small amount time, so the solution remainscontinuous on this nonzero time interval.On the other hand, Remark 4.2 from Section 4 provides a simple example where the condition(5.18) is violated, for some v ∈ S ( v ) , and where it is proved that the cascade condition musttherefore result in a jump discontinuity. In addition to the argument provided there, we remindthe reader of Figure 3, which illustrates the occurrence of the jump. In this respect, let us alsostress that there can of course be cases where t (cid:55)→ L v ( t ) only jumps for some v ∈ S ( v ) andnot for others, provided there are certain types which are not exposed to the types experiencingdefault cascades. As a particular example of this, we could amend the example behind Figure 3by imposing that ‘Core 2’ is not exposed to losses in ‘Core 1’ and ‘Periphery 2’: then we obtainan example where L l jumps for l = 1 , (‘Core 1’ and ‘Periphery 2’) while it does not jump for l = 2 , (‘Core 2’ and ‘Periphery 1’). Remark 5.8.
Notice that if the Dirichlet boundary condition V t − (0 | u, v ) = 0 is satisfied (mean-ing that lim x ↓ V t − ( x | u, v ) = 0 ), then (5.18) is definitely true. More generally, the criterionamounts to y (cid:55)→ V t (cid:57) ( y | u, v ) being sufficiently small relative to (cid:107) F (cid:107) − Lip g ( t ) − near y = 0 , depend-ing on the joint distribution ˆ (cid:36) . Starting from a nice initial condition, we will have V t (0 | u, v ) = 0 for some amount of time, but if the contagious feedback becomes too strong it may transportthe density so fast towards the origin that there is a blow-up time t (cid:63) : that is, the derivative of t (cid:55)→ L ( t ) diverges as t ↑ t (cid:63) , and the left limit density V t (cid:63) (cid:57) fails to vanish at zero, in a way suchthat the cascade condition enforces a jump discontinuity ∆ L ( t (cid:63) ) > . In this subsection we consider the McKean–Vlasov problem (5.13) when ρ = 0 , meaning thatthere is no common noise. This makes it more tractable to get a handle on the regularity ofthe loss processes. Under a mild assumption on the initial profile of the system, we are able togeneralise the arguments from Hambly et al. [2019] and thus show that the system is well-posedup until the L norm of the gradient of the losses, namely ( ∂ t L , . . . , ∂ t L k ) , explodes.The assumption on the initial profile amounts to controlling the decay of the mass near theorigin. Specifically, using the notation from Assumption 5.4, we require that the initial condition ν satisfies dν ( x | u, v ) = V ( x | u, v ) dx with V ( x | u, v ) ≤ C (cid:63) x β x
In this work we introduced the first combined model that considers an Eisenberg–Noe styleframework for interbank contagion which can be recast as an interacting particle system with awell-defined mean field limit. Therefore, we are able to draw a direct connection between thesepreviously disparate frameworks for systemic risk, focusing either on the resolution of defaultcascades in finite bank networks or stochastic dynamics with simple mean field interactions.The proposed contagion mechanism considers banks with stochastic external assets which, ifthey drop, can cause defaults before the maturity of all claims. This is handled first for a finitenumber of institutions in a purely Eisenberg–Noe style framework, thus extending Banerjeeet al. [2018] to account for early defaults. Next, we demonstrate a limiting behaviour as thenumber of banks increase, which provides justification for performing the analysis of contagionat the level of the mean field limit. In this way, one can significantly lower the parameter space,and it becomes more tractable to pursue analytical results for the regularity of the system’sevolution in time. Moreover, one can circumvent the curse of dimensionality and avoid the slowconvergence of Monte Carlo based methods, for example by implementing an analogue of thenumerical scheme from Ledger and Søjmark [2018a] as we did in Figure 3 (alternatively, onecould attempt to adapt the semi-analytical approach of Kaushansky et al. [2018c]). As regardsthe antecedent mean field literature, we provide a more convincing financial underpinning forHambly et al. [2019], Hambly and Søjmark [2019], Ledger and Søjmark [2018a], Nadtochiyand Shkolnikov [2019a,b], while also extending the analytical results of Hambly et al. [2019],Ledger and Søjmark [2018a,b] to allow for heterogeneous interactions and a more general formof contagion. Furthermore, we add to Nadtochiy and Shkolnikov [2019b] by introducing thecascade condition for the resolution of instantaneous default cascades (i.e., jump sizes) as wellas establishing results on convergence and uniqueness.The model of default cascades presented herein can be utilized to answer many questions insystemic risk that are typically intractable analytically (as well as computationally inefficient)for finite bank networks. Additionally, the mean field limit allows for a cleaner analysis of key‘systemic’ quantities, as exemplified by the mean field cascade condition that gives a precisecharacterisation of default cascades with an instantaneous ‘systemic’ impact. One importantnew strand of literature is that of network valuation adjustments Barucca et al. [2016], Banerjeeand Feinstein [2019], in which prices of securities should account for the full network effects.In this regard, the stochastic dynamics underlying the framework herein makes it well-suitedfor, e.g., pricing credit default swaps on the financial system. By further utilizing the meanfield limit, the lower parameter space can facilitate calibration of the stochastic dynamics, andthis also opens up the possibility of relying on more analytical methods. These problems areintimately related with systemic risk measures. For instance, the value-at-risk or CoVaR Adrianand Brunnermeier [2016] of the financial system are related to mappings such as a (cid:55)→ P ( L ( t ) > a ) and ( a, δ ) (cid:55)→ P (cid:0) L ( t + δ ) − L ( t ) > a (cid:1) , for a given time t . More specifically, an interesting modification of CoVaR in the mean fieldlimit for core-periphery systems, as discussed in Section 4, is for consideration of the healthof the aggregate system conditional on the stress of one of the “groups” of institutions. Infact, such structures may allow for the tractable consideration of general systemic risk measuresof Chen et al. [2013], Feinstein et al. [2017], Biagini et al. [2019b] as well. Additionally, ratherthan applying these network valuation adjustments for measuring systemic risk in exogenouslyprovided network structures, the pricing of risk in such a way may allow for considerations ofendogenous network formation. In such a setting, each financial institution would choose toinvest in external projects or engage in interbank markets so as to solve some portfolio opti-mization problem. Only with a consideration of credit pricing in a financial network would suchendogenous network formation be tractable, and we believe this points towards an importantavenue of future research. Appendix
This appendix contains the proofs of the main results from Section 5 and is organised into threesubsections. The first two subsections address the proofs of Theorems 5.9 and 5.10, respectively,while the final subsection is focused on the identification of the mean field problem (5.13) as thelarge population limit of the finite particle system (5.1). Throughout the appendix, we will beworking under Assumptions 5.3 and 5.4.
A.1 Proof of Theorem 5.9
We introduce the notation (cid:107) f (cid:107) t := (cid:107) f (cid:107) L ∞ (0 ,t ) and recall the notation S ( v ) = supp(Law( v )) from Assumption 5.4. Given this, we can consider the space of continuous maps v (cid:55)→ (cid:96) v ( · ) from S ( v ) to L ∞ (0 , t ) , denoted by C ∗ t := C (cid:0) S ( v ); L ∞ (0 , t ) (cid:1) , with respect to the supremum norm (cid:107) (cid:96) (cid:107) ∗ t := sup v ∈ S ( v ) (cid:107) (cid:96) v ( · ) (cid:107) t . Since the domain S ( v ) is a compact subset of R k , by Assumption 5.4, and the codomain L ∞ (0 , t ) is a Banach space, this norm makes C ∗ t a Banach space. In order to construct a regular solutionto the McKean–Vlasov problem (5.13, ρ = 0 ), until an explosion time, we will work with themap Γ , defined in (A.1) below. Our strategy is to identify a suitable closed subset of C ∗ t , forsmall enough t > , on which we can apply Banach’s fixed point theorem. A.1.1 Comparison argument and existence of regular solutions
Given
T > , we define the map Γ : C ∗ T (cid:55)→ C ∗ T by Γ[ (cid:96) ] v ( t ) := k (cid:88) l =1 v l (cid:90) R + × R k × R k ˆ u l P ( t ≥ τ x,(cid:96) ˆ u, ˆ v ) d(cid:36) ( x, ˆ u, ˆ v ) , (A.1)for all t ∈ [0 , T ] and v ∈ S ( v ) , where τ x,(cid:96)u,v = inf { t > X x,(cid:96)u,v ( t ) ≤ } X x,(cid:96)u,v ( t ) = x + (cid:82) t b u,v ( s ) ds + (cid:82) t σ u,v ( s ) dB s − F (cid:0)(cid:82) t g ( s ) d(cid:96) v ( s ) (cid:1) . (A.2)Note that, as long as s (cid:55)→ (cid:96) v ( s ) is continuous or of finite variation, the integral of g against (cid:96) v in(A.2) is well-defined, since g is both continuous and of finite variation by Assumption 5.3 (seee.g. [Stroock, 2011, Sect. 1.2]). Naturally, all of the results that follow are stated for inputs suchthat the mapping makes sense.The cornerstone of our analysis is the next comparison argument. It leads us to the fixedpoint argument for existence of regular solutions, and it reappears in the generic uniquenessargument of Section A.1.2 which completes the full statement of Theorem 5.9. Lemma A.1 (Comparison argument) . Fix any two (cid:96), ¯ (cid:96) ∈ C ∗ T such that s (cid:55)→ (cid:96) v ( s ) and s (cid:55)→ ¯ (cid:96) v ( s ) are increasing with (cid:96) v (0) = ¯ (cid:96) v (0) = 0 for all v ∈ S ( v ) . Fix also t > and suppose s (cid:55)→ (cid:96) v ( s ) iscontinuous on [0 , t ) for all v ∈ S ( v ) . Then we have (cid:0) Γ[ (cid:96) ] v ( t ) − Γ[¯ (cid:96) ] v ( t ) (cid:1) + ≤ C (cid:107) ( (cid:96) − ¯ (cid:96) ) + (cid:107) ∗ t (cid:90) t ( t − s ) − d Γ[ (cid:96) ] v ( s ) , for all t < t and all v ∈ S ( v ) , where C > is a fixed numerical constant (i.e., it is independentof t and v ). roof. Fix t < t . Recalling that (cid:96) v (0) = 0 , integration by parts (see e.g., [Stroock, 2011,Sect. 1.2]) gives (cid:90) t g ( s ) d(cid:96) v ( s ) = g ( t ) (cid:96) v ( t ) + (cid:90) t (cid:96) v ( s ) d ( − g )( s ) , and likewise for ¯ (cid:96) v . Using this and the assumptions on F , g , (cid:96) , and ¯ (cid:96) , we have F (cid:16)(cid:90) t g ( s ) d(cid:96) v ( s ) (cid:17) − F (cid:16)(cid:90) t g ( s ) d ¯ (cid:96) v ( s ) (cid:17) ≤ (cid:107) F (cid:107) Lip (cid:16)(cid:90) t g ( s ) d(cid:96) v ( s ) − (cid:90) t g ( s ) d ¯ (cid:96) v ( s ) (cid:17) + = (cid:107) F (cid:107) Lip (cid:16) g ( t )( (cid:96) v ( t ) − ¯ (cid:96) v ( t )) + (cid:90) t ( (cid:96) ( s ) − ¯ (cid:96) ( s )) d ( − g )( s ) (cid:17) + ≤ g ( t ) (cid:107) F (cid:107) Lip (cid:0) (cid:96) v ( t ) − ¯ (cid:96) v ( t ) (cid:1) + + (cid:107) F (cid:107) Lip (cid:90) t (cid:0) (cid:96) v ( s ) − ¯ (cid:96) v ( s ) (cid:1) + d ( − g )( s ) ≤ g (0) (cid:107) F (cid:107) Lip (cid:107) ( (cid:96) v − ¯ (cid:96) v ) + (cid:107) t . Thus, taking the difference between the two processes X x,(cid:96)u,v and X x, ¯ (cid:96)u,v , as defined in (A.2) coupledthrough the same Brownian motion, it follows that X x, ¯ (cid:96)u,v ( t ) − X x,(cid:96)u,v ( t ) ≤ g (0) (cid:107) F (cid:107) Lip (cid:107) ( (cid:96) v − ¯ (cid:96) v ) + (cid:107) t . Therefore, using the continuity of (cid:96) v , for any s ∈ [0 , t ] , it holds on the event { τ x,(cid:96)u,v = s } that X x, ¯ (cid:96)u,v ( s ) = X x, ¯ (cid:96)u,v ( s ) − X x,(cid:96)u,v ( s ) ≤ g (0) (cid:107) F (cid:107) Lip (cid:107) ( (cid:96) − ¯ (cid:96) ) + (cid:107) ∗ s . Based on this, we can replicate the arguments from [Hambly et al., 2019, Prop. 3.1], by insteadconditioning on the value of τ x,(cid:96)u,v and using the previous inequality, to deduce that P ( t ≥ τ x,(cid:96)u,v ) − P ( t ≥ τ x, ¯ (cid:96)u,v ) ≤ (cid:90) t P (cid:0) inf r ∈ [ s,t ] (cid:82) rs σ u,v ( h ) dB h > − g (0) (cid:107) F (cid:107) Lip (cid:107) ( (cid:96) − ¯ (cid:96) ) + (cid:107) ∗ s (cid:1) d P ( s ≥ τ x,(cid:96)u,v ) . Performing a time change in the Brownian integral, and using that there is a uniform (cid:15) > suchthat (cid:15) ≤ σ u,v ≤ (cid:15) − , by Assumption 5.3, it follows from the law of the infimum of a Brownianmotion that P ( t ≥ τ x,(cid:96)u,v ) − P ( t ≥ τ x, ¯ (cid:96)u,v ) ≤ C (cid:107) ( (cid:96) − ¯ (cid:96) ) + (cid:107) ∗ t (cid:90) t ( t − s ) − d P ( s ≥ τ x,(cid:96)u,v ) where the constant C > is independent of t , x , u , and v . Now fix any ˜ v ∈ S ( v ) . Multiplyingboth sides of the above inequality by (cid:80) kl =1 ˜ v l u l and recalling that this is non-negative for all u in the support of (cid:36) , by Assumption 5.4, we can then integrate both sides of the resultinginequality against (cid:36) , over ( x, u, v ) ∈ R + × R k × R k , to arrive at Γ[ (cid:96) ] ˜ v ( t ) − Γ[¯ (cid:96) ] ˜ v ( t ) ≤ C (cid:107) ( (cid:96) − ¯ (cid:96) ) + (cid:107) ∗ t (cid:90) t ( t − s ) − d Γ[ (cid:96) ] ˜ v ( t ) , for all t < t , for some fixed numerical constant C > independent of t and ˜ v . As theright-hand side is positive, this proves the lemma.For any γ ∈ (0 , / , A > , and t > , we define the space S ( γ, A, t ) ⊂ C ∗ t by S ( γ, A, t ) := (cid:8) (cid:96) ∈ C (cid:0) S ( v ); H (0 , t ) (cid:1) : (cid:96) (cid:48) v ( t ) ≤ At − γ for a.e. t ∈ [0 , t ] , v ∈ S ( v ) (cid:9) , (A.3)which is a complete metric space with the metric inherited from C ∗ t . Moreover, we define themap ˆΓ[ (cid:96) ; u, v ]( t ) := (cid:90) ∞ P ( t ≥ τ x,(cid:96)u,v ) dν ( x | u, v ) , o that Γ[ (cid:96) ] ˜ v ( t ) = (cid:80) kl =1 ˜ v l (cid:82) R k × R k u l ˆΓ[ (cid:96) ; u, v ]( t ) d ˆ (cid:36) ( u, v ) . Then, for each u and v , we can replicatethe arguments from [Hambly et al., 2019, Sect. 4] for the function t (cid:55)→ ˆΓ[ (cid:96) ; u, v ]( t ) in place of thecorresponding function considered there. Given ˆ (cid:36) and V ( ·| u, v ) satisfying (5.19), we can thusconclude (by arguing precisely as in [Hambly et al., 2019, Prop. 4.9]), that there exists A > such that, for any ε > , there is a small enough time t > for which Γ : S (cid:0) − β , A + ε , t (cid:1) → S (cid:0) − β , A + ε , t (cid:1) , (A.4)where A only depends on C (cid:63) and x (cid:63) from (5.19). Moreover, by analogy with [Hambly et al.,2019, Thm. 1.6], we can deduce from Lemma A.1 that Γ is a contraction on this space for smallenough t . Therefore, the small time existence of a regular solution L ∗ v ( t ) = (cid:80) kl =1 v l L l ( t ) to(5.13, ρ = 0 ) now follows from an application of Banach’s fixed point theorem as in the proof of[Hambly et al., 2019, Thm. 1.7]. Finally, by replicating the bootstrapping argument from theproof of [Hambly et al., 2019, Cor. 5.3], we conclude that the regular solution extends until thefirst time t (cid:63) such that the H norm of ( L , . . . , L k ) diverges on (0 , t (cid:63) ) . This proves the first partof Theorem 5.9. A.1.2 Generic uniqueness
It remains to verify that the general uniqueness result of [Hambly et al., 2019, Thm. 1.8] can beextended to the present setting, which will follow from the two lemmas below. The first lemmaconcerns a family of auxiliary McKean–Vlasov problems given by X x,(cid:15)u,v ( t ) = x x ≥ ε − ε + (cid:82) t b u,v ( s ) ds + (cid:82) t σ u,v ( s ) dB s − F (cid:0) g (0) λ εv + (cid:82) t g ( s ) d L εv ( s ) (cid:1) L εv ( t ) = (cid:80) kl =1 v l (cid:82) R + × R k × R k ˆ u l P ( τ x,ε ˆ u, ˆ v ≤ t ) d(cid:36) ( x, ˆ u, ˆ v ) τ x,εu,v = inf { t ≥ X x,εu,v ( t ) ≤ } , (A.5)for ε > , where λ εv := (cid:80) kl =1 v l (cid:82) (0 ,ε ) × R k × R k ˆ u l d(cid:36) ( x, ˆ u, ˆ v ) . This family of equations will serveas the equivalent of the ‘ ε -deleted solutions’ introduced in [Hambly et al., 2019, Sect. 5.2]. Bywriting L εv ( t ) = λ εv + ˜ L εv ( t ) with ˜ L εv ( t ) = k (cid:88) l =1 v l (cid:90) [ ε, ∞ ) × R k × R k ˆ u l P ( τ x,ε ˆ u, ˆ v ≤ t ) d(cid:36) ( x, ˆ u, ˆ v ) , (A.6)we can show that these ε -deleted problems are well-posed with regularity estimates that areuniform in ε > . Lemma A.2 (Uniformly regular ε -deleted solutions) . There is an ε > such that (A.5) hasa family of solutions { L ε } ε ≤ ε which are uniformly regular in the following sense: There exists A > and t > such that L ε ∈ S ( − β , A, t ) uniformly in ε ∈ (0 , ε ] .Proof. First of all, we can note that λ εv ≤ C (cid:63) ε β / (1+ β ) uniformly in v , for small enough ε > ,by (5.19), and clearly F ( x ) = o ( x / (1+ β ) ) as x ↓ , since F is Lipschitz with F (0) = 0 . Hencethere exists ε > such that F ( g (0) λ εv ) ≤ ε/ for all ε ∈ (0 , ε ) . Next, using (A.6) and makingthe change of variables y = x − ε/ − F ( g (0) λ εv ) in (A.5) we obtain the equivalent formulation ˜ X y,εu,v ( t ) = y + (cid:82) t b u,v ( s ) ds + (cid:82) t σ u,v ( s ) dB ( s ) − F (cid:0) g (0) λ εv + (cid:82) t g ( s ) d ˜ L εv ( s ) (cid:1) + F (cid:0) g (0) λ εv (cid:1) ˜ L εv ( t ) = (cid:80) kl =1 v l (cid:82) R + × R k × R k ˆ u l P (˜ τ y,ε ˆ u, ˆ v ≤ t ) V ε ( y | ˆ u, ˆ v ) dyd ˆ (cid:36) (ˆ u, ˆ v ) V ε ( y | u, v ) = V (cid:0) y + ε + F ( g (0) λ εv ) (cid:1) { y + ε + F ( g (0) λ εv ) ≥ ε } ˜ τ y,εu,v = inf { t ≥ X y,εu,v ( t ) ≤ } ow take ε ≤ x (cid:63) . Recalling that F ( λ εv ) ≤ ε/ for all ε ∈ (0 , ε ) , we can then observe that V ε ( y | u, v ) ≤ C (cid:63) (cid:0) y + ε/ F ( g (0) λ εv ) (cid:1) y + ε/ F ( g (0) λ εv ) ≥ ε ≤ ( y + ε/ β y ≥ ε/ ≤ β C (cid:63) y β for all y < x (cid:63) / , uniformly in u , v , and ε ∈ (0 , ε ) . Therefore, for each ε ∈ (0 , ε ) , we can indeed construct aregular solution ˜ L ε to the above system by the first part of Theorem 5.9 (as proved in SectionA.1.1). Moreover, since the boundary control on V ε ( ·| u, v ) is uniform in ε ∈ (0 , ε ) , uniformlyin u and v , it follows from the fixed point argument in Section A.1.1 that the regularity of thesolutions ˜ L ε is also uniform in ε ∈ (0 , ε ) . By (A.6), the uniform regularity of the original family L ε follows a fortiori from that of ˜ L ε , and thus the proof is complete.Armed with Lemma A.2, we can now proceed to the final ingredient of the general uniquenessresult, namely the ‘monotonicity and trapping’ argument from [Hambly et al., 2019, Sect. 5.2]. Lemma A.3 (Monotonicity and vanishing envelope) . Let L ∗ : S ( v ) → L ∞ (0 , T ) be given by L ∗ v ( t ) := k (cid:88) l =1 v l L l = k (cid:88) l =1 v l (cid:90) R + × R k × R k ˆ u l P ( t ≥ τ x ˆ u, ˆ v ) d(cid:36) ( x, ˆ u, ˆ v ) for a generic solution to (5.13) with ρ = 0 , and suppose there are no jumps of ( L , . . . , L k ) on [0 , t ) so s (cid:55)→ L ∗ v ( s ) is continuous on [0 , t ) for all v . If L ε is a continuous ‘ ε -deleted’ solution on [0 , t ) , then L εv > L v on [0 , t ) , for all v (cid:54) = 0 . Moreover, if L is regular on [0 , t ) and the family { L ε } is uniformly regular on [0 , t ) , in the sense of Lemma A.2, then there is a t ∈ (0 , t ) suchthat the envelope between the two is vanishing on [0 , t ] , that is, (cid:107) L − L ε (cid:107) ∗ t → as ε → .Proof. Noting that L εv (0) = λ (cid:15)v > L v (0) for v (cid:54) = 0 , towards a contradiction we let t ∈ (0 , t ) be the first time L εv ( t ) = L v ( t ) for some v (cid:54) = 0 . Then it holds for any s < t that g (0) λ εv + (cid:90) s g ( r ) d L εv ( r ) = (cid:90) s L εv ( r ) d ( − g )( r ) + g ( s ) L εv ( s ) ≥ (cid:90) s L v ( r ) d ( − g )( r ) + g ( s ) L v ( s ) = (cid:90) s g ( r ) d L v ( r ) . and, since F is increasing, we thus have X xu,v ( s ) − X x,εu,v ( s ) = x x<ε + ε F (cid:16) g (0) λ εv + (cid:90) s gd L εv (cid:17) − F (cid:16)(cid:90) s gd L v (cid:17) ≥ ε , (A.7)for all s ∈ (0 , t ) . Arguing as in the proof of [Hambly et al., 2019, Lemma 5.6], it follows from(A.7) that L εv ( t ) ≥ L v ( t ) + k (cid:88) l =1 v l (cid:90) R + × R k × R k ˆ u l P (cid:16) inf r ∈ [0 ,t ] X xu,v ( s ) ∈ (0 , ε/ (cid:17) d(cid:36) ( x, ˆ u, ˆ v ) > L v ( t ) , which contradicts the definition of t , thus proving the first claim.For the second claim, can now rely on the fact that L εv > L v on [0 , t ) for all v (cid:54) = 0 .Consequently, since X x,εu,v ( s ) = 0 on the event { τ x,εu,v = s } , we deduce that, on this event, X xu,v ( s ) = X xu,v ( s ) − X x,εu,v ( s ) ≤ ε + ε g (0) (cid:107) F (cid:107) Lip (cid:107) L εv − L v (cid:107) s , (A.8)where the inequality follows by the equality in (A.7) and the same estimate as in the proof ofLemma A.1. From here, (A.8) allows us to replicate the proof of [Hambly et al., 2019, Lemma5.7], only with the term ‘ g (0) (cid:107) F (cid:107) Lip (cid:107) L ε − L (cid:107) ∗ s ’ in place of the term ‘ α ( L εs − L s ) ’ appearing inthat proof. This verifies the second claim.Based on Lemmas A.1 and A.3, the uniqueness part of Theorem 5.9 now follows immediatelyby retracing the proof of [Hambly et al., 2019, Thm. 1.8] (at the very end of [Hambly et al.,2019, Sect. 5]) with the cascade condition (5.16) taking the place of the physical jump condition[Hambly et al., 2019, (1.7)]. .2 Proof of Theorem 5.10 Let us begin by proving the continuity of a given solution satisfying the smallness condition(5.20). To this end, we fix a pair ( u, v ) and write X xu,v ( t ) = x + Y ( t ) + Y ( t ) with Y ( t ) := (cid:82) t ρσ u,v ( s ) dB s so Y ( t ) is B -measurable. Letting p ( t, · ) denote the density of Y ( t ) , it followsfrom Tonelli’s theorem that (cid:90) R + P (cid:0) X xu,v ( t ) ∈ A, t < τ xu,v | B (cid:1) V ( x | u, v ) dx ≤ (cid:90) R (cid:90) A p ( t, y + x + Y ( t )) V ( x | u, v ) dydx = (cid:90) A (cid:90) R p ( t, x + y + Y ( t )) V ( x | u, v ) dxdy ≤ (cid:107) V ( ·| u, v ) (cid:107) ∞ | A | , for all A ∈ B ( R ) , since p ( t, · ) integrates to . Recalling the definition of V t from (5.17), this showsthat V t ( x | u, v ) ≤ (cid:107) V ( ·| u, v ) (cid:107) ∞ for all x ∈ (0 , ∞ ) and all times t ≥ . Therefore, the criterion(5.18) holds for all times, by the smallness condition (5.20), and hence the given solution mustbe globally continuous in time.To prove the uniqueness part of Theorem 5.10, we show how to extend the arguments behind[Ledger and Søjmark, 2018b, Thm. 2.3]. Let ( X, L ) and ( ¯ X, ¯ L ) be any two solutions to (5.13)coupled through the same Brownian drivers B and B . Then we define the increasing processes L v := k (cid:88) l =1 v l L l and ¯ L v := k (cid:88) l =1 v l ¯ L l for every v ∈ S ( v ) . Retracing the arguments of [Ledger and Søjmark, 2018b, Lemma 2.1], andapplying Fubini’s theorem, we can deduce that L v ( s ) − ¯ L v ( s ) ≤ E (cid:20) k (cid:88) l v l (cid:90) R k × R k ˆ u l (cid:90) I ˆ v ( s )¯ I ˆ v ( s ) V ( x | ˆ u, ˆ v ) dxd ˆ (cid:36) (ˆ u, ˆ v ) (cid:12)(cid:12)(cid:12) B (cid:21) , where I v ( t ) := sup s ≤ t (cid:26) F (cid:16)(cid:90) s g ( r ) d L v ( r ) (cid:17) − Z s (cid:27) , and ¯ I v ( t ) := sup s ≤ t (cid:26) F (cid:16)(cid:90) s g ( r ) d ¯ L v ( r ) (cid:17) − Z s (cid:27) , with Z s := (cid:90) s b ( r ) dr + (cid:90) s σ ( r ) d ( ρB + (cid:112) − ρ B )( r ) . By symmetry, ¯ L v ( s ) − L v ( s ) satisfies the analogous bound with I v ( s ) and ¯ I v ( s ) interchanged.Furthermore, by simply repeating the first estimate from the proof of Lemma A.1, only with L and ¯ L in place of (cid:96) and ¯ (cid:96) , we have F (cid:16)(cid:90) t g ( s ) d L v ( s ) (cid:17) ≥ F (cid:16)(cid:90) t g ( s ) d ¯ L v ( s ) (cid:17) − g (0) (cid:107) F (cid:107) Lip (cid:107) L − ¯ L (cid:107) ∗ t . Therefore, relying on this inequality together with the previous observation, we can retrace thearguments of [Ledger and Søjmark, 2018b, Theorem 2.2] and conclude that | L v ( s ) − ¯ L v ( s ) | ≤ g (0) (cid:107) F (cid:107) Lip (cid:107) L − ¯ L (cid:107) ∗ s k (cid:88) l =1 v l (cid:90) R k × R k ˆ u l (cid:107) V ( ·| ˆ u, ˆ v ) (cid:107) ∞ d ˆ (cid:36) (ˆ u, ˆ v ) . At this point, the smallness condition (5.20) gives | L v ( s ) − ¯ L v ( s ) | ≤ (1 − δ ) (cid:107) L − ¯ L (cid:107) ∗ s , so, taking a supremum over v ∈ S ( v ) on the left-hand side, we conclude that there is pathwiseuniqueness. .3 Convergence of the particle system In this section we outline how the convergence to the conditional McKean–Vlasov problem(5.13) can be established by retracing the approach of Ledger and Søjmark [2018a] after someadjustments. The arguments rely heavily on specific properties of the M1-topology for theSkorokhod space of càdlàg paths. The reader is referred to Whitt [2002] for an introduction tothis topology. For concreteness, let us restrict to random start points X i satisfying (5.19) nearthe absorbing boundary at zero, although it is possible to consider higher generality in thesearguments.Let D R denote the space of real-valued càdlàg paths on [0 , T ] , and let ( X , . . . , X n ) be theunique strong solution to the particle system (5.5) of size n in D R × · · · × D R (recall Proposition5.5). Moreover, as usual, we let u i ∈ R k and v i ∈ R k denote the type vectors from (5.4), for i = 1 , . . . , n . For simplicity of notation, we are suppressing the dependence on n ≥ in eachtriple ( u i , v i , X i ) ∈ R k × R k × D R . Now consider the empirical measures P n := 1 n n (cid:88) i =1 δ u i ⊗ δ v i ⊗ δ X i ( · ) , for n ≥ , (A.9)which are random variables valued in the space of probability measures P ( R k × R k × D R ) . For ( u, v, η ) ∈ R k × R k × D R , we define the coordinate projections π ,l ( u, v, η ) := u l , π ,l ( u, v, η ) := v l ,and π ,t ( u, v, η ) := η t as well as π t ( u, v, η ) := ( u, v, η t ) and π (1 , ( u, v, η ) := ( u, v ) . Writing P nt := P n ◦ π − t and ˆ (cid:36) n := P n ◦ π − , , we have P n = (cid:36) n ⇒ (cid:36) and ˆ (cid:36) n → ˆ (cid:36) by virtue ofAssumption 5.4. The first task is to ensure tightness of the pair ( P n , B ) so that we can extractweakly convergent subsequences. Lemma A.4 (Tightness) . The sequence of random variables ( P n , B ) is tight on the productspace P ( R k × R k × D R ) × C R . Here C R is the space of continuous real-valued paths on [0 , T ] topologized by uniform convergence, and P ( R k × R k × D R ) is topologized by weak convergence ofmeasures as induced by the M1-topology on D R .Proof. Since D R is a Polish space with the M1-topology, it is a classical result (see e.g. [Sznitman,1991, Ch.I, Prop. 2.2]) that the sequence of (random) empirical measures P n is tight if, for each ε > , we can find K ε compact in R k × R k × D R , where D R comes with the M1-topology, suchthat, for all n ≥ , E n ( K ε ) ≥ − ε, where E n ( K ε ) := 1 n n (cid:88) i =1 P (cid:0) ( u i , v i , X i ) ∈ K ε (cid:1) . To fulfil this, it is sufficient that, for every ε > , we can find a compact set K ε such that P (( u i , v i , X i ) ∈ K ε ) ≥ − ε for each i = 1 , . . . , n uniformly in n ≥ . By Assumption 5.4, wehave | u i | + | v i | ≤ C , for some C > , uniformly in i = 1 , . . . , n and n ≥ . Hence we can take K ε to be of the form K ε = ¯ B C × S ε , where ¯ B C is the closed ball of radius C in R k , and S ε is compact in ( D R , M1 ) . Consequently, writing X ni for the i ’th particle in the size- n particlesystem, it suffices to show that each sequence ( X ni ) n ≥ is tight with estimates that are uniformin i = 1 , . . . , n and n ≥ . To this end, the first crucial observation is that t (cid:55)→ F (cid:16) k (cid:88) l =1 v il (cid:90) t g ( s ) d L nl ( s ) (cid:17) is increasing. Therefore, exploiting the special nature of the M1-topology, the uniform tightnessestimates can be established by retracing the steps of [Ledger and Søjmark, 2018a, Prop. 3.9].We now turn to the problem of identifying the limit points of ( P n , B ) as n → ∞ , whereconvergent subsequences are ensured by Prokhorov’s theorem in light of the previous lemma.First of all, we define the mapping ( L l ( µ ))( t ) := (cid:10) µ, π ,l ( · ) ( ∞ , (cid:0) inf s ≤ t π ,t ( · ) (cid:1)(cid:11) , (A.10) or µ ∈ P ( R k × R k × D R ) , where the rationale is of course that ( L l ( P n ))( t ) = 1 n n (cid:88) j =1 u jl t ≥ τ j = L nl ( t ) . (A.11)Using the mappings µ (cid:55)→ L l ( µ ) , we in turn define ( M ( u, v, η, µ ))( t ) := η ( t ) − η (0) − (cid:90) t b u,v ( s ) ds − F (cid:16) k (cid:88) l =1 v l (cid:90) t g ( s ) d ( L l ( µ ))( s ) (cid:17) , (A.12)and we then intend to perform a martingale argument to identify the limit points of ( P n , B ) based on mappings of the form ( µ, w ) (cid:55)→ (cid:10) µ, Ψ (cid:0) M ( · , µ ) , w (cid:1)(cid:11) (A.13)for ( µ, w ) ∈ P ( R k × R k × D R ) × C R , for suitable functions Ψ : D R × C R → R . Indeed, we canobserve that (cid:10) P n , Ψ (cid:0) M ( · , P n ) , B (cid:1)(cid:11) = 1 n n (cid:88) i =1 Ψ (cid:16)(cid:90) t σ u i , v i ( s ) d (cid:0) ρB ( s ) + (cid:112) − ρ B i ( s ) (cid:1) , B (cid:17) , where the right-hand side is a nice average of the function Ψ applied to square integrablemartingales on [0 , T ] . This will essentially allow us to transfer suitable martingale properties tothe limit as n → ∞ , which is the machinery behind the next observations.Proceeding as in [Ledger and Søjmark, 2018a, Lemma 3.11] we can show that (A.13) and sim-ilar mappings are continuous for suitable functions Ψ (the specific mappings are defined imme-diately before [Ledger and Søjmark, 2018a, Lemma 3.11]). Fix a limit point ( P , B ) of ( P n , B ) ,realised along a convergent subsequence (due to Lemma A.4 above). Write (cid:36) = Law( u , v , X (0)) ,where we recall that (cid:36) = P is the limit of (cid:36) n = P n , as ensured by Assumption (5.4). Re-tracing the steps of [Ledger and Søjmark, 2018a, Prop. 3.12] and [Ledger and Søjmark, 2018a,Proof of Thm. 3.2, p. 26], based on the aforementioned continuity results, we can show thatthere is a probability space ( ¯Ω , ¯ F , ¯ P ) which supports our limiting random variables ( u , v , P , B ) and also carries a càdlàg process X as well as a Brownian motion B ⊥ B , for which ( B, B ) isindependent of ( u , v , X (0)) , such that ( M ( u , v , X, P ))( t ) = (cid:90) t σ u , v ( s ) d (cid:0) ρB ( s ) + (cid:112) − ρ B ( s ) (cid:1) holds for all t ∈ [0 , T ] , ¯ P -almost surely. In other words, on the background space ( ¯Ω , ¯ F , ¯ P ) , wehave X ( t ) = X (0) + (cid:90) t b u , v ( s ) ds + (cid:90) t σ u , v ( s ) d (cid:0) ρB ( s ) + (cid:112) − ρ B i ( s ) (cid:1) − F (cid:16) k (cid:88) l =1 v l (cid:90) t g ( s ) d ( L l ( P ))( s ) (cid:17) . (A.14)To avoid clouding the presentation, let us (for now) assume that the limiting random probabilitymeasure P is known to be B measurable. Intuitively, this is what one expects, as the sequence P n is subject to the common noise B , which is felt by all the particles, and hence shouldstay in the limit; whereas the effect of the idiosyncratic noise from the independent Brownianmotions B , . . . , B n will be averaged way in the limit. The situation where P is not knownto be B -measurable is dealt with separately in Remark A.5 below. Once we have that P is B -measurable, retracing the proof of [Ledger and Søjmark, 2018a, Thm. 3.2], as we did above, ot only gives (A.14), but also shows that P = Law( u , v , X | B ) . Therefore, letting ¯ E denotethe expectation operator corresponding to ¯ P , we have ( L l ( P ))( t ) = (cid:10) P , π ,l ( · ) ( ∞ , (cid:0) inf s ≤ t π ,t ( · ) (cid:1)(cid:11) = ¯ E (cid:2) u l ( −∞ , (cid:0) inf s ≤ t X s (cid:1) | B (cid:3) = ¯ E (cid:2) u l ¯ P (cid:2) inf s ≤ t X s ≤ | B , u , v , X (0) (cid:3) (cid:12)(cid:12) B (cid:3) . (A.15)Since ( u , v , X (0)) is independent of ( B, B ) , using the equation for X in (A.14) and the definitionof L in (A.10), we can conclude from (A.15) that ( L l ( P ))( t ) = (cid:90) R + × R k × R k u l P ( t ≥ τ xu,v | B ) d(cid:36) ( x, u, v ) , l = 1 , . . . , k, (A.16)where (cid:36) = P is the limit of (cid:36) n given by (5.6), and where we have defined τ xu,v := inf { t ≥ X xu,v ( t ) ≤ } , and X xu,v ( t ) := x + (cid:90) t b u,v ( s ) ds + (cid:90) t σ u,v ( s ) d ( ρB ( s ) + (cid:112) − ρ B i ( s ) (cid:1) − F (cid:16) k (cid:88) l =1 v l (cid:90) t g ( s ) d ( L l ( P ))( s ) (cid:17) , (A.17)for all realisations ( u, v ) of ( u , v ) . Consequently, we have recovered the desired mean fieldlimit (5.13), since the limit point ( P , B ) of ( P n , B ) satisfies the conditional McKean–Vlasovproblem (A.16)–(A.17). Furthermore, as in [Ledger and Søjmark, 2018a, Prop. 3.6] and the proofof [Ledger and Søjmark, 2018a, Prop. 3.9], the above tightness and continuity results, along withthe expression (A.11), give that (in the M1 topology on D R ), the loss processes L nl = L l ( P n ) converge to the desired limiting loss processes L l = L l ( P ) satisfying the conditional McKean–Vlasov problem (A.16)–(A.17). Remark A.5.
Without assuming B -measurability, we need to work with what is defined as a‘relaxed’ solution to (5.13) in [Ledger and Søjmark, 2018a, Sect. 3]. Specifically, the argumentsfrom Ledger and Søjmark [2018a] only gives that P = Law( u , v , X | B , P ) with ( B , P ) ⊥ B and ( B, ( B , P )) ⊥ X (0) , as opposed to P = Law( u , v , X | B ) which we relied on above. Thatis, P fulfils the first criteria for being the conditional law of ( u , v , X ) given B , but it is onlyknown to be ( B , P ) -measurable, and hence it may not be the true conditional law given B .Yet, it behaves in almost the same way, since it is also independent of B , which is preciselywhat we expected to happen in the limit, as the idiosyncratic noise is averaged away and thecommon noise B is independent of B . Repeating (A.15) with ( P , B ) in place of B , and usingthat ( u , v , X (0)) is independent of ( B, ( B , P )) , we instead get ( L l ( P ))( t ) = (cid:90) R + × R k × R k u l P ( t ≥ τ xu,v | B , P ) d(cid:36) ( x, u, v ) , so there is potentially some extra randomness that has survived the limiting procedure. In otherwords, we have mildly relaxed the criterion that the loss processes should strictly be conditionalon the common noise B . For this reason, the limit thus obtained is called a ‘relaxed’ solutionto (5.13). Nevertheless, in cases where we have a pathwise uniqueness argument for (5.13) suchas in Section A.2 (the proof of Theorem 5.10), we can apply a Yamada-Watanabe argument asin [Ledger and Søjmark, 2018b, Thm. 2.3] to ensure that P really is B measurable and that weare hence only conditioning on the common noise B . References
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