On fairness of systemic risk measures
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Biagini, Francesca; Fouque, Jean-Pierre; Frittelli, Marco; Meyer-Brandis, Thilo Article — Published Version On fairness of systemic risk measures Finance and Stochastics Provided in Cooperation with: Springer Nature Suggested Citation: Biagini, Francesca; Fouque, Jean-Pierre; Frittelli, Marco; Meyer-Brandis, Thilo (2020) : On fairness of systemic risk measures, Finance and Stochastics, ISSN 1432-1122, Springer, Berlin, Heidelberg, Vol. 24, Iss. 2, pp. 513-564, https://doi.org/10.1007/s00780-020-00417-4 This Version is available at: https://hdl.handle.net/10419/288396 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/
Finance Stoch (2020) 24:513–564 https://doi.org/10.1007/s00780-020-00417-4 On fairness of systemic risk measures Francesca Biagini1,2 ·Jean-Pierre Fouque3· Marco Frittelli4·Thilo Meyer-Brandis1 Received: 4 July 2019 / Accepted: 30 September 2019 / Published online: 4 February 2020 © The Author(s) 2020 Abstract In our previous paper “A unified approach to systemic risk measures via acceptance sets” (Mathematical Finance, 2018), we have introduced a general class of systemic risk measures that allow random allocations to individual banks before aggregation of their risks. In the present paper, we prove a dual representation of a particular subclass of such systemic risk measures and the existence and uniqueness of the optimal allocation related to them. We also introduce an associated utility maximisation problem which has the same solution as the minimisation problem associated to the systemic risk measure. In addition, the optimiser in the dual formulation provides a risk allocation which is fair from the point of view of the individual financial institutions. The case with exponential utilities which allows explicit computation is treated in detail. Work supported by NSF grants DMS-1409434 and DMS-1814091. Part of this research was performed while F. Biagini, M. Frittelli and T. Meyer-Brandis were visiting the University of California, Santa Barbara. BF. Biagini [email protected] J.-P. Fouque [email protected].edu M. Frittelli [email protected] T. Meyer-Brandis [email protected] 1Department of Mathematics, University of Munich, Theresienstraße 39, 80333 Munich, Germany 2Department of Mathematics, University of Oslo, Box 1053, Blindern, 0316, Oslo, Norway 3Department of Statistics & Applied Probability, University of California, Santa Barbara, CA 93106-3110, USA 4Dipartimento di Matematica, Università degli Studi di Milano, Via Saldini 50, 20133 Milano, Italy
514 F. Biagini et al. Keywords Systemic risk measures ·Random allocations ·Risk allocation ·Fairness Mathematics Subject Classification (2010) 60A99 ·91B30 ·91G10 ·93D99 JEL Classification C690 ·G1 1 Introduction Consider a vector X=(X1,...,XN)∈L0(,F,P;RN)of Nrandom variables denoting a configuration of risky (financial) factors at a future time Tassociated to asystemofNfinancial institutions/banks. One of the first proposals in the framework of risk measures to measure the systemic risk of X, see Chen et al. [16], was to consider the map ρ(X):=inf{m∈R:(X)+m∈A},(1.1) where :RN→Ris an aggregation rule that aggregates the N-dimensional risk factors into a univariate risk factor, and A⊆L0(, F,P;R)is an acceptance set of real-valued random variables. As within the framework of univariate monetary risk measures, systemic risk might again be interpreted as the minimal cash amount that secures the system when it is added to the total aggregated system loss (X), given that (X)allows a monetary loss interpretation. Note, however, that in (1.1), systemic risk is the minimal capital added to secure the system after aggregating individual risks. It might be more relevant to measure systemic risk as the minimal cash amount that secures the aggregated system by adding the capital into the single institutions before aggregating their individual risks. This way of measuring systemic risk can be expressed by ρ(X):=infN i=1 mi:m=(m1,...,mN)∈RN,(X+m)∈A.(1.2) Here, the amount miis added to the financial position Xiof institution i∈{1,...,N} before the corresponding total loss (X+m)is computed (we refer to Armenti et al. [3], Biagini et al. [7] and Feinstein et al. [27]). One of the main novelties of our paper [7] was the possibility of adding to X not merely a vector m=(m1,...,mN)∈RNof deterministic cash amounts, but more generally a random vector Y∈Cfor some given class C. In particular, the main example considered in [7], and studied further in this paper, is given by choosing the aggregation function (x)= N n=1 un(xn)(1.3) for utility functions un,n=1,...,N, the acceptance set A={Z∈L1(,F,P;R), E[Z]≥B}
On fairness of systemic risk measures 515 for a given constant B, and the class Csuch that C⊆CR∩L,where CR:=Y∈L0(,F,P;RN): N n=1 Yn∈R,(1.4) where the subspace L⊆L0(, F,P;RN)will be specified later. Here, the notation N n=1Yn∈Rmeans that N n=1Ynis P-a.s. equal to some deterministic constant in R, even though each single Yn,n=1,...,N, is a random variable. Under these assumptions, the systemic risk measure considered in [7] takes the form ρ(X):=infN n=1 Yn:Y∈C⊆CR,EN n=1 un(Xn+Yn)≥B(1.5) and can still be interpreted as the minimal total cash amount N n=1Yn∈Rneeded today to secure the system by distributing the cash at the future time Tamong the components of the risk vector X. However, while the total capital requirement N n=1Yn is determined today, contrary to (1.2), the individual allocation Yi(ω) to institution idoes not need to be decided today, but in general depends on the scenario ωrealised at time T. This total cash amount ρ(X)is computed today through the formula N n=1ρn(X)=ρ(X), where each ρn(X)∈Ris the risk allocation of each bank, as explained in Definition 1.2 below. Thus, one prominent example that can be modelled by considering random allocations is the default fund of a CCP1that is liable for any participating institution. We come back to this mechanism in Sect. 5. By considering scenario-dependent allocations, we are also taking into account possible dependencies among the banks, as the budget constraints in (1.5) do not depend only on the marginal distribution of X, as it would happen for deterministic Yn. Definition 1.1 A scenario-dependent allocation YX=(Y n X)n=1,...,N ∈Cis called a systemic optimal allocation for ρ(X)defined in (1.5) if it satisfies ρ(X)=N n=1Yn X and E[N n=1un(Xn+Yn X))]≥B. As two of the main results of the paper, – we study in Sect. 3the dual formulation of the systemic risk measure (1.5)as ρ(X)=max Q∈DN n=1 EQn[−Xn]−αB(Q),(1.6) where Q:=(Q1,...,QN), the penalty function αBand the domain Dare specified in Sect. 3. In particular, we establish existence and uniqueness of the optimiser QX∈Dof (1.6). 1A central counterparty clearing house (CCP) is an entity that helps facilitate trading in various European derivatives and equities markets in order to reduce risk for traders and introduce efficiency and stability into various financial markets.
516 F. Biagini et al. – we show in Sect. 4existence and uniqueness of the systemic optimal allocation YX for the systemic risk measure (1.5). We now associate to the risk minimisation problem (1.5) a related utility maximisation problem that plays a central role in this paper, namely π(X):=supEN n=1 un(Xn+Yn):Y∈C⊆CR, N n=1 Yn≤A.(1.7) If we interpret N n=1un(Xn+Yn)as the aggregated utility of the system after allocating Y, then π(X)can be interpreted as the maximal expected utility of the system over all random allocations Y∈Csuch that the aggregated budget constraint N n=1Yn≤Aholds for a given constant A. In the following, we may write ρ(X)=ρB(X)and π(X)=πA(X)to express the dependence on the minimal level of expected utility B∈Rand maximal budget level A∈R, respectively. We shall see in Sect. 4.1 that B=πA(X)if and only if A=ρB(X), and in these cases, the two problems πA(X)and ρB(X)have the same unique solution YX. From this, we infer that once a level ρ(X)of total systemic risk has been determined, then –the systemic optimal allocation YXfor ρmaximises the expected system utility among all random allocations of total cost less than or equal to ρ(X). Once the total systemic risk has been identified as ρ(X), the second essential question is how to allocate the total risk to the individual institutions. Definition 1.2 We say that a vector (ρn(X))n=1,...,N ∈RNis a systemic risk allocation of ρ(X)if it fulfils N n=1ρn(X)=ρ(X). The requirement N n=1ρn(X)=ρ(X)is known as the “full allocation” property; see for example Brunnermeier and Cheridito [13]. In the case of deterministic allocations Y∈RN, i.e., C=RN, the optimal deterministic YXrepresents a canonical risk allocation ρn(X):=Yn X. For general (random) allocations Y∈C⊆CR, we no longer have such a canonical way to determine ρn(X); however, we shall provide evidence that a good choice is ρn(X):=EQn X[Yn X]for n=1,...,N, (1.8) where QXis the optimiser of the dual problem (1.6). To this end, suppose a probability vector Q=(Q1,...,QN)is given for the system and consider an alternative formulation of the systemic utility maximisation problem in terms of the valuation provided by Q, namely πQ(X)=πQ A(X):=supEN n=1 un(Xn+Yn):Y∈L, N n=1 EQn[Yn]≤A.(1.9) Note that in (1.9) (as well as in (1.10) below), the allocation Ybelongs to a vector space Lof random variables (introduced later) without requiring that Y∈CR(which
On fairness of systemic risk measures 517 would mean that the componentwise sum is equal to a deterministic quantity). Thus for πQ(X), we maximise the expected systemic utility among all Y∈Lsatisfying the budget constraint N n=1EQn[Yn]≤A. Similarly, we can introduce a systemic risk measure in terms of the vector Qof probability measures by ρQ(X)=ρQ B(X):=infN n=1 EQn[Yn]:Y∈L,EN n=1 un(Xn+Yn)≥B.(1.10) For ρQ(X), we thus look for the minimal systemic cost N n=1EQn[Yn]among all Y∈Lunder the acceptability constraint E[N n=1un(Xn+Yn)]≥B. Apriori,ρand ρQdefined in (1.5) and (1.10) are quite different objects: even if they both subsume the same systemic budget constraint, ρis defined only through the computation of the cash amount N n=1Yn∈R, while in ρQthe risk is defined by calculating the value (or the cost) of the random allocations, N n=1EQn[Yn]. A similar comparison applies to πand πQ. Remark 1.3 To better understand the above comparison, we make an analogy with the classical (univariate) utility maximisation from terminal wealth in securities markets. Let K:= {(H.S)T:Hadmissible}, where (H.S)Tis the stochastic integral, and let U(x) =sup{E[u(x +K)]:K∈K}be the utility from the initial wealth x∈Rwhen optimally investing in the securities Sadopting admissible strategies H. In this case, there is no need to introduce a cost operator, as we are investing in replicable contingent claims having by definition initial value x. On the other hand, UQ(x) =sup{E[u(x +K)]:EQ[K]≤0}is the optimal utility function when a probability vector Qis given. A priori, the two problems are of different nature, unless one shows (see [6]) that for a particular probability measure Qx, the two problems have the same value and U(x)=UQx(x) =minQ∈MUQ(x), where Mis the set of martingale measures. From the mathematical point of view, once the minimax martingale measure Qxis determined, UQx(x) is easier to solve than U(x), and the solution to UQx(x) can then be used to find the solution to U(x). Also for the financial application, one may use Qxto compute the fair price (see [21] and [23, Remark 3.2.2]) of a contingent claim Cby computing EQx[C]. In view of the analogy in the above remark, we also prove in this paper that (i) the optimiser QX=(Q1 X,...,QN X)of the dual problem (1.6) satisfies ρB(X)=ρQX B(X), πA(X)=πQX A(X); (ii) all four problems have the same (unique) solution YXwhen A:=ρB(X); (iii) QXprovides a systemic risk allocation (EQ1 X[Y1 X],...,EQN X[YN X])with N n=1 EQXn[Yn X]=ρB(X);(1.11)
518 F. Biagini et al. (iv) and ρB(X)=max Q∈DρQ B(X)=ρQX B(X), where the domain Dis defined in (3.3) below and replaces, in analogy with utility maximisation, the set of martingale measures. Hence ρQX Bis a valid alternative to ρB(same value and solution), and this justifies its use to compute the systemic risk. In addition, (1.11) shows that the operator assigned by EQX[·] evaluates the risk component Yn Xof the optimal allocation according to ρB(not only to ρQX B) and proves that the definition in (1.8) provides indeed a systemic risk allocation for ρ(X). In Sect. 5, we further elaborate on this interpretation, we study in detail the properties of the systemic risk probability vector QX, and we provide in particular for the marginal risk contribution the formula d dερ(X+εV)ε=0=− N n=1 EQn X[Vn]for V∈L. We also discuss certain properties inferred from the above results that argue for the fairness of the systemic risk allocation. Based on the above exposition, we structure the remaining part of the paper as follows. In Sect. 2, we introduce the technical setting within Orlicz spaces and the main assumptions, and we show that our optimisation problems are well posed. In Sect. 3, we study the dual representation (1.6) of the systemic risk measure. Notably, existence and uniqueness of the dual optimiser QXare proved in Proposition 3.1;see also Corollary 4.13 in Sect. 4. In Sect. 4, we deal with existence and uniqueness of solutions of the primal problems (1.5), (1.7) and (1.9), (1.10). To guarantee existence, we need to enlarge the environment and consider appropriate spaces of integrable random variables. In Sect. 5, we derive cash-additivity and risk marginal contribution properties of the systemic risk measure ρ(X), and fairness properties of the optimal allocations ρn(X). The case with exponential utilities and grouping of institutions is treated in detail in Sect. 6, where additional sensitivity and monotonicity properties are established as well. We conclude this section with a literature overview on systemic risk. In Craig and von Peter [20], Boss et al. [12] and Cont et al. [19], one can find empirical studies on banking networks, while interbank lending has been studied via interacting diffusions and a mean-field approach in several papers like Fouque and Sun [30], Fouque and Ichiba [28], Carmona et al. [15], Kley et al. [37], Battiston et al. [5]. Among the many contributions on systemic risk modelling, we mention the classical contagion model proposed by Eisenberg and Noe [26], the default model of Gai and Kapadia [33], the illiquidity cascade models of Gai and Kapadia [32], Hurd et al. [36] and Lee [39], the asset fire sale cascade model by Cifuentes et al. [18] and Caccioli et al. [14], as well as the model in Weber and Weske [45] that additionally includes cross-holdings. Further works on network modelling are Amini et al. [1], Rogers and Veraart [43], Amini et al. [2], Gleeson et al. [34], Battiston and Caldarelli [4], Detering et al. [24] and Detering et al. [25]. See also the references therein. For an exhaustive overview on the literature on systemic risk, we refer the reader to the recent volumes of Hurd [35] and of Fouque and Langsam [29].
On fairness of systemic risk measures 519 2 The setting We now introduce the setting and discuss some fundamental properties of our systemic risk measures. Given a probability space (, F,P), we consider the space of random vectors L0:=L0(P;RN):={X=(X1,...,XN):Xn∈L0(,F,P;R), n =1,...,N}. The measurable space (, F)is fixed throughout the paper and does not appear in the notations. Unless we need to specify a different probability, we also suppress P from the notations and simply write L0(RN). In addition, we sometimes suppress Rd,d=1,...,N, in the notation of the vector spaces when the dimension of the random vector is clear from the context. We assume that L0(RN)is equipped with the componentwise order relation, i.e., X1≥X2if Xi 1≥Xi 2P-a.s. for i=1,...,N. When Q=(Q1,...,QN)is a vector of probability measures on (, F),we set L1(Q):={X=(X1,...,XN):Xn∈L1(Qn), n =1,...,N}. Unless differently stated, all inequalities between random vectors are meant to be P-a.s. inequalities. A vector X=(X1,...,XN)∈L0denotes a configuration of risky factors at a future time Tassociated to a system of Nentities. 2.1 Orlicz setting We consider systemic risk measures defined on Orlicz spaces; see Rao and Ren [40, Chap. III, Sect. 3.4 and Chap. IV, Sects. 4.2 and 4.4] for further details on Orlicz spaces. This presents several advantages. From a mathematical point of view, it is a more general setting than L∞, but at the same time it simplifies the analysis since the topology is order-continuous and there are no singular elements in the dual space. Furthermore, it has been shown by Biagini and Frittelli [9] that the Orlicz setting is natural to embed utility maximisation problems, as the natural integrability condition E[u(X)]>−∞ is implied by E[φ(X)]<+∞; see below. Univariate convex risk measures on Orlicz spaces have been introduced and studied by Cheridito and Li [17] and Biagini and Frittelli [10]. Let u:R→Rbe a concave and increasing function with limx→−∞ u(x) x=+∞. Consider φ(x) :=−u(−|x|)+u(0). Then φ:R→[0,+∞)is a strict Young function, meaning that it is finite-valued, even and convex on Rwith φ(0)=0 and limx→+∞ φ(x) x=+∞. The Orlicz space Lφand Orlicz heart Mφare respectively defined by Lφ:={X∈L0(R):E[φ(αX)]<+∞for some α>0}, Mφ:={X∈L0(R):E[φ(αX)]<+∞for all α>0}, and they are Banach spaces when endowed with the Luxemburg norm. The topological dual of Mφis the Orlicz space Lφ∗, where the convex conjugate φ∗of φdefined by φ∗(y) :=supx∈R(xy −φ(x)), y ∈R, is also a strict Young function. Note that E[u(X)]>−∞ if E[φ(X)]<+∞.(2.1)
520 F. Biagini et al. Remark 2.1 It is well known that L∞(P;R)⊆Mφ⊆Lφ⊆L1(P;R). In addition, from the Fenchel inequality xy ≤φ(x)+φ∗(y), we obtain for any probability measure QPthat (α|X|)λdQ dP≤φ(α|X|)+φ∗λdQ dP, and we immediately deduce that dQ dP∈Lφ∗implies Lφ⊆L1(Q;R). Given utility functions u1,...,uN:R→Rsatisfying the above conditions with associated Young functions φ1,...,φN, we define L=M:=Mφ1×···×MφN,L :=Lφ1×···×LφN.(2.2) 2.2 Assumptions and some properties of ρ We consider systemic risk measures ρ:M→[−∞,+∞]with ρ(X):=infN n=1 Yn:Y∈C⊆CR,EN n=1 un(Xn+Yn)≥B(2.3) as in (1.5), where the notation E[N n=1un(Xn+Yn)]≥Balso implicitly means that N n=1un(Xn+Yn)∈L1(P)and the linear space CRwas introduced in (1.4). Note that there is no loss of generality in assuming un(0)=0 (simply replace Bwith B−N n=1un(0)). The following are standing assumptions for the rest of the paper. Assumption 2.2 1) C0⊆CRand C=C0∩Mis a convex cone which satisfies RN⊆C⊆CR. 2) For all n=1,...,N,un:R→Ris increasing, strictly concave, differentiable and satisfies the Inada conditions un(−∞):= lim x→−∞un(x) =+∞,u n(+∞):= lim x→+∞un(x) =0. 3) B<(+∞), i.e., there exists M∈RNsuch that N n=1un(Mn)≥B. 4) For all n=1,...,N, it holds for any probability measure QPthat EvndQ dP<∞if and only if EvnλdQ dP<∞,∀λ>0, where vn(y) :=supx∈R(un(x) −xy). Also, from the Fenchel inequality un(X) ≤XdQ dP+vn(dQ dP)P-a.s., we immediately deduce that if X∈L1(Q) and E[vn(dQ dP)]<∞for some probability measure QP, then E[un(X)]<+∞. Some further useful properties of vnare collected in Lemma A.5.
On fairness of systemic risk measures 527 Proposition 4.1 (a) B=πA(X)if and only if A=ρB(X). (b) If B=πA(X),then A=ρB(X). (c) If A=ρB(X)and there exists a solution to one of the two problems πA(X)or ρB(X),then it is the unique solution to both problems. Proof (a) “⇐”LetA=ρB(X)and suppose first that πA(X)>B. Then there exists ˜ Y∈C0∩Msuch that N n=1˜ Yn≤Aand E[N n=1un(Xn+˜ Yn)]>B. The continuity of unand E[un(Zn)]>−∞ for all Z∈Mimply that there exist ε>0 and Y:= ˜ Y−ε1∈C0∩Msuch that E[N n=1un(Xn+ Yn)]≥Band N n=1 Yn<A. This is in contradiction to A=ρB(X). Suppose now that πA(X)<B. Then there must exist δ>0 such that we have E[N n=1un(Xn+Yn)]≤B−δfor all Y∈C0∩Msuch that N n=1Yn≤A.As A=ρB(X), for all ε>0, there exists Yε∈C0∩Msuch that N n=1Yn ε≤A+εand E[N n=1un(Xn+Yn ε)]≥B. For any η≥ε≥N n=1Yn ε−A, we get N n=1Yn ε−η N≤A+ε−η≤A. Due to E[un(Zn)]>−∞ for all Z∈Mand the continuity of un, we may select ε>0 and η≥εsmall enough so that E[N n=1un(Xn+Yn ε−η N)]>B−δ.As Y:=(Yn ε−η N)n∈C0∩M, we obtain a contradiction. “⇒”LetB=πA(X)and suppose first that ρB(X)<A. Then there must exist ˜ Y∈C0∩Msuch that E[N n=1un(Xn+ Yn)]≥Band N n=1 Yn<A. Then there exist ε>0 and Y:= ˜ Y+ε1∈C0∩Msuch that N n=1 Yn≤Aand E[N n=1un(Xn+ Yn)]>B. This is in contradiction to B=πA(X). Suppose now that ρB(X)>A. Then there must exist δ>0 such that we have N n=1Yn≥A+δfor all Y∈C0∩Msuch that E[N n=1un(Xn+Yn)]≥B.As B=πA(X), for all ε>0, there exists Yε∈C0∩Msuch that N n=1Yn ε≤Aand E[N n=1un(Xn+Yn ε)]>B−ε. Define ηε:=infa>0:EN n=1 unXn+Yn ε+a N≥B and note that ηε↓0ifε↓0. Take ε>0 such that ηε<δ. Then for any 0<β<δ−ηε,wehaveN n=1(Yn ε+ηε+β N)≤A+ηε+β<A+δas well as E[N n=1un(Xn+Yn ε+ηε+β N)]≥B.As(Y n ε+ηε+β N)∈C0∩M, we obtain a contradiction. (b) This follows in the same way as “⇒” in (a), replacing Mwith L1(P,QX). (c) Suppose there exists Y∈C0∩Mwhich is a solution to problem (1.5). As A:=ρB(X), then N n=1Yn=Aand the constraint in problem (1.7) is fulfilled for Y.By(a),B=πA(X)≥E[N n=1un(Xn+Yn)]≥Band we deduce that Yis a solution to problem (1.7). Suppose there exists Y∈C0∩Mwhich is a solution to problem (1.7) and set B:= πA(X). Then E[N n=1un(Xn+Yn)]=Band the constraint in problem (1.5) is fulfilled for Y.By(a),A=ρB(X)≤N n=1Yn≤A
528 F. Biagini et al. and we deduce that Yis a solution to problem (1.5). As ρB(X)admits at most one solution by Proposition 2.4, the same must be true for πA(X). Proposition 4.2 (a) B=πQ A(X)if and only if A=ρQ B(X). (b) If B=πQ A(X),then A=ρQ B(X).Similarly,if B=πQ A(X),then A=ρQ B(X). (c) If A=ρQ B(X)and B=πQ A(X)and there exists a solution to one of the two problems πQ A(X)or ρQ B(X),then it is the unique solution to both problems. (d) In (c), we may replace πQ A,ρQ Bwith πQ A,ρQ Bor with πQ A,ρQ B. Proof Use step by step the same arguments as in the proof of Proposition 4.1,replacing N n=1Ynwith N n=1EQn[Yn]. The uniqueness in (c) is a consequence of Remark 4.9. When using Q=QX, we have already proved that ρB(X)=ρQX B(X). Similarly: Corollary 4.3 Let A:=ρB(X).Then πA(X)=πQX A(X). Proof As A=ρB(X)∈R, Proposition 3.3 gives A=ρB(X)=ρQX B(X). By Proposition 4.1 (a), respectively Proposition 4.2 (a), we deduce that B=πA(X), resp. B=πQX A(X), hence πA(X)=πQX A(X). 4.2 On the optimal values The main contribution of this section is to show that the optimal values coincide, see (4.8) and (4.9) below, and that, see (4.11) below, πQ A(X)=max N n=1an=A N n=1 Un(an), A ∈R, where Un(an):=sup{E[un(Xn+W)]:W∈Mφn,EQn[W]≤an}(4.3) and a∈RN. In the sequel, we write UQn n(an)when we need to emphasise the dependence on Qn. Note that E[un(Xn+W)]≤un(E[Xn+W])<+∞ for all Xn,W ∈Mφn⊆L1(P;R). The conditions Xn,W ∈Mφnimply that we have E[un(Xn+W)]>−∞, from which it follows that Un(an)>−∞.AsdQ dP ∈L∗, W∈Mφnimplies W∈L1(Qn)and the problem (4.3) is well posed. Due to the monotonicity and concavity of un, the function Unis monotone increasing, concave and continuous on Rand we may replace in its definition the inequality with an equality sign. However, in general, the solution to (4.3) only exists on a larger domain, as suggested by the well-known result reported in Proposition A.6. This leads us to introduce the auxiliary problems Un(an):=sup{E[un(Xn+W)]:W∈L1(Qn),EQn[W]≤an}, Un(an):=sup{E[un(Xn+W)]:W∈L1(P,Qn),EQn[W]≤an},
On fairness of systemic risk measures 529 where L1(P,Qn)is defined as in (4.1). The following proposition is a multi-dimensional version of well-known utility maximisation problems. Its proof is based on the extended Namioka–Klee theorem and deferred to Appendix A.4. Proposition 4.4 We have that Un(an)= Un(an)= Un(an)<+∞,(4.4) if Un(an)<u n(+∞), then Un:R→Ris differentiable, Un(−∞)=−∞,U n>0,U n(−∞)=+∞,U n(+∞)=0,(4.5) and Un(an)=inf λ>0λ(EQn[Xn]+an)+EvnλdQn dP .(4.6) We now show that the optimal values are the same. Lemma 4.5 Let A:=ρQ B(X)and πQ A(X)<+∞.Then πQ A(X)=supEN n=1 un(Xn+Yn):Y∈M, N n=1 EQn[Yn]=A =:πQ,= A(X)(4.7) and πQ A(X)=sup N n=1an=A N n=1 Un(an)=πQ A(X)=πQ A(X), (4.8) ρQ B(X)=ρQ B(X)=ρQ B(X). (4.9) Proof Clearly, +∞ >π Q A(X)≥πQ,= A(X). By way of contradiction, suppose that πQ A(X)>π Q,= A(X)and take ε>0 such that πQ A(X)−ε>π Q,= A(X). By the definition of πQ A(X), there exists Y∈Msatisfying N n=1EQn[Yn]<A as well as E[N n=1un(Xn+Yn)]>πQ A(X)−ε.Take Yn=Yn+δ,δ∈R+, such that N n=1EQn[ Yn]=A. Then πQ,= A(X)≥EN n=1 un(Xn+ Yn)≥EN n=1 un(Xn+Yn) >πQ A(X)−ε>πQ,= A(X), which is a contradiction. Hence (4.7) holds true. Note that M={Y=a+Z:a∈RNand Z∈Msuch that EQn[Zn]=0 for each n}.
530 F. Biagini et al. Indeed, take Y∈Mand let an:=EQn[Yn]∈Rand Zn:=Yn−an∈Mφn. Then πQ A(X)=supEN n=1 un(Xn+Yn):Y∈M, N n=1 EQn[Yn]=A =sup N n=1an=A,Zn∈Mφn,EQn[Zn]=0∀n EN n=1 un(Xn+an+Zn) =sup N n=1an=A N n=1 sup Yn∈Mφn,EQn[Yn]=anE[un(Xn+Yn)] =sup N n=1an=A N n=1 Un(an), (4.10) which shows the first equality in (4.8). Then πQ A(X)=πQ A(X)=πQ A(X)are consequences of (4.4) and the decompositions analogous to the one just obtained for πQ A(X)in (4.10). If A:=ρQ B(X)>−∞, then B=πQ A(X)by Proposition 4.2 (a). Hence B=πQ A(X)=πQ A(X)=πQ A(X), and from Proposition 4.2 (b), we obtain A:=ρQ B(X)=ρQ B(X), hence (4.9). Proposition 4.6 Let A:=ρQ B(X)and πQ A(X)<+∞.There exists a solution a∗∈RN to problem (4.8), namely πQ A(X)=sup a∈RNwith N n=1an=A N n=1 Un(an)= N n=1 Un(an ∗)and N n=1 an ∗=A. (4.11) Proof Fix δ>0 and let am=(a1 m,...,aN m)m∈Nbe an approximating sequence for the supremum in (4.11). Then N n=1Un(an m)≥πQ A(X)−δ=:Cand N n=1an m=A for large enough m. Then (4.11) is a consequence of the continuity of Unand of Lemma 4.7 below, which guarantees that ambelongs to a compact set in RN. Lemma 4.7 Set K:= {a∈RN:N n=1an≤A, N n=1Un(an)≥B}for arbitrary constants A,B∈R.Then Kis a bounded closed set in RN. Proof See Appendix A.4. We now turn to the uniqueness of the solution to problem (3.2). The proof is in Appendix A.4 and uses the same arguments as in the proof of Proposition 2.4. Lemma 4.8 The penalty function can be written as αB(Q)=supN n=1 EQn[−Zn]:Z∈M, N n=1 E[un(Zn)]=B =supN n=1 EQn[−Zn]:Z∈L1(P,Q), EN n=1 un(Zn)≥B,(4.12)
On fairness of systemic risk measures 531 and there exists at most one Z∈L1(P,Q)satisfying αB(Q)= N n=1 EQn[−Zn]and N n=1 E[un(Zn)]≥B. (4.13) Remark 4.9 From (4.9) and (3.5), we have ρQ B(X)=ρQ B(X)=ρQ B(X)=− N n=1 EQn[Xn]−αB(Q). Hence with a proof similar to the one of Lemma 4.8, we may replace the inequality with an equality sign in the budget constraint in the definition of ρQ B(X),ρQ B(X)and ρQ B(X), and show the uniqueness of the optimiser Yin ρQ B(X),ρQ B(X)and ρQ B(X). 4.3 On the solution of ρQand comparison of solutions Theorem 4.10 Suppose αB(Q)<+∞.Consider the random vector YQgiven by Yn Q:=−Xn−v nλ∗dQn dP, where λ∗is the unique solution to (3.9). Then Yn Q∈L1(Qn),un(Xn+ Yn Q)∈L1(P), E[N n=1un(Xn+ Yn Q)]=Band ρQ B(X)=infN n=1 EQn[Yn]:Y∈M,EN n=1 un(Xn+Yn)≥B = N n=1 EQn[ Yn Q](4.14) =minN n=1 EQn[Yn]:Y∈L1(Q), EN n=1 un(Xn+Yn)≥B =ρQ B(X), (4.15) so that YQis the solution for ρQ B(X). Proof Note that ρQ B(X)>−∞ as αB(Q)<+∞. The integrability conditions hold thanks to the results stated in Appendix A.3.From(3.5) and the expression (3.8)for the penalty, we compute ρQ B(X)=− N n=1 EQn[Xn]−αB(Q) = N n=1 EQn−Xn−v nλ∗dQn dP= N n=1 EQn[ Yn Q].
532 F. Biagini et al. We show that Yn Qsatisfies the budget constraint N n=1 E[un(Xn+ Yn Q)]= N n=1 Eun−v nλ∗dQn dP = N n=1 Evnλ∗dQn dP−λ∗N n=1 EQnv nλ∗dQn dP =B due to u(−v(y)) =v(y) −yv(y) by Lemma A.5 and (3.9). Finally, from (4.9), it follows that ρQ B(X)=ρQ B(X), and Remark 4.9 implies uniqueness. When solutions to both problems ρB(X)and ρQX B(X)exist, they coincide. Proposition 4.11 Let YX∈C0∩Mbe the optimal allocation for ρB(X)and QXa solution to the dual problem (3.1). Then YX= YQX,i.e., Yn X= Yn QX:=−Xn−v nλ∗dQn X dP. Proof Note that YXsatisfies EN n=1 un(Xn+Yn X)≥B, (4.16) N n=1 Yn X=ρB(X), (4.17) N n=1 EQn X[Yn X]≤ N n=1 Yn X,(4.18) as YX∈Cand QX∈D.From(4.14), (3.5), (3.4) and (4.17), we deduce that N n=1 EQn X[ Yn QX]=ρQX B(X)=− N n=1 EQn X[Xn]−αB(QX) =ρB(X)= N n=1 Yn X.(4.19) As YXsatisfies (4.16), the definition of ρQX B(X)gives N n=1 Yn X=ρB(X)=ρQX B(X)≤ N n=1 EQn X[Yn X],
On fairness of systemic risk measures 533 which shows together with (4.18) that N n=1 Yn X= N n=1 EQn X[Yn X].(4.20) From (4.19) and (4.20), we then deduce that αB(QX)=− N n=1 EQn X[Xn+ Yn QX], αB(QX)=− N n=1 (EQn X[Xn]+Yn X)=− N n=1 EQn X[Xn+Yn X]. As both X+YXand X+ YQXsatisfy the budget constraints associated to αB(QX) in (4.13), this implies that αB(QX)is attained by both X+YXand X+ YQX.The uniqueness shown in Lemma 4.8 allows us to conclude that YX= YQX. Remark 4.12 Theorem 4.19 below proves the existence of ˜ YX∈C0∩L1(P,QX) satisfying (4.16)–(4.18) with ˜ YXinstead of YX. Then the above proof shows that ˜ YX= YQX. Similarly, Corollary 4.13 below holds for such ˜ YX∈C0∩L1(P,QX). We now show that the maximiser of the dual representation is unique. Corollary 4.13 Suppose there exists an optimal allocation YXto ρB(X).Then the solution QX=(Q1 X,...,QN X)of the dual problem (3.1)is unique. Proof Suppose Q1,Q2are two optimisers of the dual problem (3.1). Then we have αB(Q1)<+∞,αB(Q2)<+∞ and by Proposition 4.11 and Remark 4.12,for each n, −Xn−v nλ∗ 1 dQn 1 dP= Yn Q1=Yn X= Yn Q2=−Xn−v nλ∗ 2 dQn 2 dPP-a.s. As v nis invertible, we conclude that λ∗ 1 dQn 1 dP=λ∗ 2 dQn 2 dPP-a.s., which then implies Qn 1=Qn 2as E[dQn 1 dP]=E[dQn 2 dP]=1. 4.4 On the existence of the optimal allocation for ρB 4.4.1 A first step We first show that ρBreaches its infimum at some Y∈L1(P;RN).
534 F. Biagini et al. Theorem 4.14 For C⊆CR∩Mand for any X∈M,there exist Yin L1(P;RN) such that N n=1 Yn∈R,EN n=1 un(Xn+Yn)≥B, ρB(X):=infN n=1 Zn:Z∈C,EN n=1 un(Xn+Zn)≥B= N n=1 Yn and a sequence (Yk)k∈N⊆Cwith E[N n=1un(Xn+Yn k)]≥Band Yk→YP-a.s. Remark 4.15 We note that the random vector Yin Theorem 4.14 satisfies all the conditions for being the optimal allocation for ρB(X), except for the integrability condition Y∈M, which is replaced by Y∈L1(P;RN). Furthermore, Y=limk→∞Yk P-a.s. for Yk∈C0∩M. If we assume that C0is closed in L0(P), which is a reasonable assumption and holds true if C=C(n), in which case C(n) 0is defined in (2.4), then Yalso belongs to C0, but in general not to C(as Mis in general not closed for P-a.s. convergence). A special case is when the cardinality of is finite and the set Cis closed for P-a.s. convergence; under these assumptions, Ybelongs to Cand Y=YX= YQX. In Sect. 4.4.2, we show when Yalso belongs to C0∩L1(QX;RN). Proof of Theorem 4.14 Take a sequence (Vk)k∈N∈C⊆CR∩M⊆L1(P;RN)such that Rck:=N n=1Vn k↓ρB(X)as k→∞and E[N n=1un(Xn+Vn k)]≥B.Thesequence (Vk)k∈Nis bounded for the L1(P;RN)-norm if and only if so is the sequence (X+Vk)k∈N. Given the decomposition into positive and negative parts N n=1 E[|Xn+Vn k|]= N n=1 E[(Xn+Vn k)+]+ N n=1 E[(Xn+Vn k)−],(4.21) we define the index sets N+ ∞=n∈{1,...,N}:limsup k→∞ E[(Xn+Vn k)+]=+∞, N+ b=n∈{1,...,N}:limsup k→∞ E[(Xn+Vn k)+]<+∞ and similarly N− ∞and N− bfor the negative parts. We can split (4.21)as n∈N+ ∞ EP[(Xn+Vn k)+]+ n∈N+ b EP[(Xn+Vn k)+] + n∈N− ∞ EP[(Xn+Vn k)−]+ n∈N− b EP[(Xn+Vn k)−]. If the sequence (X+Vk)k∈Nis not L1(P;RN)-bounded, then one of the sets N+ ∞ or N− ∞must be nonempty and then, because of the constraint N n=1Vn k=ck, both
On fairness of systemic risk measures 535 N+ ∞and N− ∞must be nonempty. From Lemma A.1 (a), Jensen’s inequality and (4.2) give B≤ N n=1 E[un(Xn+Vn k)]≤ N n=1 un(E[Xn+Vn k]) = N n=1 unE[(Xn+Vn k)+]+ N n=1 un−E[(Xn+Vn k)−] ≤b n∈N+ ∞ E[(Xn+Vn k)+]+ n∈N+ b E[(Xn+Vn k)+] −2b n∈N− ∞ E[(Xn+Vn k)−]+ n∈N− b E[(Xn+Vn k)−]+const. =bck+ N n=1 E[Xn]+const. −b n∈N− ∞ E[(Xn+Vn k)−]+ n∈N− b E[(Xn+Vn k)−], which is a contradiction as the second sum in the last term is not bounded from above. Hence our minimising sequence (Vk)k∈Nhas bounded L1(P;RN)-norm and we may apply a Komlós compactness argument as in [22, Theorem 1.4]. Applying this to the sequence (Vk)k∈N⊆C, we can find for all ksome Yk∈conv(Vi,i ≥k) ⊆C, as Cis convex, such that (Yk)converges P-a.s. to some Y∈L1(P;RN). Observe that by construction, N n=1Yn kis P-a.s. a real number, and as a consequence, so is N n=1Yn.AsE[N n=1un(Xn+Vn k)]≥B,alsotheYksatisfy this constraint and therefore ρB(X)≤N n=1Yn k. Recall that Yk=i∈Jkλk iVi∈conv(Vi,i ≥k); so there are convex weights (λk i)i∈Jkwith λk i>0 and i∈Jkλk i=1, where Jkis a finite subset of {k,k +1,...}. For any fixed k, we compute N n=1 Yn k= N n=1 i∈Jk λk iVn ij= i∈Jk λk iN n=1 Vn i = i∈Jk λk ici≤ck i∈Jk λk i=ck,(4.22) and from ρB(X)≤N n=1Yn k≤ck, we then deduce that N n=1Yn=ρB(X). We now show that Yalso satisfies the budget constraint. If all utility functions are bounded from above, this is an immediate consequence of Fatou’s lemma
536 F. Biagini et al. since N n=1 E[−un(Xn+Yn)]= N n=1 Eliminf k→∞ −un(Xn+Yn k) ≤liminf k→∞ N n=1 E[−un(Xn+Yn k)]≤−B. In the general case, recall first that the sequence (Vk)is bounded in L1(P;RN), and the argument used in (4.22) shows that X+Yk1≤X1+sup k∈NVk1, hence supk∈NX+Yk1<∞. We now need to exploit the Inada condition at +∞. Applying Lemma A.1 (b) to the utility functions un, assumed null in 0, we get −un(x) +εx++b(ε) ≥0,∀x∈R. Plugging X+Yinto the expression above and applying Fatou’s lemma, we have EN n=1−un(Xn+Yn)+ε(Xn+Yn)++b(ε) =Eliminf k→∞ N n=1−un(Xn+Yn k)+ε(Xn+Yn k)++b(ε) ≤liminf k→∞ N n=1 E[−un(Xn+Yn k)+ε(Xn+Yn k)++b(ε)] ≤−B+εsup k∈NX+Yk1+b(ε). As the term b(ε) cancels in the above inequality, we conclude that for all ε>0, EN n=1−un(Xn+Yn)≤−B+εsup k∈NX+Yk1− N n=1 E[(Xn+Yn)+], and since supk∈NX+Yk1<∞, we obtain E[N n=1−un(Xn+Yn)]≤−Bso that Ysatisfies the constraint. 4.4.2 Second step: the optimal allocation is in L1(QX) We now prove further integrability properties of the random vector Yin Theorem 4.14. Lemma 4.16 The random vector Yin Theorem 4.14 satisfies Y−∈L1(QX).
On fairness of systemic risk measures 543 Example 5.2 For the set C(n)in Definition 2.5,ρis cash-additive on WC(n)=C(n). The latter equality holds because we are not imposing any restrictions on the vector d=(d,...,dm)∈Rmwhich determines the grouping. Remark 5.3 Under Assumption 2.2,wehaveRN⊆WCand then (5.1) holds for all V∈RN. The marginal risk contribution d dερ(X+εV)|ε=0was also considered in [13] and [3] and is an important quantity which describes the sensitivity of the risk of Xwith respect to the impact V∈L0(RN). The property (5.1) cannot be immediately generalised to the case of random vectors Vas N n=1Vn/∈Rin general. In the following, we obtain the general local version of cash-additivity, which extends (5.1)toarandom setting. Proposition 5.4 Let Xand V∈M.Let QXbe the solution to the dual problem (3.1)associated to ρ(X)and assume that ρ(X+εV)is differentiable with respect to εat ε=0, and that dQX+εV dP→dQX dPin σ(L∗,M)as ε→0. Then d dερ(X+εV)ε=0=− N n=1 EQn X[Vn].(5.2) Proof As the penalty function αBdoes not depend on X,(3.4) yields d dερ(X+εV)ε=0=d dεN n=1 EQn X+εV[−Xn−εV n]−αB(QX+εV)ε=0 =d dεN n=1 EQn X+εV[−Xn]−αB(QX+εV)ε=0 + N n=1 d dε(εEQn X+εV[−Vn])ε=0 (5.3) =0+ N n=1 lim ε→0EQn X+εV[−Vn]= N n=1 EQn X[−Vn],(5.4) where the equality between (5.3) and (5.4) is justified by the optimality of QXand the differentiability of ρ(X+εV), while the last equality is guaranteed by the convergence of (dQX+εV dP). Remark 5.5 We emphasise that the generalisation (5.2)of(5.1) holds because we are computing the expectation with respect to the vector QX. The assumptions of Proposition 5.4 are satisfied for exponential utility functions, which are considered in Sect. 6. 5.2 Interpretation and implementation of ρ(X) Going back to the definition (1.5), we see that ρ(X)represents the minimal total cash amount needed to make the system acceptable at time T. For notational simplicity,
544 F. Biagini et al. we write in the sequel YXfor the solution of ρB(X), i.e., do not distinguish YXand ˜ YX. As already mentioned in Sect. 1and as a result of Proposition 4.1, one economic justification for ρis that the optimal allocation YXof ρ(X)maximises the expected system utility among all random allocations of cost less than or equal to ρ(X). We notice also that the class Cmay determine the level of risk sharing (as explained below in (b)) between the banks, ranging from no risk sharing in the case C=RNof deterministic allocations to the case C=CRof full risk sharing, and other constraints in between as in the Definition 2.5 of grouping. We now discuss two features of our systemic risk measure. Implementation of the scenario-dependent allocation (a) In practice, the scenario-dependent allocation can be described as a default fund as in the case of a CCP (see [3]). The amount ρ(X)is collected at time 0 according to some systemic risk allocation ρn(X),n=1,...,N, which satisfies N n=1ρn(X)=ρ(X). Then at time T, this exact same amount is redistributed among the banks according to the optimal scenario-dependent allocations Yn Xsatisfying N n=1Yn X=ρ(X), so that the fund acts as a clearing house, assuming that each bank fulfils its commitment. (b) An alternative interpretation and implementation of the scenario-dependent allocation more in the spirit of monetary risk measures is in terms of capital requirements together with a risk sharing mechanism. Consider again a given systemic risk allocation ρn(X),n =1,...,N. At time 0, a capital requirement ρn(X)is imposed on each bank n=1,...,N. Then at time T, a risk sharing mechanism takes place: each bank provides (if negative) or collects (if positive) the amount Yn X−ρn(X), assuming as before that each bank fulfils its commitment. Note that in sum, the financial position of bank nat time Tis Xn+ρn(X)+(Y n X−ρn(X)) =Xn+Yn Xas required. This risk sharing mechanism is made possible by the clearing property N n=1(Yn X−ρn(X)) =0, which follows from N n=1Yn X=ρ(X)and the full risk allocation requirement N n=1ρn(X)=ρ(X). The incentive for a single bank to enter in such a mechanism is made clear below after we introduce the choice of a fair risk allocation in Sect. 5.3. Total risk reduction and dependence structure of X From a system-wide point of view, considering the optimal random allocation YX implies a reduction of the total amount needed to secure the system (compared with the optimal deterministic allocation). This reduction is also a consequence of our framework of scenario-dependent allocations that allows taking into account the dependence structure of X. An example showing these features can be found in [7, Example 7.1]. If the aggregation function is a sum of utility functions as in (1.3), one can see directly that the dependence structure of Xis taken into account from the constraint E[N n=1un(Xn+Yn)]≥Bin (1.5), which depends only on the marginal distributions of Xin the case of deterministic Yn. 5.3 Fair systemic risk allocation ρn(X) We now address the problem of choosing a systemic risk allocation (ρn(X))n=1,...,N in RN(or individual contributions at time zero) as introduced in Definition 1.2.Note
On fairness of systemic risk measures 545 that in our setting, besides providing a ranking of the institutions in terms of their systemic riskiness, a risk allocation ρn(X)can be interpreted as a capital contribution/requirement for institution nin order to secure the system. From (5.2), we see that EQX[·] defined by EQX[Y]=N n=1EQn X[Yn]already appeared as a multivariate valuation operator, and on the other hand, we have obtained in (4.20) that the minimiser YXand the maximiser QXof the dual problem satisfy ρ(X)= N n=1 Yn X= N n=1 EQn X[Yn X], which shows that ρn(X)=EQn X[Yn X],n=1,...,N, gives a systemic risk allocation. Any vector Q=(Qn)n=1,...,N of probability measures gives rise to a valuation operator EQ[·] and to the systemic risk measure ρQgiven by (1.10). Note, however, that in (1.10), the clearing condition N n=1Yn=ρ(X)is not guaranteed since the optimisation is there performed over all Y∈M. Now, using the valuation EQX[·] given by the dual optimiser, we know by Proposition 4.11 that the optimal allocation in (1.10) fulfils the clearing condition YX∈CR, and is in fact the same as the optimal allocation for the original systemic risk measure in (1.5). From (4.19) and (4.20), we obtain N n=1 Yn X=ρ(X)=ρQX(X)= N n=1 EQn X[Yn X], which shows that the valuation by EQX[·] agrees with the systemic risk measure ρ(X). This supports the introduction of EQX[·]as a suitable systemic valuation operator. The essential question for a financial institution is now whether its allocated share of the total systemic risk given by the risk allocation (EQ1 X[Y1 X],...,(EQN X[YN X]),is fair. With the choice Q=QX, Corollary 4.3, Lemma 4.5 and (4.11) lead to πA(X)=πQX A(X)=max N n=1an=A N n=1 sup EQn X[Yn]=anE[un(Xn+Yn)].(5.5) Choose A=ρB(X). Then Proposition 4.2 and the fact that YXis then the solution of πQX A(X)yield EQXn[Yn X]=an ∗,N n=1EQXn[Yn X]=A, and (5.5) can be rewritten as πA(X)=πQX A(X)= N n=1 sup EQn X[Yn]=EQn X[Yn X]E[un(Xn+Yn)]. This means that by using QXfor valuation, the system utility maximisation in (1.9) reduces to individual utility maximisation for the banks without the “systemic” constraint Y∈C, i.e., to sup{E[un(Xn+Yn)]:Ynsuch that EQn X[Yn]=EQn X[Yn X]} for all n.
546 F. Biagini et al. The optimal allocation Yn Xand its value EQn X[Yn X]can thus be considered fair by the nth bank as Yn Xmaximises its individual expected utility among all random allocations (not constrained to be in CR) with value EQn X[Yn X]. In particular, it is clear then that for individual banks, it is more advantageous to use random rather than cash-valued allocations as the supremum will be larger, as previously stated in Sect. 5.2 (a) and (b). This finally argues for the fairness of the risk allocation (EQ1 X[Y1 X],...,EQN X[YN X])as fair valuation of the optimal scenario-dependent allocation (Y1 X,...,YN X). 6 The exponential case In this section, we focus on a relevant case under Assumption 2.2, that is, we set C=C(n), see Definition 2.5 and Example 3.5, and choose un(x) =−e−αnx/αn, αn>0, n=1,...,N, as in Example 3.6. Then vn(y) =1 αn(y lny−y) and v n(y) =1 αnlny. We select B<N n=1un(+∞)=0. Under these assumptions, φn(x) :=−un(−|x|)+un(0)=1 αn(eαn|x|−1), Mφn=Mexp :={X∈L0(R):E[ec|X|]<+∞for all c>0}, the Orlicz hearts Mφn,n=1,...,N, coincide with the single Orlicz heart Mexp associated to the exponential Young function x→e|x|−1, and the random variable X:=nXn∈Mexp is well defined. The systemic risk measure ρ:(Mexp)N→R from (2.3) becomes ρ(X)=infN n=1 Yn:Y∈C(n),E− N n=1 1 αn exp−αn(Xn+Yn)=B.(6.1) Recall that each set C(n)is closed in probability and closed by truncation. From Proposition 2.4 and Corollary 4.23, we deduce Proposition 6.1 The map ρin (6.1)is finite-valued,monotone decreasing,convex, continuous and subdifferentiable on the Orlicz heart M=(Mexp)N,and the problem ρ(X)admits the unique solution ˜ YXgiven in Corollary 4.23. For a given partition nand allocations C(n), we can explicitly compute the value ρ(X), the unique optimal allocation of (6.1) and the unique optimiser QXof the corresponding dual problem (3.10). Note that in the present exponential case, the vector ˜ YX=YX∈(Mexp)Nis the solution for ρ(X)and ρ(X). Theorem 6.2 For m=1,...,hand k∈Im,we have dm=βmln−β BEexp−Xm βm,(6.2) Yk m=−Xk+1 βmαk Xm+1 βmαk dm∈Mexp,(6.3)
On fairness of systemic risk measures 547 where Xm=k∈ImXk,βm=k∈Im 1 αk,β=N i=11 αiand ρ(X)= N i=1 Yi= h m=1 dm. The vector QXof probability measures with densities dQm X dP:= e−1 βmXm E[e−1 βmXm],m=1,...,h, (6.4) is the solution of the dual problem (3.10), i.e., ρ(X)= h m=1 EQm X[−Xm]−αB(QX), (6.5) and EQm X[Yn X],m=1,...,h,n∈Im,is a systemic risk allocation as in Definition 1.2. Proof By (3.11), we note that QXdefined in (6.4) belongs to D.UsingQXand selecting λ∗=−B βfrom Example 3.6, it is easy to verify that the random variable Yn X:=−Xn−v n(λ∗dQn X dP)from Corollary 4.23 coincides with the expression in (6.3) and n∈ImYn X=dm. We prove below that h m=1dm=h m=1EQm X[−Xm]−αB(QX). A priori, these equations are not sufficient to prove that (YX,QX)are indeed the solutions to the primal and dual problems, as one needs to know that one of the two is indeed an optimiser of the corresponding problem. The proof that YXdefined in (6.3)isthe optimiser of ρ(X)uses the Lagrange method and several estimates of lengthy computations; it is omitted.3 Assuming that YXis the optimiser of the problem associated to ρ, so that we have ρ(X)=YI=dm, we now prove (6.5). First notice that H(Qm X|P)=EQm Xln dQm X dP=1 βmEQm X[−Xm]−lnE[e−1 βmXm]. By (3.13), αB(QX)can be rewritten as αB(QX)= h m=1 i∈Im1 αi H(Qm X|P)+1 αi ln−B β = h m=1EQm X[−Xm]−βmln−β BE[e−1 βmXm] = h m=1 (EQm X[−Xm]−dm)= h m=1 EQm X[−Xm]−ρ(X), as ρ(X)=N i=1Yi=h m=1dm. Then (3.12) concludes the proof. 3The proof can be obtained upon request from the authors.
548 F. Biagini et al. Remark 6.3 Note that if we arbitrarily change the components of the vector X,but keep fixed the components in one given subgroup, say Im0, then the risk measure ρ(X)will of course change, but dm0and Yk m0for k∈Im0remain the same. 6.1 Sensitivity analysis Let X∈(Mexp)N,V∈(Mexp)Nand set Vm:=k∈ImVkfor m=1,...,h. We consider a perturbation εV,ε∈R, and perform a sensitivity analysis. Consider the optimal allocations Yi X+εVand the solution QX+εVof the dual problem associated to ρ(X+εV);see(6.4). By (6.3) and (6.2), we have Yn X+εV=−Xn−εV n+1 βmαn (Xm+εV m)+1 βmαn dm(X+εV), where dm(X+εV)=βmln−β BEexp−Xm+εV m βm. Proposition 6.4 Let ρbe the systemic risk measure defined in (6.1). Then we have: 1) The marginal risk contribution of group mis d dεdm(X+εV)ε=0=EQm X[−Vm],m=1,...,h. 2) The local causal responsibility is d dεEQm X[Yn X+εV]ε=0=EQm X[−Vn],n∈Im. 3) d dεEQm X+εV[Z]|ε=0=−1 βmCovQm X(V m,Z)for any Z∈Mexp. 4) The marginal risk allocation of institution n∈Imis d dεEQm X+εV[Yn X+εV]ε=0=EQm X[−Vn]− 1 βm CovQm X(V m,Yn X)(6.6) =EQm X[−Vn]+ 1 βm CovQm X(V m,Xn) −1 αn 1 β2 m CovQm X(V m,Xm). (6.7) 5) The sensitivity of the penalty function is d dεαB(QX+εV)ε=0= h m=1 1 βm CovQm X(V m,Xm).
On fairness of systemic risk measures 549 6) The systemic marginal risk contribution is d dερ(X+εV)ε=0= h m=1 i∈Im EQm X[−Vi]= h m=1 EQm X[−Vm]. Proof The proof is the result of lengthy computations and is omitted.4 The interpretation of the above formulas is not simple because we are dealing with the systemic probability measure Qm Xand not with the “physical” measure P. Think of the difference between the physical measure Pand a martingale measure. If we replace Qm Xwith P, none of the results of Proposition 6.4 will hold in general. The first term EQm X[−Vn]in (6.6)or(6.7) is easy to interpret: EQm X[−Vn]is the contribution to the marginal risk allocation of bank nregardless of any systemic influence. The sign of the increment Vnin the first term of (6.6) is here relevant; an increment (positive) corresponds to a risk reduction, regardless of the dependence structure. If Vis deterministic, the marginal risk allocation to bank nis exactly EQm X[−Vn]=−Vnand no other terms are present. To understand the other terms in (6.6)or(6.7), take V=Vjejwith j=n. Then thefirsttermin(6.6) disappears (Vn=0) and we obtain d dεEQm X+εV jej[Yn X+εV jej]ε=0=1 βm CovQm X(V j,Xn)−1 αn 1 β2 m CovQm X(V j,Xm). To fix ideas, suppose that CovQm X(V j,Xn)<0 and examine for the moment only the contribution of 1 βmCovQm X(V j,Xn). This component does not depend on the specific αn, but it depends on the dependence structure between (V j,Xn). If the systemic risk probability Qm Xattributes negative correlation to (V j,Xn), then from the systemic perspective, this is good (independently of the sign of Vj); indeed, a decrement in bank jis balanced by bank n, and vice versa. If bank nis negatively correlated (as seen by Qm X) with the increment of bank j, then the risk allocation of bank n should decrease. Therefore, bank ntakes advantage of this as its risk allocation is reduced ( 1 βmCovQm X(V j,Xn)<0). Since the overall marginal risk allocation of the group mis fixed (and equal to EQm X[−Vm]=EQm X[−Vj]from 1)), someone else has to pay for this advantage to bank n. This is the last term in (6.7), which is discussed next. For the third component in (6.7), we distinguish between the systemic component −1 β2 m CovQm X(V j,Xm), which only depends on the aggregate group Xm, and the systemic relevance 1 αnof bank n. The systemic quantity is therefore distributed among the various banks according to 1 αn. In addition, this term must compensate for the possible risk reduction (the second term in (6.7)) as the overall risk allocation to group mis determined by EQm X[−Vm]=EQm X[−Vj]. Finally, 1) and 6) express the same property (which holds in general, as shown in Proposition 5.4) for one group or for the entire system, respectively. 4The proof can be obtained upon request from the authors.
550 F. Biagini et al. 6.2 Monotonicity Another desirable fairness property is monotonicity.IfC1⊆C2⊆CR, then we have ρ1(X)≥ρ2(X)for the corresponding systemic risk measures ρi(X):=infN n=1 Yn:Y∈Ci,EN n=1 un(Xn+Yn)≥B,i=1,2. The two extreme cases occur for C1:= RN(the deterministic case) and C2:= CR (the unconstrained scenario-dependent case). Hence we know that when going from deterministic to scenario-dependent allocations, the total systemic risk decreases. It is then desirable that each institution profits from this decrease in total systemic risk in the sense that also its individual risk allocation should decrease, i.e., ρn 1(X)≥ρn 2(X)for each n=1,...,N. (6.8) The opposite would clearly be perceived as unfair. In the next result (see in particular (6.11)), we prove that (6.8) holds true in the context of the Definition 2.5 of grouping when the risk allocation ρn(X)=EQn X[Yn X]is computed using QX.Ifwewere to select a vector of probability measures Rdifferent from QXto compute the risk allocation with the formula ERn[Yn X], the property (6.8) would be lost in general. For a given partition nand C=C(n),letYk r,k∈Ir,r=1,...,h, be the corresponding optimal allocations of the primal problem (6.1) and Qr X,r=1,...,h,the solutions of the corresponding dual problem (3.10) (in this section, we suppress the label Xfrom the optimal allocation YXto ρ(X)). Consider for some m∈{1,...,h}a nonempty subgroup I mof the group Imand set I m:=Im\I m. Then the h+1 groups I1,I2,...,Im−1,I m,I m,Im+1,...,Ihcorrespond to a new partition n. The optimal allocations of the primal problem (6.1) with C=C(n)coincide with Yk r,k∈Ir,forr=m.Forr=m,i∈I m,wehavethe following. Proposition 6.5 Denote by (Yi m),i∈I m,the optimal allocation to the primal problem with C=C(n).Then EQm X i∈I m Yi m≤ i∈I m (Yi m):=d m.(6.9) In particular,if the group I mconsists of only one single element {i},then (Y i m)is deterministic and EQm X[Yi m]≤(Y i m)for each i∈Im.(6.10) If we compare the deterministic optimal allocation Y∗(corresponding to C=RN) with the random optimal allocations Yassociated to one single group (i.e., with C=CR∩(Mexp)N), we conclude that EQX[Yn]≤(Y ∗)nfor each n=1,...,d, (6.11) where QXis the unique solution of the dual problem with C=CR∩(Mexp)N.
On fairness of systemic risk measures 551 Proof Given the subgroup I m, define β m:= k∈I m 1 αk. Then the value with respect to C(n)is given by d m=β mln−β BEexp−1 β m k∈I m Xk. Summing the components of the solutions relative to C(n)over k∈I m, we get k∈I m Yk m= k∈I m1 βmαk Xm−Xk+ k∈I m 1 βmαk dm =β m βm Xm− k∈I m Xk+β m βm dm. Using Jensen’s inequality, we obtain EQm X k∈I m Yk m=β mlnexp1 β m EQm Xβ m βm Xm− k∈I m Xk +β m βm βmln−β BEexp−Xm βm ≤β mlnEQm Xexp1 βm Xm−1 β m k∈I m Xk +β mln−β BEexp−Xm βm =β mlnEexp(−Xm βm)exp(1 βmXm)exp(−1 β mk∈I mXk) E[e−1 βmXm] +β mln−β BEexp−Xm βm =β mln−β BEexp−1 β m k∈I m Xk=d m. Then (6.10) and (6.11) follow directly by (6.9). Acknowledgements Open Access funding provided by Projekt DEAL. The third author would like to thank Enea Monzio Compagnoni for very helpful discussions and relevant insights on the whole paper during the preparation of his Laurea thesis, his Laurea student Giacomo Bizzarrini, as well as his Ph.D. student Alessandro Doldi for his careful reading and decisive contribution to Sect. 4.4.1. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
552 F. Biagini et al. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Appendix A A.1 Properties of the utility functions and proof of Proposition 2.4 Lemma A.1 Under Assumption 2.2 and if we have limx→−∞ un(x) x=+∞and limx→+∞ un(x) x=0, then: (a) There exist c∈Rand b∈R+such that (i) un(x) ≤bx +cfor all x≥0and all n, (ii) un(x) ≤2bx +cfor all x≤0and all n. (b) For all ε>0, there exists b=b(ε) > 0such that un(x) ≤εx +bfor x≥0and all n. Proof From 2) in Assumption 2.2, we know that dom(un)=Rfor each n. Hereafter, the left derivatives of the concave increasing functions unare denoted by un;they satisfy un(x) ≥0 for all x∈R. (a) For (i), the concavity of each unimplies that un(x) ≤un(0)x +cnfor all x∈R(for some cn), and setting b:=maxn=1,...,N un(0)≥0 and c≥maxn=1,...,N cn therefore gives un(x) ≤bx +cfor all x≥0. For (ii), we prove that for every M>0, there exists a constant d>0 with un(x) ≤Mx +dfor all nand x≤0. By taking M=2b, we obtain (ii). The assumption limx→−∞ un(x) x=+∞implies that there exists K>0 (which depends on M) such that un(x) ≤Mx for x≤−Kand for all n. Hence Mx −un(x) ≥0 for x∈(−∞,−K). As the function Mx −un(x) is continuous on [−K,0],wemay add a properly chosen d>0 to get Mx +d−un(x) ≥0 for all x∈(−∞,0]and all n. (b) The assumption limx→+∞ un(x) x=0 guarantees the existence of a constant K>0, which depends on ε, such that un(x) ≤εx for x≥Kand all n. Hence un(x) ≤εx +Kε +max n=1,...,N sup [0,K]un(s), ∀x≥0. Proof of Proposition 2.4 To show ρ>−∞, we suppose by way of contradiction that ρ(X)=−∞for some X∈M⊆L1(P;RN).Let(Ym)⊆Csatisfy N n=1Yn m↓−∞ as m→∞and (X+Ym)∈Afor each mfor from (1.3). The first condition implies N n=1E[Yn m]↓−∞as m→∞.Note also that by Jensen’s inequality, B≤E[(X+Ym)]≤(E[X+Ym])= N n=1 un(E[Xn]+E[Yn m]). (A.1)
On fairness of systemic risk measures 559 By applying the classical convex duality theory for real-valued functions (see [41, Sects. 12 and 26]), we get Lemma A.5 The convex conjugate function v:R→(−∞,+∞] of ugiven by v(y) =supx∈R(u(x) −xy) is a proper lower semicontinuous convex function,equal to +∞on (−∞,0),bounded from below on R,finite-valued,strictly convex,continuously differentiable on (0,+∞)and satisfying v(+∞)=+∞,v(0+)=u(+∞), v(0+)=−∞,v (+∞)=+∞, u(x) =(v)−1(−x), u−v(y)=−yv(y) +v(y), ∀y≥0, where the usual rule 0·∞=0is applied. Proposition A.6 (Biagini et al. [11, Proposition 3.6]) Let QP.For all c∈R,the optimiser λ(c;Q) of min λ>0EvλdQ dP+λc is the unique positive solution of the first order condition EQvλdQ dP+c=0. If sup{E[u(g)]:g∈L1(Q) and EQ[g]≤c}<u(+∞),then the random variable g:=−v(λ(c;Q)dQ dP)belongs to the set {g∈L1(Q) :EQ[g]=c}and satisfies u(g) ∈L1(P)and min λ>0EvλdQ dP+λc=sup{E[u(g)]:g∈L1(Q) and EQ[g]≤c} =E[u(g)]<u(+∞). A.4 Proofs for Sect. 4.2 Proof of Proposition 4.4 From Mφn⊆L1(P,Qn)⊆L1(Qn), we clearly have Un(an)≤ Un(an)≤ Un(an)≤un(+∞)so that if Un(an)=u(+∞), then Un(an)= Un(an)= Un(an)=un(+∞). (A.10) By the Fenchel inequality, we get E[un(Xn+W)]≤λ(EQn[Xn]+EQn[W])+EvnλdQn dP and hence Un(an)≤ Un(an)≤ Un(an) ≤inf λ>0λ(EQn[Xn]+an)+EvnλdQn dP <+∞ (A.11) as E[vn(λdQn dP )]<+∞. Therefore (4.4) is a consequence of (A.10) and (4.6).
560 F. Biagini et al. To show (4.6), we consider the integral functional I:Mφn→Rdefined by I(Xn)=E[un(Xn)]. It is finite-valued, monotone increasing and concave on Mφn (as E[un(Xn)]≤un(E[Xn])<+∞), and therefore by Theorem A.2, it is normcontinuous on Mφn. We can then follow the well-known duality approach (see for example [11]), as follows. Consider the convex cone D0:= {W∈Mφn:EQn[W]≤0}which is the polar cone of the one-dimensional cone D:={λdQn dP :λ≥0}, so that the bipolar D00 coincides with D.LetδD0:Mφn→R∪{+∞}be the support functional of D0. By Kozek [38], or directly by hand, the concave conjugate I∗:Lφ∗ n→R∪{−∞} is given by I∗(ξn)=E[−vn(ξn)], and so by the Fenchel duality theorem, Un(an)=sup W∈D0E[un(Xn+an+W)]= sup Z∈D0+Xn+anE[un(Z)] =sup Z∈MφnE[un(Z)]−δD0+Xn+an(Z) =min ξn∈Lφ∗ nδ∗ D0+Xn+an(ξn)−E[−vn(ξn)] =min ξn∈Lφ∗ nE[ξn(Xn+an)]+δD00 (ξn)+E[vn(ξn)] =min ξn∈D00 E[ξn(Xn+an)]+E[vn(ξn)] =min λ>0λ(EQn[Xn]+an)+EvnλdQn dP , where we used δ∗ D0=δD00 ,D00 =Dand the fact that the minimum is obtained at λ>0. The last fact follows because if λ=0, then Un(an)=E[vn(0)]=un(+∞),in contradiction to the assumption. We complete the proof by showing (4.5). From the inequality (A.11), it is clear that Un(−∞)=−∞. Define Vn(λ) :=EvnλdQn dP+λEQn[Xn]. When Un(an)<u n(+∞),wehaveUn(an)=infλ>0(Vn(λ)+λan)from (4.6), which shows that Unand Vnare conjugate to each other, i.e., we have Vn(λ) =sup an>0Un(an)−λan. From Lemmas A.4 and A.5, we know that the convex function Vnis differentiable on (0,+∞)and so Unis differentiable on (−∞,+∞)and U n(a) =(V n)−1(−a) > 0. We only need to show that U n(+∞)=0 and U n(−∞)=+∞.Wehave Vn(0+)=+∞because vn(0+)=un(+∞)=+∞. Since v n(0+)=−∞, we get V n(0+)=−∞and U n(+∞)=0. Moreover, by Jensen’s inequality, V n(+∞)=lim λ→+∞ Vn(λ) λ=lim λ→+∞ 1 λEvnλdQn dP+EQn[Xn] ≥lim λ→+∞ 1 λvn(λ)+EQn[Xn]=v n(+∞)+EQn[Xn]=+∞, which implies U n(−∞)=+∞.
On fairness of systemic risk measures 561 Proof of Lemma 4.7 The set Kis clearly closed. We show that it is bounded. For N=1, this is true. Let N>1. First we prove that for all j=1,...,N, Uj(a)1+n=jUn(A −(N −1)a) Uj(a) −→ −∞ as a↓−∞.(A.12) Recall that Un(−∞)=−∞and Un(+∞)≤un(+∞)for all n. Suppose that for some k∈{1,...,N},wehaveuk(+∞)<+∞. Then Uk(+∞)<+∞ and for all j=1,...,N, lim a→−∞ Uk(A −(N −1)a) Uj(a) =0.(A.13) Now suppose that for some k∈{1,...,N},wehaveuk(+∞)=+∞. Then Proposition 4.4 shows that Uk(ak)<+∞ = uk(+∞),U k>0, U k(−∞)=+∞and U k(+∞)=0. By l’Hôpital’s rule, we obtain again for all j=1,...,N that lim a→−∞ Uk(A −(N −1)a) Uj(a) =lim a→−∞ −(N −1)U k(A −(N −1)a) U j(a) =0.(A.14) From (A.13) and (A.14), we deduce that (A.12) holds true. We conclude that for any constant B, there exists a constant Rsuch that for all j=1,...,N and a<R,wehave Uj(a)1+n=jUn(A −(N −1)a) Uj(a) <B. Let a∈Kand take iwith ai=min{a1,...,aN}. Note that for all j=1,...,N,we have aj≤A−(N −1)aibecause N n=1an≤A. Assume that ai<R. Then B≤ N n=1 Un(an)≤Ui(ai)1+n=iUn(A −(N −1)ai) Ui(ai) which is a contradiction. Therefore aj≥Rfor all j=1,...,N, and then also aj≤A−(N −1)R for all j=1,...,N because N n=1an≤A. This proves the claim. Let X∈Mand consider the function F(δ):=E[N n=1un(Xn+Yn−δ)]with δ∈R.IfY∈M, then Fis finite-valued and concave on R, hence continuous on R (see the discussion at the beginning of Sect. 4.2). However, when Y∈L1(Q)satisfies E[N n=1un(Xn+Yn)]>B (with the understanding that un(Xn+Yn)∈L1(P)for each n), it is not any more evident if Fis continuous on Ras one has to guarantee that E[N n=1un(Xn+Yn−δ)]>−∞for δ>0. Lemma A.7 If X∈Mand Z∈L1(Q)satisfy E[N n=1un(Xn+Zn)]>B, then there exists Z∈L1(Q)which satisfies N n=1EQn[ Zn]<N n=1EQn[Zn]and E[N n=1un(Xn+ Zn)]=B.
562 F. Biagini et al. Proof Set An:={Xn+Zn>k n}and let kn∈Rsatisfy P[An]>0 and Qn[An]>0. For any δ>0, consider the random variable Z∈L1(Q)with Zn:=Zn−δ1Anand define G(δ) :=E[N n=1un(Xn+Zn−δ1An)]. Then G(δ) =EN n=1 un(Xn+Zn)1Ac n+EN n=1 un(Xn+Zn−δ)1An ≥EN n=1 un(Xn+Zn)1Ac n+EN n=1 un(kn−δ)1An>−∞, which implies that Gis continuous on R+and the result follows. Proof of Lemma 4.8 From (3.5) and ρQ B(X)=ρQ B(X), note that the penalty function can also be written as αB(Q)=− N n=1 EQn[Xn]−ρQ B(X)=− N n=1 EQn[Xn]−ρQ B(X) =supN n=1 EQn[−Zn]:Z∈L1(P,Q),E[(Z)]≥B, for from (1.3). Set c=(Q):=infN n=1 EQn[Zn]:Z∈L1(P,Q),E[(Z)]=B. Similarly to the proof of (A.3), we show that Z∈L1(P,Q)and E[(Z)]>B =⇒ N n=1 EQn[Zn]>c =(Q). (A.15) Indeed, Lemma A.7 implies the existence of Z∈L1(P,Q)satisfying E[( Z)]=B and N n=1EQn[ Zn]<N n=1EQn[Zn], and therefore we have c=(Q)≤ N n=1 EQn[ Zn]< N n=1 EQn[Zn]. It follows that c(Q):=−αB(Q)=infN n=1 EQn[Zn]:Z∈L1(P,Q),E[(Z)]≥B=c=(Q). Indeed, −∞<c(Q)≤c=(Q); so assume c(Q)<c =(Q). By the definition of c(Q), there exist ε>0 and Z∈L1(P,Q)with N n=1EQn[Zn]≤c(Q)+ε<c =(Q)and E[(Z)]>B, which contradicts (A.15).
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