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1

RISK SHARING UNDER HETEROGENEOUS BELIEFS WITHOUT

CONVEXITY

FELIX-BENEDIKT LIEBRICH

Institute of Actuarial and Financial Mathematics & House of Insurance,

Leibniz Universitat Hannover, Germany

Abstract. We consider the problem of finding Pareto-optimal allocations of risk among finitely

many agents. The associated individual risk measures are law invariant, but with respect to

agent-dependent and potentially heterogeneous reference probability measures. Moreover, indi-

vidual risk assessments are assumed to be consistent with the respective second-order stochastic

dominance relations. We do not assume their convexity though. A simple sufficient condition

for the existence of Pareto optima is provided. Its proof combines local comonotone improve-

ment with a Dieudonne-type argument, which also establishes a link of the optimal allocation

problem to the realm of “collapse to the mean” results. Finally, we extend the results to capital

requirements with multidimensional security markets.

1. Introduction

This paper addresses the problem of finding optimal allocations of risk among finitely many

agents, i.e., the optimisation problem∑

i∈I ρi(Xi) −→ min

subject to∑

i∈I Xi = X.(1.1)

The agents under consideration form a finite set I. Each agent i ∈ I measures the risk of net

losses Y in a space X with a risk measure ρi. Given the total loss X collected in the system, an

allocation attributes a portion Xi ∈ X to each agent, i.e., the condition∑

i∈I Xi = X holds. We

do not impose any restriction on the notion of an allocation, i.e., every vector whose coordinates

sum up to X is hypothetically feasible. An allocation of X is optimal (or Pareto-optimal) if it

minimises the aggregated risk∑

i∈I ρi(Xi) in the system.

This risk-sharing problem is of fundamental importance for the theory of risk measures, for capital

allocations in capital adequacy, and in the design and discussion of regulatory frameworks; cf.

Tsanakas [42] and Weber [44].

E-mail address: felix.liebrich@insurance.uni-hannover.de.

Date: August 31, 2021.

Acknowledgements: I would like to thank Cosimo Munari, Gregor Svindland, Ruodu Wang, and the participants

of the Seminar on Risk Management and Actuarial Science at the University of Waterloo for invaluable comments

and discussions related to this work.

1

2 F.-B. LIEBRICH

However, the space of net losses appearing in (1.1) usually is an infinite-dimensional space of

random variables. This complicates finding optima. Beside convexity of the individual monetary

risk measures, an assumption which has enabled a powerful solution theory is law invariance: The

risk of a random variable X on a probability space (Ω,F ,P) merely depends on its distribution

under the reference measure P. A rich strand of literature explores the wide-ranging analytic

consequences of law invariance in conjunction with convexity; see, e.g., Bellini et al. [9], Chen et

al. [14], Filipovic & Svindland [22], Jouini et al. [26], Leung & Tantrawan [29], Svindland [40], and

the references therein. Studies of the risk sharing problem for convex monetary risk measures (or

equivalently, concave monetary utility functions) are Acciaio [1], Barrieu & El Karoui [7], Filipovic

& Svindland [21], and Jouini et al. [27]. For risk sharing problems with (special) law-invariant,

but not necessarily convex functionals, see Embrechts et al. [18], Liebrich & Svindland [32], and

Liu et al. [35].

The solution theory in the law-invariant case can be split in two steps. First, the set of

optimisation-relevant allocations is reduced to comonotone allocations. This is achieved by

comonotone improvement results based on Landsberger & Meilijson [28]; see also Carlier et

al. [12], Filipovic & Svindland [21], and Ludkovski & Ruschendorf [36].1 In a second step, one

finds suitable bounds which prove that relevant comonotone allocations form a compact set. That

enables approximation procedures and the selection of converging optimising sequences.2

The previous arguments tend to rely fundamentally on the existence of a single “objective” ref-

erence probability measure which is shared and accepted by all agents in I. A more recent and

quickly growing strand of literature in finance, insurance, and economics, however, dispenses with

this paradigm. Two economic considerations motivate these heterogeneous reference probability

measures. Firstly, different agents may have access to different sources of information, resulting

in information asymmetry and in different subjective beliefs. Secondly, agents may entertain het-

erogeneous probabilistic subjective beliefs as a result of their preferences or their use of different

internal models.3,4

Applying heterogeneous probabilistic beliefs in risk sharing and related problems, Amarante

et al. [4] explore the demand for insurance when the insurer exhibits ambiguity, whereas the

insured is an expected-utility agent. Under heterogeneous reference probabilities for insurer

and insured, Boonen [10] provides optimal reinsurance designs, and Ghossoub [25] generalises

Arrow’s “Theorem of the Deductible”. Boonen & Ghossoub [11] study bilateral risk sharing

1 Another desirable property of comonotone allocations is that they rule out “ex post moral hazard issues thatcould arise from the possibility that the insurer could misreport the actual amount of loss suffered”, as Amaranteet al. [4] point out.2 These two steps are separated explicitly in, e.g., Liebrich & Svindland [32], where also numerous economicoptimisation problems are analysed according to that scheme. Under cash-additivity of the involved functionals,the second step usually poses no problem as one can “rebalance cash” among the involved agents. For the details,see for instance [21].3 Ghossoub [25] suggests that the prevalence of an “objective probability” is a “heritage of the von Neumann andMorgenstern approach to ... uncertainty”.4 While heterogeneous subjective beliefs and information asymmetry may not be synonymous, the distinction willnot play a role for the purposes of the present paper.

RISK SHARING UNDER HETEROGENEOUS BELIEFS WITHOUT CONVEXITY 3

with exposure constraints and admit a very general relation between the two involved reference

probabilities. Asimit et al. [5, Section 6] consider optimal risk sharing under heterogeneous

reference probabilities affected by exogenous triggering events, while Dana & Le Van [16] study

the existence of Pareto optima and equilibria in a finite-dimensional setting when optimality of

allocations is assessed relative to individual and potentially heterogeneous sets of probabilistic

priors.

In the realm of monetary risk measures, Embrechts et al. [19] study risk sharing for Value-at-Risk

and Expected Shortfall agents endowed with heterogeneous reference probability measures, and

provide explicit formulae for the shape of optimal allocations. Liu [34] makes similar contributions

to weighted risk sharing with distortion risk measures under heterogeneous beliefs. Up to a sign

change, Acciaio [2] considers two convex monetary risk measures which are law invariant with

respect to equivalent probability measures. It is then shown that their inf-convolution (cf. (3.2))

has the Fatou property. In contrast to the present paper, the existence of optimal allocations—or

exactness of the inf-convolution—does not play a role. Finally, up to the aforementioned sign

change, Acciaio & Svindland [3] study the existence of optimal allocations among two agents

with heterogeneous reference models, each measuring risk with a law-invariant convex monetary

risk measure. The risk sharing problem is constrained to random variables over a finite σ-algebra

though, which reduces the optimisation problem to a finite-dimensional space. Moreover, one

of the two reference probabilities involved only takes rational values on the finite σ-algebra in

question.

Against the outlined backdrop of the existing literature, several key features of our results stand

out.

Infinite-dimensional setting: We do not deal with heterogeneous reference probabilities by restric-

tion to a finite-dimensional setting. Instead, we consider problem (1.1) for the space X = L∞ of

all bounded random variables over an atomless probability space (Ω,F ,P), a bona fide infinite-

dimensional space.5

True generalisation of most known results: As already stated, we consider a finite set I of agents,

each measuring risk with a functional ρi : L∞ → R which is law invariant with respect to a

probability Qi ≈ P. The probability measure P only plays the role of a gauge and we are free to

assume P = Qj for some j ∈ I. Our main results in Section 3 provide mild sufficient conditions for

the existence of optimal risk allocations without a rationality condition on involved probabilities

as in [3]. Mathematically, we pursue the procedure of comonotone improvement and adapt it

to the case of heterogeneous reference probabilities. However, “rebalancing cash” as in, e.g.,

[21] is not an option in this heterogeneous case anymore. Instead, the second step of the two-

step procedure explained above will be crucial. While optimal allocations will usually not be

5 We believe that the results can be suitably generalised to rearrangement-invariant function spaces L∞⊂ X ⊂ L1

using the well-developed tool box of extension methods for law-invariant functionals; see [14, 22, 32, 38, 40].However, this goes beyond the scope of the present note and is left for future research.

4 F.-B. LIEBRICH

comonotone in the heterogeneous case, our procedure nevertheless presents a unified theory for

the cases of a single and multiple reference probability measures alike.

Nonconvex risk measures: Thirdly, we drop (quasi)convexity/(quasi)concavity of the involved

functionals as a necessary requirement. Instead, we work with three axiomatic properties of risk

measures introduced and studied in the very recent literature as alternatives to (quasi)convexity.

(a) The main ingredient is the assumption that individual risk measures are consistent, and we

shall impose this requirement throughout our study. Consistent risk measures have recently

been axiomatised by Mao & Wang [37].6 While many desirable characterisations are shown to

be equivalent, their eponymous characteristic feature is monotonicity with respect to second-

order stochastic dominance. Each law-invariant convex monetary risk measure is a consistent

risk measure (up to an affine transformation), but the converse implication does not hold.

(b) The second ingredient—which is mostly of technical relevance and makes the assumptions in

the main results particularly satisfiable—is star-shapedness. Star-shaped risk measures have

been studied systematically in the recent working paper Castagnoli et al. [13]. They are for

instance motivated by the fact that empirical support for the general feasibility of mergers—

which underpins the translation of the economic diversification paradigm as (quasi)convexity

of risk measures—is questionable. Instead, (quasi)convexity is replaced by the demand that

decreasing exposition to an acceptable loss profile does not lead to a loss of acceptability:

∀X ∈ X ∀λ ∈ [0, 1] : ρ(X) ≤ 0 =⇒ ρ(λX) ≤ 0.

Further economic motivation is discussed in [13, Section 2].

(c) Third, we shall require a certain compatibility of probabilistic beliefs and assume the exis-

tence of a finite measurable partition π ⊂ F such that the agents agree on the associated

conditional distributions; see Lemma 2.2. In Section 2.2, we explain that the individual risk

measures therefore fall in a particularly relevant class of scenario-based risk measures re-

cently introduced in Wang & Ziegel [43]. Another desirable feature of this perspective is that

scenario-basedness is preserved under the infimal convolution operation (cf. Corollary 3.7),

while the individual law-invariance is lost in the general heterogeneous case.

Dispensing with law invariance: Last, we will also drop law invariance of the involved functionals ex

post. In Theorem 3.10, we consider capital requirements computed on the basis of the acceptance

set of a consistent risk measure and a multidimensional frictionless security market. The former

poses a capital adequacy test that the agent attempts to pass with hedging instruments from

the security market. These capital requirements follow the spirit of Frittelli & Scandolo [24] and

are studied in, e.g., Baes et al. [8], Farkas et al. [20], and Liebrich & Svindland [31]. They are

typically neither law-invariant nor cash-additive functions. Our approach nevertheless yields a

simple sufficient condition for the existence of Pareto-optimal allocations in this case.

6 [37, Section 4] already provides a preliminary study of risk sharing with consistent risk measures under a singlereference probability measure.

RISK SHARING UNDER HETEROGENEOUS BELIEFS WITHOUT CONVEXITY 5

The paper unfolds as follows. Section 2 collects preliminaries, introduces consistent risk measures

and characterises their admissibility for our main results. The latter are collected in Section 3,

where we also discuss the necessity of the involved assumptions. Theorem 3.1 covers the case

of agents endowed with consistent risk measures, which are cash-additive and law invariant by

definition. Theorem 3.10, goes one step further and addresses capital requirements with mul-

tidimensional security markets and acceptance sets induced by consistent risk measures, thus

dispensing with cash-additivity and law invariance. All proofs and auxiliary results can be found

in Appendices A–E.

2. Preliminaries

2.1. Basic terminology. We first outline terminology, notation, and conventions adopted through-

out the paper:

• The effective domain of a function f : S → [−∞,∞] defined on a nonempty set S is

dom(f) := s ∈ S | f(s) ∈ R.

• For an arbitrary natural number K ∈ N, we denote the set 1, . . . ,K by [K].

• Bold-faced symbols denote vectors of objects.

• (Ω,F ,P) is an atomless probability space, i.e., it admits a random variable whose cumu-

lative distribution function under P is continuous.

• L0 denotes the space of equivalence classes up to P-almost-sure (P-a.s.) equality of real-

valued random variables. Similarly, (L∞, ‖ ·‖∞) and (L1, ‖ ·‖1) denote the Banach spaces

of equivalence classes of bounded and P-integrable random variables in L0, respectively.

All these spaces are canonically equipped with the P-a.s. order ≤, and all appearing

(in)equalities between random variables are to be understood in this sense. We denote

the respective positive cones by L∞+ and L1

+.

• If we consider the spaces L∞ and L1 with respect to a probability measure Q 6= P on

(Ω,F), we shall write L∞Q and L1

Q.

• Expectations and conditional expectations, given a sub-σ-algebra G ⊂ F , computed with

respect to P are denoted by E[·] and E[·|G]. If instead a probability measure Q 6= P is

used, we employ the notation EQ[·] and EQ[·|G], respectively.

• For a probability measure Q on (Ω,F) and an event B ∈ F with Q(B) > 0, we define the

conditional probability measure

QB :F → [0, 1],

A 7→ Q(A∩B)Q(B) .

• We denote the absolute continuity relation between two probability measures P and Q on

(Ω,F) by Q ≪ P, and equivalence of probability measures by Q ≈ P.

• If for two elements X,Y ∈ L0 and a probability measure Q ≪ P the distributions QY −1

of Y under Q agrees with the distribution Q X−1 of X under Q, we shall write Xd=Q Y .

6 F.-B. LIEBRICH

• A subset A ⊂ L0 is law invariant with respect to a probability measure Q ≪ P (or Q-law

invariant) if, for all X ∈ A,

Y ∈ L0 | Xd=Q Y ⊂ A.

Given nonempty sets A ⊂ L0 and S, a function f : A → S is law invariant with respect to

a probability measure Q ≪ P if f(X) = f(Y ) holds for all X,Y ∈ A satisfying Xd=Q Y .

• Π denotes the set of finite measurable partitions π ⊂ F of Ω with minB∈π P(B) > 0.7

• Denoting the space of all bounded linear functionals on L∞ by (L∞)∗, the convex conjugate

of a (not necessarily convex) function f : L∞ → [−∞,∞] is the function

f∗ :(L∞)∗ → [−∞,∞],

φ 7→ supφ(X) − f(X) | X ∈ L∞.

• A monetary risk measure ρ : L∞ → R is a map that is:

(a) monotone, i.e., for X,Y ∈ L∞ with X ≤ Y , ρ(X) ≤ ρ(Y ) holds.

(b) cash-additive, i.e.,

∀X ∈ L∞ ∀m ∈ R : ρ(X +m) = ρ(X) +m.

Its acceptance set Aρ collects all loss profiles that bear neutral risk:

Aρ := X ∈ X | ρ(X) ≤ 0.

All proofs accompanying results in Section 2 are collected in Appendix C.

2.2. Heterogeneous reference measures. As in most of the main results, we consider in this

subsection a vector (Q1, . . . ,Qn) of n ≥ 2 probability measures on (Ω,F) which are all absolutely

continuous with respect to P, i.e., P acts as a dominating measure for Q1, . . . ,Qn. The following

assumption will be crucial for our results.

Assumption 2.1. Each density dQi

dP , i ∈ [n], has a version which is a simple function.

There are alternative formulations of this assumption.

Lemma 2.2. For a vector (Q1, . . . ,Qn) of nonatomic probability measures on (Ω,F), the following

assertions are equivalent:

(1) There is a dominating probability measure P such that (Q1, . . . ,Qn) satisfies Assump-

tion 2.1.

(2) There is a dominating probability measure P and a measurable partition π ∈ Π of Ω such

that each density dQi

dP has a σ(π)-measurable version.

(3) There is a finite measurable partition π of Ω such that, for all B ∈ π, all i, k ∈ [n] with

the property Qi(B),Qk(B) > 0, and all X,Y ∈ L∞,

Xd=QB

iY ⇐⇒ X

d=QB

kY. (2.1)

7 While this set technically depends on the choice of P, we suppress this dependence in the notation.

RISK SHARING UNDER HETEROGENEOUS BELIEFS WITHOUT CONVEXITY 7

It has already been anticipated in the introduction that Assumption 2.1 embeds nicely in the

extant literature. We shall explain this fact in the following.

(1) By Lemma 2.2, the role of Assumption 2.1 is similar to the “exogenous environments” studied

in Asimit et al. [5]. Each event B ∈ π in the finite measurable partition π can be understood

as the occurrence of an exogenous shock, and agents disagree about the likelihood of those

shocks, but not about their respective relevance or conditional distributional implications.

(2) Suppose a vector of atomless probability measures (Q1, . . . ,Qn) on (Ω,F), a dominating

measure P, and a measurable partition π ∈ Π are as described in Lemma 2.2(2). Consider

P := PB | B ∈ π,

a finite set of mutually singular probability measures. If two random variables X,Y ∈ L∞

satisfy Xd=PB Y for all B ∈ π, one infers on the basis of equation (C.1) in the proof of

Lemma 2.2 that

∀ i ∈ [n] : Xd=Qi

Y.

Hence, if for each i ∈ [n] a Qi-law-invariant risk measure is given (as in, e.g., Theorem 3.1),

the family (ρ1, . . . , ρn) consists of P-based risk measures, a notion recently introduced in

[43]. Whenever two random variables X,Y ∈ L∞ satisfy Xd=

PY for all P ∈ P, then

ρi(X) = ρi(Y ), i ∈ [n]. Recall that the measures in P above are mutually singular. This is a

case of particular importance for [43], cf. [43, Section 3].

(3) Assume now, like in the main results in Section 3 below, that Q1, . . . ,Qn are all equivalent to

P. Let π ∈ Π be as constructed in Lemma 2.2(3). Up to the fact that we work on an infinite

state space, the assertions show that all Q1, . . . ,Qn are π-concordant to P, a property which

is crucial for the results of [39], cf. [39, Definition 1]. As the authors remark, the notion of

Π-concordancy defines an equivalence relation on the set of all probability measures µ on

(Ω,F) with minB∈π µ(B) > 0.

2.3. Consistent risk measures, admissibility, and compatibility. Amonetary risk measure

ρ : L∞ → R is normalised if ρ(0) = 0. As usual, we can express a monetary risk measure in terms

of its acceptance set as

ρ(X) = infm ∈ R | X −m ∈ Aρ, X ∈ L∞.

By monotonicity and cash-additivity, ρ is a norm-continuous function and the acceptance set Aρ

is closed.

This paper proves the existence of optimal risk sharing schemes for consistent risk measures in-

troduced in [37]. Consistency means that the risk assessment respects the second-order stochastic

dominance relation between arguments. Given random variables X,Y ∈ L∞ and a probability

measure Q ≪ P, recall that Y dominates X in Q-second-order stochastic dominance relation if

EQ[v(X)] ≤ EQ[v(Y )] holds for all convex and nondecreasing functions v : R → R.8

8 In view of the references we cite, we shall mostly work with the convex order defined in Appendix D. In fact, ifY dominates X with respect to the Q-convex order, Y also second-order stochastically dominates X under Q.

8 F.-B. LIEBRICH

Definition 2.3 (cf. [37]). Let Q ≪ P be a probability measure. A monetary risk measure

ρ : L∞ → R is a Q-consistent risk measure if:

(a) ρ is normalised.

(b) If Y ∈ L∞ dominates X ∈ L∞ in second-order stochastic dominance relation under Q, then

also

ρ(X) ≤ ρ(Y ).

Each normalised, convex, Q-law-invariant monetary risk measure is a Q-consistent risk measure,

but also infima of such risk measures (cf. [37, Proposition 3.2]). By [37, Theorem 3.3], for each

consistent risk measure ρ there even is a set T of Q-law-invariant and convex monetary risk

measures τ such that

ρ(X) = infτ∈T

τ(X), X ∈ L∞. (2.2)

A direct consequence of (2.2) is the following formula for the convex conjugate ρ∗:

∀φ ∈ (L∞)∗ : ρ∗(φ) = supY ∈Aρ

φ(Y ) = supτ∈T

τ∗(φ). (2.3)

Every Q-consistent risk measure satisfies

ρ(X) ≥ ρ(EQ[X]) = EQ[X] = E[dQdPX], X ∈ L∞,

which means that dQdP ∈ dom(ρ∗).

Given a monetary risk measure ρ, we recall that the asymptotic cone of the acceptance set Aρ is

A∞ρ = U ∈ L∞ | ∃ (sk)k∈N ⊂ (0,∞), (Yk)k∈N ⊂ Aρ : lim

k→∞sk = 0 & lim

k→∞skYk = U. (2.4)

More information on asymptotic cones in a finite-dimensional setting can be found in [6, Chapter

2].

Definition 2.4. A Q-consistent risk measure ρ : L∞ → R is admissible if there is Z∗ ∈ L1 with

the following two properties:

(a) ρ∗(Z∗) < ∞.

(b) If U ∈ A∞ρ satisfies E[Z∗U ] = 0, then U = 0.

C(ρ) denotes the set of all compatible Z∗ ∈ L1, i.e., Z∗ has properties (a)–(b) above.

In order to better appreciate condition (b) in Definition 2.4, note that by Lemma A.2(1),

∀U ∈ A∞ρ ∀φ ∈ dom(ρ∗) : φ(U) ≤ 0.

The notions of admissibility and compatibility are of central importance for the second step in the

quest for optimal allocations outlined in the introduction: extracting a convergent optimising se-

quence. They thus appear prominently in Theorems 3.1 and 3.10 below. However, a Q-consistent

risk measure can only be admissible if Q ≈ P.

The following proposition compactly characterises admissible consistent risk measures.

Proposition 2.5. Let Q ≈ P be a probability measure and ρ : L∞ → R a Q-consistent risk

measure.

RISK SHARING UNDER HETEROGENEOUS BELIEFS WITHOUT CONVEXITY 9

(1) For every Z ∈ dom(ρ∗) ∩ L1, every Q ∈ C(ρ), and every λ ∈ (0, 1],

λQ+ (1− λ)Z ∈ C(ρ). (2.5)

(2) The following assertions are equivalent:

(a) ρ is admissible.

(b) dQdP ∈ C(ρ).

(c) For all Z ∈ dom(ρ∗) ∩ L1 and λ ∈ (0, 1],

λdQdP + (1 − λ)Z ∈ C(ρ).

(d) ρ 6= EQ[·].

Remark 2.6. The equivalence between items (a) and (d) in Proposition 2.5(2) establishes a

link to the realm of “collapse to the mean” results. The latter subsumes incompatibility results

between law invariance of a functional and the existence of “directions of linearity” (as are, for

instance, collected in the asymptotic cone of a consistent risk measure). In the theory of risk

measures, these go back at least to [23]. We refer to the more detailed discussion in [30].

The next proposition further illustrates which elements in dom(ρ∗) are compatible.

Proposition 2.7. Suppose Q ≈ P is a probability measure and that a Q-consistent risk measure

ρ : L∞ → R is admissible. Then the set

I(ρ) := Q ∈ dom(ρ∗) ∩ L1 | ∀Z ∈ dom(ρ∗) ∩ L1 ∃ ε > 0 : Q+ ε(Q− Z) ∈ dom(ρ∗)

satisfies

I(ρ) ⊂ C(ρ).

However, it is possible that an admissible consistent risk measure only possesses a single compat-

ible element without reducing to an expectation.

Example 2.8. Define

τ1(X) := E[X] + 1,

τ2(X) := ess sup(X) = infm ∈ R | P(X ≤ m) = 1,

ρ(X) := minτ1(X), τ2(X),

X ∈ L∞.

ρ is a consistent risk measure. In view of (2.3) and the affinity of τ1, dom(ρ∗) = 1 holds, which

means that C(ρ) ⊂ 1. However,

ρ(1A) = minP(A) + 1, 1 = 1 6= P(A) = E[1A],

for all events A ∈ F with 0 < P(A) < 1, which means that ρ is admissible by Proposition 2.5(2)

and that C(ρ) = 1.

Richness of compatible elements can be guaranteed under the additional demand that a consistent

risk measures is star shaped. For such risk measures, the assumptions of Theorems 3.1 and 3.10

are particularly mild.

10 F.-B. LIEBRICH

Definition 2.9 (cf. [13]). A monetary risk measure ρ : L∞ → R is star shaped if

(a) ρ is normalised.

(b) The acceptance set Aρ is a star-shaped set, i.e.,

∀Y ∈ Aρ ∀ s ∈ [0, 1] : sY ∈ Aρ.

Star-shaped risk measures have recently been studied in detail in [13], and in our definition we

implicitly invoke the characterisation of star-shapedness provided in [13, Proposition 2]. Each

normalised convex monetary risk measure is star shaped, which means that, like consistency, star-

shapedness is a weaker property than convexity. We also remark that the class of star-shaped

consistent risk measures is discussed in [13, Theorem 11].

If ρ is a star-shaped risk measure, we have the following substantially simpler characterisation of

the asymptotic cone in comparison to (2.4):

A∞ρ = U ∈ L∞ | ∀ t ≥ 0 : tU ∈ Aρ.

Proposition 2.10. Let Q ≈ P be a probability measure and ρ : L∞ → R a Q-consistent, star-

shaped, and admissible risk measure. Then

dQdP ( C(ρ) (2.6)

and C(ρ) is uncountable.

Remark 2.11. Note that Example 2.8 shows that the preceding Proposition 2.10 fails without

the assumption of star-shapedness.

Example 2.12.

(1) C(ρ) ( dom(ρ∗)∩L1 is well possible. As an example, fix a parameter p ∈ (0, 1) and consider

the Expected Shortfall

ρ(X) := ESp(X) = supQ∈Q

E[QX], X ∈ L∞,

where

Q = Q ∈ L∞+ | Q ≤ 1

1−p , E[Q] = 1.

ρ is is convex, normalised, and P-consistent. Recalling that (Ω,F ,P) is atomless, we may fix

an event A ∈ F with P(A) = 1− p. Let Q∗ := 11−p1A and U := −1Ac . For all t ≥ 0,

ρ(tU) = t supQ∈Q

−E[Q1Ac] ≤ 0.

Moreover, for Q ∈ Q, E[QU ] = 0 is equivalent to Q1Ac = 0 P-a.s., whence we conclude that

Q = Q∗. Hence, U ∈ A∞ρ and E[Q∗U ] = 0, but U 6= 0. We obtain Q∗ /∈ C(ρ).

The same example shows that the set I(ρ) in Proposition 2.7 can agree with all convex

combinations of shape λdQdP + (1 − λ)Z, 0 < λ ≤ 1, Z ∈ dom(ρ∗), which are verified to

be compatible in Proposition 2.5. A priori, the proof of Proposition 2.7 shows that each

RISK SHARING UNDER HETEROGENEOUS BELIEFS WITHOUT CONVEXITY 11

element of I(ρ) has the aforementioned shape. For the converse inclusion, let Z ∈ dom(ρ∗)

and 0 < λ ≤ 1 be arbitrary, and abbreviate Q∗ := λ + (1 − λ)Z.9 Choose ε > 0 such that

both

ε(s − λ) ≤ λ (2.7)

and

ε(λ+ (1− λ)s) ≤ λ(s− 1). (2.8)

It is known that dom(ρ∗) = Q. Hence, (2.7) implies for all Q ∈ dom(ρ∗) ∩ L1 that

Q∗ + ε(Q∗ −Q) ≥ λ+ ε(λ+ (1− λ)Z −Q) ≥ λ+ ε(λ− s) ≥ 0.

and by (2.8),

Q∗ + ε(Q∗ −Q) ≤ (1 + ε)(λ+ (1− λ)s) ≤ λ+ (1− λ)s+ λ(s− 1) = s.

Both inequalities together yield Q∗ + ε(Q∗ −Q) ∈ dom(ρ∗).

(2) Consider the entropic risk measure

ρ(X) := log(E[eX ]

)= sup

Q∈L1+:E[Q]=1

E[QY ]− E [Q log(Q)] , X ∈ L∞.

ρ is a normalised, convex, and P-consistent risk measure which satisfies A∞ρ = −L∞

+ . In

particular, each Q ∈ L1+ which satisfies E[Q] = 1 and P(Q > 0) = 1 lies in C(ρ), but does not

necessarily have a representation Q = (1 − λ)Q∗ + λ for some Q∗ ∈ dom(ρ∗) ∩ L1 and some

0 < λ ≤ 1. The latter representation cannot exist if P(Q ≥ s) > 0 for all s > 0. Moreover,

I(ρ) = ∅.

(3) Examples (1)–(2) consider convex risk measures. For a bona fide P-consistent risk measure,

consider the convex monetary risk measures

τ1(X) = 12ess sup(X) + 1

2E[X],

τ2(X) = 2∫ 11/2 qX(s)ds,

X ∈ L∞,

where ess sup(X) is defined as in Example 2.8. In [37, Example 3.3] it is verified that the

consistent risk measure ρ := minτ1, τ2 is not convex. However, using, for instance, [13,

Theorem 5] or a direct verification, one shows that ρ is star shaped. Moreover,

dom(τ∗1 ) ∩ L1 = Q ∈ L1 | E[Q] = 1, Q ≥ 12,

dom(τ∗2 ) ∩ L1 = Q ∈ L∞ | E[Q] = 1, 0 ≤ Q ≤ 2.

Hence, dom(ρ∗) ∩ L1 = Q ∈ L∞+ | E[Q] = 1, 1

2 ≤ Q ≤ 2, an observation that illustrates

Proposition 2.10.

9 In the present case, Q = P and therefore dQ

dP= 1.

12 F.-B. LIEBRICH

3. The main results

3.1. Risk sharing with consistent risk measures. All results of this section are proved in

Appendix D. For X ∈ L∞, we denote by

AX := X ∈ (L∞)n | X1 + · · ·+Xn = X

the set of all allocations of X. Our problem of interest concerns Qi-consistent risk measures

ρi : L∞ → R, where Qi ≪ P is some probability measure, i ∈ [n]. For X ∈ L∞, we aim to solve

the problem ∑ni=1 ρi(Xi) −→ min

subject to X ∈ AX .(3.1)

The associated infimal convolution

ρ := i∈[n]ρi (3.2)

gives precisely the optimal value of (3.1). The functional ρ is known to be cash-additive and

monotone, i.e., ρ is a monetary risk measure. An allocation X ∈ AX , X ∈ L∞, is optimal if

ρ(X) =(i∈[n]ρi

)(X) =

n∑

i=1

ρi(Xi).

That is, X is an optimiser in problem (3.1). If an optimal allocation of X exists, we also say that

ρ is exact at X.

Theorem 3.1. Suppose:

(i) A vector (Q1, . . . ,Qn) of probability measures equivalent to P satisfies Assumption 2.1.

(ii) For each i ∈ [n], ρi : L∞ → R is a Qi-consistent risk measure.

(iii) The risk measures ρ1, . . . , ρn−1 are admissible, and for all i ∈ [n− 1] there is Zi ∈ C(ρi)

such that

maxk∈[n]

ρ∗k(Zi) < ∞.

Then, for each X ∈ L∞, there is an optimal allocation X ∈ AX .

Remark 3.2. Let us compare Theorem 3.1 to [37, Theorem 4.1]. The latter proves the existence

of optimal allocations in case of homogeneous beliefs, i.e., Q1 = · · · = Qn = P. Mathematically,

a converging optimising sequence can be found using a simple “rebalancing of cash” argument

which is not available in the heterogeneous case anymore. However, [37, Theorem 4.1] can also be

interpreted as a special case of Theorem 3.1. Indeed, if Q1 = · · · = Qn = P, assumptions (i)–(iii)

are met if and only if each ρi is a P-consistent risk measure and if at most one agent i ∈ [n]

satisfies ρi = E[·]; cf. Proposition 2.5(2). However, the latter can be further relaxed because of

the homogeneity of beliefs. If ρn = ρn+1 = · · · = ρn+k = E[·], one obtains an optimal allocation

of X ∈ L∞ in the following three steps:

• Allocate X ∈ L∞ optimally among ρ1, . . . , ρn with Y ∈ AX .

• Choose Xn, . . . ,Xn+k ∈ L∞ with the property Yn =∑n+k

i=n Xi.

• Augment this to an optimal allocation X among ρ1, . . . , ρn+k setting Xi = Yi, i ∈ [n− 1].

RISK SHARING UNDER HETEROGENEOUS BELIEFS WITHOUT CONVEXITY 13

Hence, the only necessary requirement is that all risk measures are P-consistent.

Remark 3.3. The homogeneous case also shows that law invariance, alone or combined with star-

shapedness, is too weak to guarantee the existence of optimal allocations.10 As an illustration,

consider constants α, β > 0 with α+ β < 1. Let ρi : L∞ → R, i = 1, 2, be defined by

ρ1(X) := infx ∈ R | P(X ≤ x) > 1− α,

ρ2(X) := 1β

∫ 11−β qX(s) ds.

Here q• denotes a version of the quantile function under P. Both functionals are law invariant

with respect to the reference measure P, cash-additive, normalised, positively homogeneous and

thus star-shaped. However, while ρ2 is convex and therefore a consistent risk measure, ρ1 is not

consistent. To see this, suppose events A,B ∈ F satisfy A ⊂ B and 1− α < P(A) < P(B) < 1.

Set X := −1A, G := ∅, B,Bc,Ω, and

Y := E[X|G] = −P(A)P(B)1B .

Jensen’s inequality implies that X dominates Y in second-order stochastic relation. However,

ρ2(X) = −1 < −P(A)P(B) = ρ1(Y ).

At last, [33, Proposition 1] shows that no X ∈ L∞ with continuous distribution can be allocated

Pareto optimally between ρ1 and ρ2.

Remark 3.4. If all risk measures involved are admissible, the much simpler condition

∀ i, k ∈ [n] : ρ∗k(dQi

dP ) < ∞ (3.3)

is sufficient for assumption (iii) in Theorem 3.1 to hold. However, there are examples of admissible

risk measures ρ1 and ρ2 on L∞ that satisfy all assumptions of Theorem 3.1, but fail to satisfy

(3.3); i.e., (3.3) is not necessary for assumption (iii) to hold.

Indeed, fix an event A ∈ F with P(A) = 12 and consider the probability measure Q ≈ P defined

bydQdP = 1

21A + 321Ac .

Moreover, define ρi : L∞ → R, i = 1, 2, by

ρ1(X) := ESP1/5(X) = sup

Z∈dom(ρ∗1)E[ZX],

ρ2(X) := ESQ

4/9(X) = supZ∈dom(ρ∗

2)E[ZX],

where one identifies

dom(ρ∗1) = Z ∈ L∞+ | E[Z] = 1, Z ≤ 1.25,

dom(ρ∗2) = Z ∈ L∞+ | E[Z] = 1, Z ≤ 0.91A + 2.71Ac.

10 I am indebted to Ruodu Wang for pointing out the corresponding result in [33].

14 F.-B. LIEBRICH

It is obvious that ρ∗2(1) = ∞ and that ρ∗1(dQdP) = ∞. However,

Z := 0.81A + 1.21Ac

lies in dom(ρ∗1) ∩ dom(ρ∗2) and satisfies Z ∈ C(ρ1) in light of the discussion in Example 2.12(1).

The following examples continue the illustration of the necessity of the assumptions of Theo-

rem 3.1 in case n = 2.

Example 3.5.

(1) All appearing reference probability measures have to be equivalent: As a counterexample,

suppose Q ≪ P is arbitrarily chosen with the property Q 6≈ P. Consider the convex monetary

risk measures ρ1(X) := log(E[eX ]

)and ρ2(X) := EQ[X] = E[QX], X ∈ L∞. By [27, Example

6.1], ρ1ρ2 is not exact at all X ∈ L∞.

(2) P-a.s. positive densities of reference probability measures are not enough: To illustrate this,

consider the convex monetary risk measures defined in (1) and assume Q := dQdP satisfies

P(Q > 0) = 1. If ρ1ρ2 were exact at each X ∈ L∞, one could follow the reasoning of

[27, Example 6.1] and conclude that, if ρ1ρ2(X) = ρ1(X1) + ρ2(X2), Q would have to be a

subgradient of ρ1 at X1, i.e.

ρ1(X1) = E[QX1]− ρ∗1(Q).

By [41, Lemma 6.1], the identity

Q =eX1

E[eX1 ]

must hold. As X1 ∈ L∞, log(Q) has to be bounded from above and below, that is, there

have to be constants 0 < s < S such that P(Q ∈ [s, S]) = 1.

(3) Assumption (iii) cannot be dropped, even if Assumption 2.1 holds: In case n = 2, assumption

(iii) of Theorem 3.1 reads as C(ρ1)∩dom(ρ∗2) 6= ∅. Here we use [3, Example 4.3]. Consider the

probability measure Q ≈ P from Remark 3.4 and define the convex monetary risk measures

ρ1(X) := 12E[X] + 1

2 log(E[eX ]),

ρ2(X) := EQ[X],X ∈ L∞.

By [3, Example 4.3], ρ1ρ2 is not exact at all X ∈ L∞. However, as dom(ρ∗2) is a singleton,

C(ρ1) ∩ dom(ρ∗2) 6= ∅ would hold if and only if dQdP ∈ C(ρ1). Recall that A = dQ

dP = 12

and consider U := −1A + 131Ac . A direct computation shows that U ∈ A∞

ρ1 . As, however,

EQ[U ] = 0, dQdP cannot be compatible for ρ1.

Remark 3.6. The proofs in Appendix D demonstrate that optimal risk allocations will usually

not be comonotone allocations of an aggregated quantity. This is a substantial difference between

the cases of law-invariant risk sharing with heterogeneous and homogeneous reference probability

measures. As an example, consider an arbitrary Q ≈ P as well as the convex risk measures

ρ1(X) := 1β1

log(E[eβ1X ]),

ρ2(X) := 1β2

log(EQ[eβ2X ]),

X ∈ L∞.

RISK SHARING UNDER HETEROGENEOUS BELIEFS WITHOUT CONVEXITY 15

Here, β1, β2 > 0 are positive constants. Then ρ1ρ2 is exact at each X ∈ L∞, and the unique

optimal risk allocation is given by(

β2

β1+β2X + 1

β1+β2log(dQdP),

β1

β1+β2X − 1

β1+β2log(dQdP)

).

Clearly, the dependence on the density dQdP precludes comonotonicity unless dQ

dP ≡ 1, which is the

case if and only if Q = P.

In view of the preceding remark and the lack of comonotonicity of optimal allocations, it is worth

pointing out that the infimal convolution ρ in Theorem 3.1 preserves P-basedness of the individual

risk measures discussed in Section 2.2.

Corollary 3.7. In the situation of Theorem 3.1 let π ∈ Π be a finite measurable partition as in

Lemma 2.2(2). Consider the set

P := PB | B ∈ π

of conditioned probability measures. Then the infimal convolution ρ = ni=1ρi is P-based: If for

X,Y ∈ L∞ and all B ∈ π we have Xd=PB Y , then ρ(X) = ρ(Y ).

3.2. Risk sharing with capital requirements. Proofs and auxiliary results relating to this

subsection can be found in Appendix E.

The second main result of the paper concerns risk sharing with capital requirements based on mul-

tidimensional security spaces—sometimes also called multi-asset risk measures. The formulation

of this problem—which is thoroughly studied in [31] under the assumption of convexity—requires

risk measurement regimes, a notion we adopt from the aforementioned paper to the present set-

ting.

• For a vector (Q1, . . . ,Qn) of probability measures equivalent to P, ρi : L∞ → R are Qi-

consistent risk measures, i ∈ [n].

• Security spaces are finite-dimensional subspaces Si ⊂ L∞ which contain a nontrivial

Zi ∈ L∞+ , i ∈ [n]. Elements Z ∈ Si are called securities.

• pi : Si → R are linear and positive pricing functionals, mapping at least one positive

security to a positive price.

• If these assumptions are satisfied and

∀X ∈ L∞ : suppi(Z) | Z ∈ Si, X + Z ∈ Aρi < ∞,

Ri := (Aρi ,Si, pi) is a risk measurement regime, i ∈ [n].

• These risk measurement regimes induce capital requirements defined—in the spirit of

[8, 20, 24]—as

ηi(X) := infpi(Z) | Z ∈ Si, X − Z ∈ Aρi.

The value ηi(X) has an immediate operational interpretation as the infimal amount of capital

which needs to be raised and invested in suitable securities Z ∈ Si available in the security

market at price pi(Z) such that the secured net loss profile X − Z passes the capital adequacy

16 F.-B. LIEBRICH

test posed byAρi . Note also that the risk measures ρi merely play the role of gauges determinining

acceptability. Their precise values do not matter for the computation of the ηi’s.

In contrast to monetary risk measures, such capital requirements may neither only attain finite

values nor be cash-additive. Moreover, law invariance of the acceptance set is not inherited by

the capital requirement beyond trivial cases.

Assumption 3.8. There is Q∗ ∈⋂n

i=1C(ρi) such that, for all i ∈ [n] and all Z ∈ Si,

pi(Z) = E[Q∗Z].

Remark 3.9. If each ρi is an admissible Qi-consistent risk measure, then⋂n

i=1 C(ρi) 6= ∅ if

∀ i, k ∈ [n] : dQi

dP ∈ dom(ρ∗k).

Indeed, (2.5) in Proposition 2.5 implies that, for all choices of positive convex combination pa-

rameters δi > 0, i ∈ [n],n∑

i=1

δidQi

dP ∈n⋂

i=1

C(ρi).

At last, the global security market (M, π) is given by

M :=∑n

i=1 Si,

π(Z) = E[Q∗Z], Z ∈ M.

Like in Theorem 3.1, we define the infimal convolution

η := ni=1ηi :

L∞ → [−∞,∞],

X 7→ infX∈AXηi(Xi),

to find optimal allocations, and ask if η takes finite values and is exact at a given X ∈ L∞. The

following theorem generalises [31, Theorem 5.6].

Theorem 3.10. Suppose:

(i) The underlying vector (Q1, . . . ,Qn) of probability measures satisfies Assumption 2.1.

(ii) The risk measurement regimes (R1, . . . ,Rn) satisfy Assumption 3.8.

(iii) U ∈∏n

i=1 Si such that U :=∑n

i=1 Ui satisfies π(U) = 1 is fixed.

Then for all X ∈∑n

i=1 dom(ηi) there is N ∈∏n

i=1 Si and A ∈∏n

i=1Aρi such that N :=∑ni=1 Ni ∈ ker(π), and

Xi := Ai +Ni + η(X)Ui, i ∈ [n],

satisfies

η(X) =

n∑

i=1

ηi(Xi). (3.4)

The assumptions of Theorem 3.10 cannot be relaxed, as the following example demonstrates.

RISK SHARING UNDER HETEROGENEOUS BELIEFS WITHOUT CONVEXITY 17

Example 3.11. Consider the homogeneous case Q1 = Q2 = P. Moreover, let

ρ1 = ρ2 :L∞ → R,

X 7→ 12 log

(E[e2X ]

),

be entropic risk measures. By [41, Example 2.9], ρ := ρ1ρ2 is given by ρ(X) = log(E[eX ]).

Suppose Q∗ ∈ L1+ satisfies E[Q∗] = 1 and E[Q∗1A] = 0 for some A ∈ F with P(A) > 0. Let

S1 := span1Ω,

S2 := span1Ω,1Ac.

As M = S2, we also have

ker(π) = R · 1A.

The associated risk sharing functional η is easily seen to be finite. Assume for contradiction that

η is exact at all X ∈ dom(η) = dom(ρ1) + dom(ρ2) = L∞. As η is convex and continuous at X,

we can find some subgradient Z at X. This subgradient has to satisfy Z ∈ L1+ because η has the

Lebesgue property. We obtain

η(X) = E[ZX]− η∗(Z) =2∑

i=1

E[ZXi]− ρ∗i (Z) ≤2∑

i=1

ρi(Xi) = η(X).

Hence, Z is a subgradient of ρi at X, i = 1, 2. Now, [20, Theorem 3] on the one hand shows that

E[Z·]|ker(π) ≡ 0, or equivalently,

E[Z1A] = 0. (3.5)

On the other hand, by [41, Lemma 6.1],

Z = eX2

E[eX2 ].

No Z of this shape can satisfy (3.5).

Appendix A. Auxiliary results

Before we give the proofs of results from Sections 2–3, this appendix collects structural properties

of Q-law-invariant risk measures where the probability measure Q may not agree with the gauge

probability measure P. While the first set of results in Lemma A.1 is standard if Q = P, we shall

provide the short proof in the general case for the convenience of the reader.

Lemma A.1. Suppose that ρ : L∞ → R is a Q-consistent risk measure for a probability measure

Q ≈ P.

(1) For all Q ∈ L1,

ρ∗(Q) = ρ♯((dQdP)

−1Q),

where

ρ♯(Z) = supX∈L∞

EQ[ZX]− ρ(X), Z ∈ L1Q. (A.1)

Moreover, the function ρ♯ is Q-law invariant.

18 F.-B. LIEBRICH

(2) If G ⊂ F is a sub-σ-algebra and X ∈ L∞, then

ρ(EQ[X|G]) ≤ ρ(X).

(3) For all Q ∈ L1 ∩ L1Q and all sub-σ-algebras G ⊂ F such that dQ

dP has a G-measurable

version,

ρ∗(E[Q|G]) ≤ ρ∗(Q).

Proof.

(1) Let Q ∈ L1 and compute

ρ∗(Q) = supX∈L∞

E[QX]− ρ(X) = supX∈L∞

EQ

[(dQdP)

−1QX]− ρ(X) = ρ♯

((dQdP)

−1Q).

Law invariance with respect to Q of the function ρ♯ follows, for instance, from [9, Lemma

3.2].

(2) The statement is a direct consequence of the definition of a Q-consistent risk measure.

(3) Let G ⊂ F be a sub-σ-algebra such that dQdP has a G-measurable version. Let Q ∈ L1. Using

the Bayes rule for conditional expectations,

dQdPEQ[Q|G] = E[dQdPQ|G] = dQ

dPE[Q|G].

Q ≈ P implies that EQ[Q|G] = E[Q|G]. The function ρ♯ defined in (A.1) is convex, Q-law

invariant, proper (i.e., dom(ρ♯) 6= ∅), and σ(L1Q, L

∞Q )-lower semicontinuous by definition.

Hence, [9, Theorem 3.6] admits to infer for all Q ∈ L1 that

ρ∗(Q) = ρ♯((dQdP)

−1Q)≥ ρ♯

(EQ

[(dQdP)

−1Q|G])

= ρ♯((dQdP)

−1EQ[Q|G])

= ρ∗(EQ[Q|G]) = ρ∗(E[Q|G]).

Lemma A.2. Let ρ : L∞ → R be a monetary risk measure.

(1) For all φ ∈ dom(ρ∗) and all U ∈ A∞ρ , φ(U) ≤ 0 holds.

(2) Suppose Q ≈ P is a probability measure and ρ is a Q-consistent risk measure. Then A∞ρ

is closed, Q-law-invariant, and closed under taking conditional expectations with respect

to Q.

Proof.

(1) Let φ and U be as described. Let (sk)k∈N ⊂ (0,∞) be a null sequence and (Yk)k∈N ⊂ Aρ

such that limk→∞ skYk = U . Recall that a consequence of cash-additivity is that

∀φ ∈ dom(ρ∗) : ρ∗(φ) = supY ∈Aρ

φ(Y ) ≥ 0.

Hence,

φ(U) = limk→∞

skφ(Yk) ≤ ρ∗(φ) · limk→∞

sk = 0.

RISK SHARING UNDER HETEROGENEOUS BELIEFS WITHOUT CONVEXITY 19

(2) It is straightforward to verify that A∞ρ is closed. In order to see that A∞

ρ is closed under

taking conditional expectations with respect to Q, fix U ∈ A∞ρ and an arbitrary sub-σ-

algebra G ⊂ F . Let (sk)k∈N ⊂ (0,∞) and (Yk)k∈N ⊂ Aρ be sequences as in the proof of (1)

with limk→∞ skYk = U . By Jensen’s inequality and the monotonicity of ρ in second-order

stochastic dominance relation with respect to Q, EQ[Yk|G] ∈ Aρ holds for all k ∈ N. Hence,

EQ[U |G] = limk→∞

EQ[skYk|G] = limk→∞

skEQ[Yk|G] ∈ A∞ρ .

The latter property of A∞ρ is called Q-dilatation monotonicity in the literature. The function

F : L∞ = L∞Q → [0,∞),

X 7→ infU∈A∞ρ‖X − U‖∞,

is continuous (because A∞ρ is closed) and Q-dilatation monotone. Indeed, for every X ∈ L∞

and every sub-σ-algebra G ⊂ F ,

F (EQ[X|G]) ≤ infU∈A∞

ρ

‖EQ[X|G] − EQ[U |G]‖∞

= infU∈A∞

ρ

‖E[X − U |G]‖∞

≤ infU∈A∞

ρ

‖X − U‖∞ = F (X).

Hence, by [15, Theorem 1.1], F is Q-law invariant. This translates to Q-law-invariance of

A∞ρ = F−1(0).

Lemma A.3. Suppose:

(i) Q ≈ P is a probability measure.

(ii) G ⊂ F is a sub-σ-algebra such that dQdP has a G-measurable version.

(iii) ρ : L∞ → R is a Q-consistent risk measure.

Then, for all Z ∈ C(ρ),

E[Z|G] ∈ C(ρ).

Proof. As there is nothing to show if C(ρ) = ∅, we may assume that ρ is admissible. Let G ⊂ F

be a sub-σ-algebra as in (ii). By Lemma A.1(3), E[Z|G] ∈ dom(ρ∗). Moreover, as Q ≈ P, we may

apply the Bayes rule for conditional expectations to compute

EQ[Z|G] =E[dQdPZ|G]

E[dQdP |G]=

dQdPE[Z|G]

dQdP

= E[Z|G]. (A.2)

Now suppose U ∈ A∞ρ satisfies E[E[Z|G]U ] = 0. E[Z|G] ∈ C(ρ) follows if we can show U = 0.

Switching the conditioning and arguing as in (A.2) yields

0 = E[E[Z|G]U ] = E[ZEQ[U |G]],

20 F.-B. LIEBRICH

and EQ[U |G] ∈ A∞ρ holds by Lemma A.2(2). As Z ∈ C(ρ), we infer EQ[U |G] = EQ[U ] = 0. As

dQdP ∈ C(ρ) by Proposition 2.5(2) below, U = 0 has to hold.

Appendix B. Local comonotone improvement

In the following, we denote by C(n) the set of all comonotone n-partitions of the identity, i.e.

functions f : R → Rn such that each coordinate fi is non-decreasing and∑n

i=1 fi = idR. Con-

sequently, each coordinate function fi of f ∈ C(n) is 1-Lipschitz continuous. For f ∈ C(n), we

set

f := f − f(0).11

Definition B.1. Let X ∈ L∞. A vector Y ∈ AX is a locally comonotone allocation of X over a

finite measurable partition π ∈ Π if there are (fB)B∈π ⊂ C(n) such that

∀ i ∈ [n] : Yi =∑

B∈π

fBi (X)1B .

In the following, we denote by Q the Q-convex order on L∞: X Q Y holds if and only if,

for all convex test functions v : R → R such that both expectations are well defined, EQ[v(X)] ≤

EQ[v(Y )].

Lemma B.2. Suppose:

(i) A vector (Q1, . . . ,Qn) of probability measures satisfies Assumption 2.1.

(ii) π ∈ Π is a partition such that each dQi

dP has a σ(π)-measurable version.

(iii) ρi : L∞ → R is a Qi-consistent risk measure, i ∈ [n].

Let X ∈ L∞ and X ∈ AX be arbitrary. Then there exists a locally comonotone allocation Y ∈ AX

over π such that:

(1) ∀ i ∈ [n] : Yi QiXi.

(2) ∀ i ∈ [n] : ρi(Yi) ≤ ρi(Xi).

Proof. For each B ∈ π consider the nonatomic probability space (B,GB ,PB), where GB :=

B ∩A | A ∈ F. Asn∑

i=1

Xi|B = X|B ,

there is a comonotone n-partition of the identity fB ∈ C(n) such that

∀ i ∈ [n] : fBi (X|B) PB Xi|B ;

cf. [12, Theorem 3.1]. In particular, setting

Yi :=∑

B∈Π

fBi (X)1B , i ∈ [n],

11 Each coordinate fi of f could be interpreted as an indemnity function.

RISK SHARING UNDER HETEROGENEOUS BELIEFS WITHOUT CONVEXITY 21

defines an allocation Y ∈ AX which we claim to be as described in (1). In order to verify this,

recall that for each i ∈ [n],dQi

dP=

∑

B∈π

Qi(B)P(B) 1B

by assumption (i). Let v : R → R be an arbitrary convex function, assume v(0) = 0 without loss

of generality, and compute

EQi[v(Yi)] = EQi

[∑

B∈π

v(fBi (X)

)1B

]=

∑

B∈π

Qi(B)EPB

[v(fBi (X|B)

)]

≤∑

B∈π

Qi(B)EPB [v(Xi|B)] =∑

B∈π

EQi[v(Xi)1B ] = EQi

[v(Xi)].

The same allocation also has the property described in (2) as each consistent risk measure ρi is

monotone with respect to Qi.

The previous lemma is the first step in the adaptation of the comonotone improvement procedure

to our setting of heterogeneous reference probability measures. The next proposition provides

the bounds necessary for the second step.

Proposition B.3. Suppose:

(i) A vector (Q1, . . . ,Qn) of probability measures Qi ≈ P satisfies Assumption 2.1.

(ii) π ∈ Π is a measurable partition of Ω as in Lemma 2.2(2).

(iii) ρi : L∞ → R is a Qi-consistent risk measure, i ∈ [n].

(iv) The risk measures ρ1, . . . , ρn−1 are admissible and, for all i ∈ [n− 1] there is Zi ∈ C(ρi)

such that

maxk∈[n]

ρ∗k(Zi) < ∞.

Let (Yk)k∈N ⊂∑n

i=1Ai be a bounded sequence and let Yk ∈∏n

i=1 Ai be a locally comonotone

allocation of Yk over π, k ∈ N. Then the sequence (Yk)k∈N is bounded as well.

Proof. Let π be the measurable partition of Ω from (ii). For k ∈ N, there are (fB,k)B∈π ∈ C(n)

such that, for all i ∈ [n],

Y ki :=

∑

B∈π

fB,ki (Yk)1B . (B.1)

Let us first discuss boundedness of the sequence (Y k1 )k∈N in detail. Towards a contradiction,

suppose (Y k1 )k∈N is unbounded in norm. As

Y k1 =

∑

B∈π

fB,k1 (Yk)1B +

∑

B∈π

fB,k1 (0)1B (B.2)

and the first summand is uniformly bounded in k by Lipschitz continuity of fB,k1 , we obtain that

βk := maxB∈π

|fB,k1 (0)| =

∥∥∥∑

B∈π

fB,k1 (0)1B

∥∥∥∞

22 F.-B. LIEBRICH

satisfies

limk→∞

‖Y k1 ‖∞βk

= 1.

Moreover, up to passing to a subsequence, we can assume that for a suitable u ∈ Rπ with

maxB∈π |uB | = 1,

V := limk→∞

∑

B∈π

fB,k1 (0)

βk1B =

∑

B∈π

uB1B 6= 0. (B.3)

By definition of the asymptotic cone, V ∈ A∞ρ1 . By Lemma A.2(1), for Z1 ∈ C(ρ1) chosen as in

assumption (iv),

E[Z1V ] ≤ 0. (B.4)

As max2≤i≤n ρ∗i (Z1) < ∞ by (iv), we further obtain

E[Z1V ] = − limt→∞

E[Z1(−V )] = − limk→∞

n∑

i=2

E[Z1

Y ki

βk

]≥ − lim

k→∞

1βk

n∑

i=2

ρ∗i (Z1) = 0.

Combining the latter estimate with (B.4), E[Z1V ] = 0 follows. Using that V ∈ A∞ρ1 and that

Z1 ∈ C(ρ1), V = 0 is a consequence in direct contradiction to (B.3). The assumption that

(Y k1 )k∈N is unbounded has to be absurd.

Arguing analogously for the coordinate sequences (Y ki )k∈N, i = 2, . . . , n−1, one infers their norm

boundedness. At last,

supk∈N

‖Y kn ‖∞ ≤ sup

k∈N

‖Yk‖∞ +n−1∑

i=1

supk∈N

‖Y ki ‖∞ < ∞.

Remark B.4. The proof of Proposition B.3 justifies the reliance on Assumption 2.1 of our

approach, which is itself based on comonotone improvement. A priori, the comonotone func-

tions (fB)B∈π which describe the allocation on events B ∈ π have no apparent relation to each

other. This problem is circumvented by the use that equation (B.3) makes of the compactness

of the finite-dimensional unit ball in Rπ. We do not see an immediate generalisation to infinite

dimensions, i.e., completely arbitrary heterogeneous reference measures.

Appendix C. Proofs from Section 2

Proof of Lemma 2.2. (1) implies (2): If probability measures (Q1, . . . ,Qn) satisfy Assumption 2.1,

we may fix a vector Q of versions Qi ofdQi

dP which are simple functions. The set

Σ := q ∈ Rn | P(Q = q) > 0

is finite. The event⋂

q∈ΣQ = qc is a null set which we can assume to be empty by suitably

manipulating the choice of densities. Hence, setting π := Q = q | q ∈ Σ ∈ Π, each density

has a σ(π)-measurable version.

RISK SHARING UNDER HETEROGENEOUS BELIEFS WITHOUT CONVEXITY 23

(2) implies (3): For i ∈ [n], let Qi be a σ(π)-measurable version of dQi

dP . Each Qi is constant on

B, B ∈ π. Hence, if Qi(B) > 0, X ∈ L∞, and A ∈ B(R) is an arbitrary Borel set,

QBi (X ∈ A) = EP[Qi1A(X)1B ]

EP[Qi1B] = EPB [1A(X)1B ] = PB(X ∈ A). (C.1)

The right-hand side does not depend on i, and we conclude (2.1).

(3) implies (1): Set P := 1n

∑ni=1 Qi. Without loss of generality, we can assume minB∈π P(B) > 0,

i.e., π ∈ Π. Let B ∈ π be arbitrary, but fixed. Set IB := i ∈ [n] | Qi(B) > 0. By (2.1), (QBi )i∈IB

is a vector of nonatomic probability measures, all equivalent to PB . Also, for each A ∈ F ,

PB(A) =1

nP(B)

∑

i∈IB

QBi (A)Qi(B). (C.2)

Together with (2.1), (C.2) can be used to show thatdQB

i

dPB , i ∈ IB, is constant PB-a.s. At last, for

i ∈ [n], the representationdQi

dP=

∑

B∈π: i∈IB

Qi(B)

P(B)·dQB

i

dPB1B

shows that dQi

dP may be chosen as σ(π)-measurable (and thus simple) function.

Proof of Proposition 2.5.

(1) For the verification of (2.5), let Q,Z, and λ be as described. Abbreviate

R := λQ+ (1− λ)Z.

For property (a) in Definition 2.4, convexity of dom(ρ∗) implies R ∈ dom(ρ∗). As for property

(b), suppose U ∈ A∞ρ satisfies E [RU ] = 0. By Lemma A.2(1), both E[QU ] ≤ 0 and E[ZU ] ≤

0. This forces E[QU ] = E[ZU ] = 0. As Q ∈ C(ρ), U = 0 follows.

(2) (b) implies (c) because of (2.5). (c) trivially implies (a).

For the equivalence of (b) and (d), recall from [30, Theorem 5.10] that a Q-consistent risk

measure ρ satisfies ρ = EQ[·] if and only if one can find a nonzero U ∈ L∞ with EQ[U ] = 0

and

supt≥0

ρ(tU) ≤ 0.

The latter implies U ∈ A∞ρ . Rephrasing this result in our terminology, ρ = EQ[·] holds if and

only if QdP /∈ C(ρ).

At last, we show that (a) implies (b). Assume towards a contradiction that C(ρ) 6= ∅ anddQdP /∈ C(ρ). The latter means that we can select a nonconstant U ∈ A∞

ρ such that EQ[U ] = 0.

In particular, U is not constant Q-a.s. because Q ≈ P. Let Z ∈ L1Q be such that Z dQ

dP ∈ C(ρ);

cf. Lemma A.1. By assumption, Z cannot be constant. Using, for instance, [30, Lemma 3.2],

0 = EQ[Z]EQ[U ] < supEQ[V U ] | V ∈ L1Q, V

d=Q Z.

We may therefore select V ∈ L1Q such that V

d=Q Z and

0 < EQ[V U ] = E[V dQ

dPU].

24 F.-B. LIEBRICH

But V dQdP ∈ dom(ρ∗) by Lemma A.1, whence E[V dQ

dPU ] ≤ 0 follows with Lemma A.2(1). This

is a contradiction, and U cannot be nonconstant.

Proof of Proposition 2.7. Let Q∗ ∈ I(ρ). As dQdP ∈ dom(ρ∗), we can find some ε > 0 such that

Q := Q∗ + ε(Q∗ − dQdP) ∈ dom(ρ∗). Rearranging the defining equality yields

Q∗ = 11+εQ+ ε

1+εdQdP .

By Proposition 2.5(2), Q∗ ∈ C(ρ).

Proof of Proposition 2.10. By Proposition 2.5(2), the admissible risk measure ρ satisfies ρ 6= EQ[·].

Consider the function ρ♯ : L1Q → (−∞,∞] defined in (A.1). By [30, Theorem 5.10],

1 ( dom(ρ♯).

By Lemma A.1(1), we find a nonconstant Z ∈ L1Q such that

ρ∗(dQdPZ

)= ρ♯(Z) < ∞.

Using Proposition 2.5(2) again,dQdP(λ + (1 − λ)Z)

∣∣ 0 < λ < 1

is an uncountable subset of

C(ρ) \ dQdP.

Appendix D. Proofs of Theorem 3.1 and Corollary 3.7

Proof of Theorem 3.1. First of all, we prove that ρ does not attain the value −∞. To this end,

let Z1 ∈ C(ρ1) be as described in assumption (iii), let X ∈ L∞, and fix Y ∈ AX . By definition

of the convex conjugate,

n∑

i=1

ρi(Yi) ≥n∑

i=1

E[Z1Yi]− ρ∗i (Z1) = E[Z1X]−n∑

i=1

ρ∗i (Z1).

Taking the infimum over Y ∈ AX on the left-hand side proves

ρ(X) ≥ E[Z1X]−n∑

i=1

ρ∗i (Z1) > −∞.

Next, we show that A+ :=∑n

i=1 Aρi is closed. Let (Yk)k∈N ⊂ A+ be a sequence that converges

to Y ∈ L∞ as k → ∞. We need to prove that Y ∈ A+. To this effect, for all k ∈ N let Yk be a

locally comonotone allocation as in (B.1) such that, for all i ∈ [n], Y ki ∈ Aρi . By Proposition B.3,

supk∈N

maxi∈[n]

‖Y ki ‖∞ < ∞.

By (B.2)

κ := supk∈N

maxB∈π

maxi∈[n]

|fB,ki (0)| ≤ sup‖Y k‖∞ +max

i∈[n]‖Y k

i ‖∞ | k ∈ N < ∞.

RISK SHARING UNDER HETEROGENEOUS BELIEFS WITHOUT CONVEXITY 25

The set C(n)κ := f ∈ C(n) | maxi∈[n] |fi(0)| ≤ κ is sequentially compact in the topology of

pointwise convergence; cf. [21, Lemma B.1]. Hence, there are (gB)B∈π ⊂ C(n)κ such that, up to

switching to a subsequence |π| times,

∀B ∈ π ∀x ∈ R : limk→∞

fB,k(x) = gB(x).

For each i ∈ [n], we further observe that∥∥∑

B∈π

(fB,ki (Y k)− gBi (Y ))1B

∥∥∞

≤∑

B∈π

∥∥(fB,ki (Y k)− gBi (Y )

)1B

∥∥∞

≤∑

B∈π

∥∥(fB,ki − gBi )(Y

k)∥∥∞

+ ‖Y k − Y ‖∞ + |fB,ki (0)− gBi (0)|.

As fB,k converges to gB uniformly on the compact interval [− supk∈N ‖Yk‖∞, supk∈N ‖Yk‖∞], we

infer that

limk→∞

∥∥ ∑

B∈π

(fB,ki (Yk)− gBi (Y ))1B

∥∥∞

= 0.

As Aρi is closed,∑

B∈π gBi (Y )1B ∈ Aρi must hold, i ∈ [n]. It remains to observe that the latter

defines a vector in AY and that therefore Y ∈ A+.

The remainder of the proof is standard, but shall be included here for the sake of completeness.

In its next step, we verify A+ = Aρ. The inclusion of the left-hand set in the right-hand set

is immediate. Conversely, assume without loss of generality ρ(X) = 0 and consider a sequence

(Yk)k∈N ⊂ AX such that

limk→∞

n∑

i=1

ρi(Yki ) = 0.

Let Uki := Y k

i − ρi(Yki ) ∈ Ai, i ∈ [n]. As A+ is closed,

A+ ∋ limk→∞

n∑

i=1

Uki = lim

k→∞X −

n∑

i=1

ρi(Yki ) = X.

Now, let X ∈ L∞. As ρ (X − ρ(X)) = 0, X − ρ(X) ∈ A+ and we can find Y ∈∏n

i=1 Aρi such

that X − ρ(X) =∑n

i=1 Yi. Setting Xi := Yi +1nρ(X), we obtain

ρ(X) ≤n∑

i=1

ρi(Xi) =

n∑

i=1

ρi(Yi +1nρ(X)) =

n∑

i=1

ρi(Yi) + ρ(X) ≤ ρ(X).

Hence, X is an optimal allocation of X.

Remark D.1. Mathematically, the role that Proposition B.3—and thereby admissibility of in-

dividual risk measures—plays in the proof of Theorem 3.1 is best understood as adaptation of

the spirit of Dieudonne’s [17] famous theorem to the present nonconvex situation.

26 F.-B. LIEBRICH

Proof of Corollary 3.7. LetX,Y ∈ L∞ and suppose thatXd=PB Y holds for all B ∈ π. Moreover,

let (fB)B∈π ⊂ C(n) and A ⊂ R an arbitrary Borel set. We then compute for i ∈ [n] that

Qi

(∑

B∈π

fBi (X)1B ∈ A

)=

∑

B∈π

Qi(B) · PB(fBi (X) ∈ A

)=

∑

B∈π

Qi(B) · PB(fBi (Y ) ∈ A

)

= Qi

( ∑

π∈B

fBi (Y )1B ∈ A

).

Given the Qi-law-invariance of each ρi and the proof of Theorem 3.1,

ρ(X) = min(fB)B∈π⊂C(n)

n∑

i=1

ρi

(∑

π∈B

fBi (X)1B

)= min

(fB)B∈π⊂C(n)

n∑

i=1

ρi

(∑

π∈B

fBi (Y )1B

)= ρ(Y ).

Appendix E. Proof of Theorem 3.10

Before the next preparatory result, Lemma E.1, we introduce the operation of computing the

star-shaped hull of a normalised monetary risk measure ρ : L∞ → R.

ρ⋆(X) := infm ∈ R | X −m ∈⋃

s∈[0,1] sAρ.

Lemma E.1. Suppose ρ : L∞ → R is a normalised monetary risk measure.

(1) ρ⋆ is a star-shaped risk measure which satisfies

Aρ⋆ = cl( ⋃

s∈[0,1]

sAρ

). (E.1)

If ρ is consistent, then ρ⋆ is also consistent.

(2) The asymptotic cones satisfies

A∞ρ⋆ = A∞

ρ . (E.2)

(3) If ρ is consistent, we have

C(ρ) = C(ρ⋆).

Proof.

(1) For star-shapedness, note that for X ∈ L∞ and λ ∈ (0, 1),

λ ·m | m ∈ R, ∃ s ∈ (0, 1] : X −m ∈ sAρ ⊂ k ∈ R | ∃ s ∈ (0, 1] : λX − k ∈ sAρ,

which means ρ⋆(λX) ≤ λρ⋆(X).

Next, ρ⋆(sY ) ≤ 0 holds for all s ∈ (0, 1] and all Y ∈ Aρ, which means

cl( ⋃

s∈[0,1]

sAρ

)⊂ Aρ⋆

RISK SHARING UNDER HETEROGENEOUS BELIEFS WITHOUT CONVEXITY 27

by closedness of the right-hand set. Conversely, suppose ρ⋆(X) ≤ 0, which means we must

be able to find (sn)n∈N ∈ (0, 1] such that X − 1n ∈ snAρ, n ∈ N. Hence,

X = limn→∞

X − 1n ∈ cl

( ⋃

s∈[0,1]

sAρ

).

If ρ is consistent, suppose X is dominated by X ′ in the second-order stochastic dominance

relation and (m, s) ∈ R× (0, 1] is such that X ′ −m = sY for some Y ∈ Aρ. One verifies that1s (X −m) is dominated by 1

s (X′ −m) = Y ∈ Aρ in the second-order stochastic dominance

relation. Hence, X −m ∈ sAρ as well. We infer that

ρ⋆(X) = infk ∈ R | ∃ s ∈ (0, 1] : X − k ∈ sAρ

≤ infm ∈ R | ∃ s ∈ (0, 1] : X ′ −m ∈ sAρ = ρ⋆(X′),

which proves consistency of ρ⋆.

(2) In order to verify (E.2), note first that A∞ρ ⊂ A∞

ρ⋆ follows from identity (E.1). Conversely,

suppose that U ∈ A∞ρ⋆ , which is the case if and only if kU ∈ Aρ⋆ for all k ∈ N. By (1), we

can find tk ∈ [0, 1] and Yk ∈ Aρ such that ‖kU − tkYk‖∞ ≤ 1. This implies that

U = limk→∞

tkk Yk,

which means U ∈ A∞ρ .

(3) For all φ ∈ (L∞)∗ and each normalised risk measure τ , we have τ∗(φ) ≥ 0. Using (1),

ρ∗⋆(φ) = supR∈Aρ⋆

φ(R) = supY ∈Aρ

sups∈(0,1]

φ(sY )

= supY ∈Aρ: φ(Y )≥0

sups∈(0,1]

φ(sY )

= supY ∈Aρ

φ(Y ) = ρ∗(φ).

In particular, dom(ρ∗⋆) = dom(ρ∗). The assertion now follows from (2).

Lemma E.2. In the situation of Theorem 3.10 set

ρ := ni=1ρi,

τi := (ρi)⋆, i ∈ [n].

Then

A∞ρ ⊂

n∑

i=1

A∞τi =

n∑

i=1

A∞ρi . (E.3)

Proof. The second identity in (E.3) is a direct consequence of (E.2). For the first inclusion, let

U ∈ A∞ρ , (sk)k∈N ⊂ (0, 1) be a null sequence and (Yk)k∈N ⊂ Aρ such that limk→∞ skYk = U . By

the proof of Theorem 3.1,

Aρ =

n∑

i=1

Aρi .

28 F.-B. LIEBRICH

Hence, skYk ∈∑n

i=1 Aτi , k ∈ N. Take (fB,k)B∈π ⊂ C(n), k ∈ N, be such that the associated

locally comonotone allocation satisfies∑

B∈π

fB,ki (skYk)1B ∈ Aτi , i ∈ [n].

Applying the argument from the proof of Theorem 3.1 to the consistent risk measures τ1, . . . , τn,

there is a subsequence (kλ)λ∈N and (gB)B∈π such that

Ui :=n∑

i=1

gBi (U)1B = limλ→∞

∑

B∈π

fB,kλi (skλYkλ)1B , i ∈ [n].

Hence, U =∑n

i=1 Ui ∈∑n

i=1Aτi .

As A∞ρ is a cone, we can repeat this argument for each element of the set kU | k ∈ N to obtain

(gB,k)B∈π ⊂ C(n) such that, for all k ∈ N,∑

B∈π

gB,ki (kU)1B ∈ Aτi , i ∈ [n].

In particular, for gB,k := 1kg

B,k(k·) ∈ C(n), we also have∑

B∈π

gB,k(U)1B ∈ Aτi , i ∈ [n].

Using the above argument once more, there is a subsequence (kλ)λ∈N and (hB)B∈π ⊂ C(n) such

that

limλ→∞

∑

B∈π

gB,kλi (U)1B =

∑

B∈π

hBi (U)1B ∈ A∞τi , i ∈ [n].

This is sufficient to show that

U =n∑

i=1

∑

B∈π

hBi (U)1B ∈n∑

i=1

A∞τi .

Proof of Theorem 3.10. First, we prove that η(X) > −∞ holds for all X ∈ L∞. Let Q∗ be the

probability density from Assumption 3.8. For all X ∈ AX ,

n∑

i=1

ηi(Xi) =

n∑

i=1

infE[Q∗Zi] | Zi ∈ Si, Xi − Zi ∈ Aρi

=

n∑

i=1

infE[Q∗Xi]− E[Q∗(Xi − Zi)] | Zi ∈ Si, Xi − Zi ∈ Aρi

≥n∑

i=1

infE[Q∗Xi]− E[Q∗Yi] | Yi ∈ Aρi = E[Q∗X]−n∑

i=1

ρ∗i (Q∗).

Now recall that the risk measure ρ := ni=1ρi satisfies Aρ =

∑ni=1Aρi by the proof of Theorem 3.1.

We shall verify now that Aρ + ker(π) is closed.

RISK SHARING UNDER HETEROGENEOUS BELIEFS WITHOUT CONVEXITY 29

Let (Ak)k∈N ⊂ Aρ and (Nk)k∈N ⊂ ker(π) such that Y k := Ak + Nk → Y ∈ L∞ as k → ∞.

Assume towards a contradiction that the sequence (Ak)k∈N is unbounded. This also implies that

(Nk)k∈N is unbounded and that

limk→∞

‖Nk‖∞‖Ak‖∞

= 1.

As ker(π) is of finite dimension, there is N∗ ∈ ker(π) such that

limk→∞

Nk

‖Ak‖∞= N∗ = − lim

k→∞

Ak

‖Ak‖∞.

By definition, −N∗ ∈ A∞ρ . Set τi := (ρi)⋆, an operation introduced in the context of Lemma E.1.

By Lemma E.2, we can find Ki ∈ A∞τi , i ∈ [n], such that

−N∗ =

n∑

i=1

Ki.

Using Lemma A.2(1) in the first and second estimate, we find for all j ∈ [n] that

0 ≥ E[Q∗Kj] ≥n∑

i=1

E[Q∗Ki] = E[Q∗(−N∗)] = 0.

As Q∗ ∈⋂n

i=1C(τi) by Lemma E.1(3), Ki = 0 for all i ∈ [n], whence N∗ = 0 follows. The

assumption that (Ak)k∈N is unbounded has to be absurd. Consequently, the sequence (Nk)k∈N =

(Y k−Ak)k∈N is bounded. As ker(π) is of finite dimension and Aρ is closed, there are N ∈ ker(π)

and A ∈ Aρ such that, for a suitably chosen subsequence (kλ)λ∈N, Akλ → A and Nkλ → N ,

λ → ∞. We have verified that

Y = A+N ∈ Aρ + ker(π).

Let X ∈∑n

i=1 dom(ηi) = dom(η) and U, U as in the assertion. For all k ∈ N we may choose

Xk ∈ AX such thatn∑

i=1

ηi(Xki ) ≤ η(X) + 1

k .

Moreover, for all i ∈ [n] we may select Zki ∈ Si such that

pi(Zki ) ≤ η(Xk

i ) +1kn ,

Xki − Zk

i ∈ Aρi .

For Zk :=∑n

i=1 Zki ∈ M, we observe

X − π(Zk)U = (X − Zk) + (Zk − π(Zk)U) ∈ Aρ + ker(π).

As limk→∞ π(Zk) = η(X),

X − η(X)U ∈ Aρ + ker(π)

holds by closedness of the latter Minkowski sum. For appropriate N ∈∏n

i=1 Si, setting N :=∑ni=1 Ni ∈ ker(π), X − η(X)U −N ∈ Aρ. Using Lemma B.2(2), we may select (fB)B∈π ⊂ C(n)

30 F.-B. LIEBRICH

such that, for all i ∈ [n], ∑

B∈π

fBi (X − η(X)U −N)1B ∈ Aρi .

(3.4) is now an immediate consequence of Si-additivity of ρi, i.e., for all X ∈ L∞, Z ∈ Si,

ρi(X + Z) = ρi(X) + pi(Z).

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