Budget-constrained optimal insurance with belief heterogeneity

Insurance: Mathematics and Economics 89 (2019) 79–91

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Insurance: Mathematics and Economics journal homepage: www.elsevier.com/locate/ime

Budget-constrained optimal insurance with belief heterogeneity✩ Mario Ghossoub University of Waterloo, Department of Statistics and Actuarial Science, 200 University Ave. W., Waterloo, ON, N2L 3G1, Canada

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Article history: Received January 2019 Received in revised form May 2019 Accepted 9 September 2019 Available online 28 September 2019 JEL classification: C02 D86 G22 MSC: 91B30 Keywords: Optimal insurance Retention function Deductible Heterogeneous beliefs Monotone likelihood ratio Monotone hazard ratio

a b s t r a c t We re-examine the problem of budget-constrained demand for insurance indemnification when the insured and the insurer disagree about the likelihoods associated with the realizations of the insurable loss. For ease of comparison with the classical literature, we adopt the original setting of Arrow (1971), but allow for divergence in beliefs between the insurer and the insured; and in particular for singularity between these beliefs, that is, disagreement about zero-probability events. We do not impose the no sabotage condition on admissible indemnities. Instead, we impose a state-verification cost that the insurer can incur in order to verify the loss severity, which rules out ex post moral hazard issues that could otherwise arise from possible misreporting of the loss by the insured. Under a mild consistency requirement between these beliefs that is weaker than the Monotone Likelihood Ratio (MLR) condition, we characterize the optimal indemnity and show that it has a simple two-part structure: full insurance on an event to which the insurer assigns zero probability, and a variable deductible on the complement of this event, which depends on the state of the world through a likelihood ratio. The latter is obtained from a Lebesgue decomposition of the insured’s belief with respect to the insurer’s belief. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Arrow’s (1971) analysis of the optimal insurance coverage under uncertainty is foundational, and it is entirely rooted in the von Neumann and Morgenstern (vNM) approach to decision-making under uncertainty: A risk-averse Expected-Utility (EU) maximizing decision maker (DM) is subject to a random but insurable loss, which he can partially transfer to an insurer in return for a premium payment. Arrow shows that if the insurer is a risk-neutral EU-maximizer, if the premium principle depends on the actuarial value of the indemnity, and if the two parties have the same beliefs about the realizations of the random loss, then full insurance above a constant deductible maximizes the DM’s expected utility of wealth, subject to a premium constraint. This result, which came to be known as Arrow’s Theorem of the Deductible, was subsequently extended in various ways, and the vast majority of the literature maintained the assumption of belief homogeneity between the DM and the insurer. However, one can argue that belief heterogeneity is pervasive in insurance markets. Indeed, ✩ I am grateful to Tim Boonen, Yichun Chi, and three anonymous reviewers for comments and suggestions. Financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC) is gratefully acknowledged (Grant No. 2018-03961). E-mail address: [email protected]. https://doi.org/10.1016/j.insmatheco.2019.09.002 0167-6687/© 2019 Elsevier B.V. All rights reserved.

on a theoretical level, in the Subjective Expected Utility (SEU) theory of De Finetti and Savage, disagreements about (subjective) beliefs follow from divergence in preferences over alternatives. Moreover, in game theory, disagreement about (posterior) beliefs can be a direct consequence of relaxing the controversial and heavily criticized Common Priors Assumption (Morris, 1995). Furthermore, on a practical level, primary and secondary insurance markets can display belief heterogeneity for various reasons, and we refer to Ghossoub (2017) for a more detailed discussion. Despite the pervasiveness of belief heterogeneity in insurance markets and the importance of accounting for divergence of beliefs in insurance contract design, little attention has been given to the study of optimal insurance design when the insurer and the DM entertain different subjective beliefs over a common state space. We refer to Ghossoub (2017) for a review of the relevant literature, and we only mention here the part of the literature that is most directly related to the present work. Specifically, Ghossoub (2017) shows that if the two parties have different subjective beliefs that satisfy a certain compatibility condition that is weaker than the Monotone Likelihood Ratio (MLR) condition, then optimal indemnity schedules exist and are nondecreasing in the loss. However, Ghossoub (2017) only gives a characterization of these optimal indemnity schedules in the special case of a MLR: he shows that, in that case, the optimal indemnity schedule is a variable deductible schedule, with

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M. Ghossoub / Insurance: Mathematics and Economics 89 (2019) 79–91

a state-contingent deductible that depends on the state of the world only through the likelihood ratio. Arrow’s classical result is then obtained as a special case. Ghossoub (2016) extends the analysis done in Ghossoub (2017) by allowing for disagreement about zero-probability events. Under a condition of compatibility between the two beliefs, Ghossoub (2016) fully characterizes the class of all optimal indemnity schedules that are nondecreasing in the loss, in terms of their distribution under the DM’s probability measure, and he obtains Arrow’s classical result as well as one of the results of Ghossoub (2017) as corollaries. However, Ghossoub (2016) does not provide a closed-form characterization of the optimal indemnity in the general case, which we do in this paper. Chi (2019) considers a similar setting to Ghossoub (2016, 2017) but imposes the no sabotage condition. That is, he restricts the set of admissible indemnities to those that are such that the indemnity and the retention function are both nondecreasing functions of the loss. Under an assumption of a Monotone Hazard Ratio (MHR), which is weaker than the MLR assumption, he shows optimality of a linear deductible schedule. Boonen (2016) provides an implicit characterization of the optimal indemnity that relies on the hazard ratio, similarly to Chi (2019). Amarante and Ghossoub (2016) and Amarante et al. (2015) consider belief heterogeneity with more general premium principles than the expected value premium principle used in Arrow’s model and in Ghossoub (2016, 2017). Amarante et al. (2015) examine the case of ambiguity on the side of the insurer, where the insurer’s preferences admit a Choquet-Expected Utility (CEU) representation, as in Schmeidler (1989). They show that if the ambiguous beliefs of the insurer are compatible with the DM’s non-ambiguous beliefs in a certain sense, then optimal indemnity schedules exist and are monotonic. In addition, in the case where the insurer is either ambiguity-seeking or ambiguity-averse, they show that the problem of determining the optimal indemnity schedule reduces to that of solving an auxiliary problem that is simpler than the original one in that it does not involve ambiguity. Under additional assumptions, they provide an explicit characterization of the optimal indemnity schedule for the insured, and show how their results naturally extend the classical result of Arrow (1971) on the optimality of the deductible indemnity schedule. Because of the properties of the Choquet integral, in the case where the insurer is a CEU-maximizer as in Schmeidler (1989), a problem of insurance design given a premium constraint is not equivalent to a problem of insurance design given a minimum expected retention constraint, where the expectation is in the sense of Choquet, that is, it is an ambiguous (or non-additive) expectation. The latter problem is examined by Amarante and Ghossoub (2016) who show that when the insurer is ambiguityseeking in the sense of Schmeidler (1989), the optimal indemnity schedule is a state-contingent deductible schedule, in which the deductible depends on the state of the world only through the insurer’s distortion function. A related problem to the one examined in this paper is that of budget-constrained optimal reinsurance design, where an insurer seeks a reinsurance indemnification that minimizes the insurer’s risk measure, subject to a premium constraint. Recent work that examine situations in which the insurer’s risk measure and/or the reinsurer’s premium principle is a Choquet integral with respect to a distortion of a probability measure include, for example, Assa (2015), Balbás et al. (2009), Boonen (2016), Cai et al. (2017), Cheung and Lo (2017), Cheung et al. (2011), Cui et al. (2013), Lo (2017) and Zhuang et al. (2016). However, with the exception of Balbás et al. (2009) and Boonen (2016), the contracting parties distort the same underlying probability measure, even if they use different distortion functions. Therefore, using a quantile transformation approach, these problems can all be reduced to optimization problems with perfect belief homogeneity. Boonen

(2016) considers dual utilities (that is, CEU preferences with a distortion of a probability and a linear utility function) and, as in Chi (2019), he imposes a priori the no sabotage condition. He uses the marginal indemnification function approach of Assa (2015) to characterize the optimal indemnity in terms of its derivative. Boonen’s (2016) characterization of the optimal indemnity relies crucially on the hazard ratio, similarly to Chi (2019). In this paper, we re-examine Arrow’s model, but we allow for belief heterogeneity between the DM and the insurer. Specifically, we assume that both agents are SEU-maximizers, but with different subjective probability measures over the relevant state space. In particular, we allow for singularity between the beliefs, that is, disagreement about zero-probability events. We do not assume that the DM’s actions influence the realization of the random loss under consideration. Moreover, we do not impose the no sabotage condition on admissible indemnities, that is, we do not restrict the set of admissible indemnities to those that are such that the indemnity and the retention function are both nondecreasing functions of the loss. Instead, we impose a stateverification cost that the insurer can incur in order to verify the loss severity, which rules out ex post moral hazard issues that could otherwise arise from possible misreporting of the loss by the insured. Our main interest in this paper is the effect of the divergence in beliefs and the singularity between the beliefs on optimal indemnities. To achieve such a characterization we use a Lebesgue decomposition of the insured’s subjective belief P with respect to the insurer’s subjective belief Q . We write P = Pac + Ps , ac , and Ps ⊥ Q , that is, there where Pac ≪ Q , with h := dP dQ exists some event A such that Q is concentrated on A and Ps is concentrated on the complement of A. Hence, it is precisely the shape of optimal indemnities as a function of the likelihood ratio h and the singular event A that is this paper’s main focus. Our main result (Theorem 3.6) characterizes the shape of the optimal indemnity as a function of h and A, under a mild consistency requirement between the two probability measures that is weaker than the Monotone Likelihood Ratio (MLR) condition. Specifically, we show that optimal indemnity schedules have a simple and intuitive two-part structure: (i) full insurance on an event to which the insurer assigns zero probability; and, (ii) a variable deductible on the complement of this event, which depends on the state of the world through the likelihood ratio h. This variable deductible is increasing in the likelihood ratio h. When the likelihood ratio h itself displays some monotonicity with respect to the insurable random loss X , this translates into monotonicity of the optimal indemnity as a function of the loss X . This occurs, for instance, in the special cases of a MLR and a Constant Likelihood Ratio (CLR), which we examine. In each case, we provide a closed-form characterization of optimal indemnity schedules. However, this monotonicity with respect to X (which is the main concern of Ghossoub (2017) and Chi (2019)) is not our primary focus in this paper; and we allow ourselves this shift in focus precisely because of the insurer’s ability to verify the state of the world, that is, the loss severity. The rest of this paper is organized as follows. Section 2 presents the setting and basic framework for the problem examined in this paper. Section 3 examines the DM’s problem of demand for insurance indemnification, derives the optimal indemnity function, and examines the special cases of a MLR and a CLR. Section 4 concludes. Related analysis and most proofs are presented in the Appendices A–E. 2. Model setup 2.1. Uncertainty Uncertainty is represented by a nonempty set of states of the world S, equipped with a σ -algebra of events F . The insurable

M. Ghossoub / Insurance: Mathematics and Economics 89 (2019) 79–91

random loss facing the DM is represented by a random variable X on the measurable space (S , F ), taking values on the closed interval [0, M ], for some M ∈ R+ . We denote by Σ the σ -algebra σ {X } over S generated by X . The DM can purchase insurance against the loss X from an insurer, for a premium calculated according to a premium principle set by the insurer. The insurance coverage provides indemnification against the value X (s) of the loss X in each state of the world s ∈ S. Specifically, an indemnity function is a random variable Y = I (X ) on (S , Σ ), for some bounded, Borel-measurable map I : X (S ) → R that pays off the amount I (X (s)) ∈ R in state of world s ∈ S. As is commonly assumed in the literature (e.g., Cheung et al., 2015; Lo, 2017), the DM has a fixed total insurance budget of Π > 0. The insurer uses an expected value premium principle, based on the insurer’s beliefs about the realizations of X . Let B (Σ ) be the vector space of all bounded, R-valued, and Σ -measurable functions on (S , Σ ), and let B+ (Σ ) be its positive cone. By Doob’s measurability theorem (Aliprantis and Border, 2006, Theorem 4.41), for any Y ∈ B (Σ ) there exists a bounded, Borel-measurable map I : R → R such that Y = I ◦ X . Moreover, Y ∈ B+ (Σ ) if and only if the function I is nonnegative. Hence, by Doob’s measurability theorem, we will hereafter identify the collection of possible indemnity functions with B (Σ ). Definition 2.1. Two functions Y1 , Y2 ∈ B (Σ ) are said to be comonotonic (resp., anti-comonotonic) if

[

( )][

Y1 (s)− Y1 s′

( )]

Y2 (s)− Y2 s′

≥ 0 (resp., ≤ 0), for all s, s′ ∈ S .

For instance any Y ∈ B (Σ ) is comonotonic and anti-comonotonic with any c ∈ R. Moreover, if Y1 , Y2 ∈ B (Σ ), and if Y2 is of the form Y2 = I ◦ Y1 , for some Borel-measurable function I, then Y2 is comonotonic with Y1 if and only if the function I is nondecreasing. 2.2. DM’s preferences As in Ghossoub (2017), we assume that both the DM and the insurer have preferences over indemnity functions admitting a Subjective Expected-Utility (SEU) representation. The DM’s preferences induce a utility function u : R → R and a subjective probability measure P on (S , Σ ). Similarly, the insurer’s preferences induce a utility function v : R → R and a probability measure Q on (S , Σ ). For any V ∈ B (Σ ), we denote by FV ,Q the cumulative distribution function of V with respect to the probability measure Q , defined by FV ,Q (t ) := Q {s ∈ S : V (s) ≤ t } , ∀t ≥ 0,

(

)

and we denote by FV−,1Q (t ) the left-continuous inverse of the distribution function FV ,Q (that is, the quantile function of V w.r.t. Q ), defined by

{

}

FV−,1Q (t ) = inf z ∈ R+ : FV ,Q (z ) ≥ t , ∀t ∈ [0, 1] .

Assumption 2.2. The DM’s utility function u is strictly increasing, strictly concave, continuously differentiable, and satisfies lim

x→+0

u

(x) ≥ W0 − Π and

lim

( ′ )−1

x→+∞

u

1 The usual Inada conditions are lim u x→+0

W (s) := W0 − Π − X (s) + Y (s) , ∀s ∈ S . We make the standard assumption that DM is well-diversified so that the particular loss exposure X against which he is seeking an insurance coverage is sufficiently small. Such an assumption can also be interpreted as a limited liability constraint, since it guarantees a nonnegative terminal wealth for indemnity functions that pay no more than the value of the loss. Assumptions of limited liability are commonly used in the literature (e.g., Bernard et al., 2015; Cheung et al., 2015; Xu et al., 2018). Specifically, in our setting, we assume the following. Assumption 2.3. X ≤ W0 − Π , Q -a.s. 2.3. Insurer’s preferences The insurer’s preferences induce a utility function v : R → R and a probability measure Q on (S , Σ ). Similarly to the classical literature, we assume that the insurer is risk-neutral, that is, the utility function v is the identity function. We assume that, in each state of the world s ∈ S, the insurer is able to observe the realization X (s) of the insurable loss, by incurring a monitoring cost, or State-Verification Cost (SVC). We represent this SVC by a random variable C ∈ B+ (Σ ), given exogenously. In particular, C is σ {X }-measurable and can therefore be written in the form C = c (X ), for some nonnegative Borelmeasurable function c. This is in line with reality, since typically an insurer determines the level of indemnification based on his appraisal of the loss, rather than on the DM’s reported loss value. However, loss appraisal is costly, and this is what we call SVC here. We assume that c (0) = 0, but do not impose continuity of c. In particular, c could ( ) be discontinuous at 0, accounting for an initial fixed cost c 0+ > 0. Moreover, the insurer incurs an indemnification or administrative cost, related to the cost of handling the indemnity payment. Similarly to the literature, we assume that for a given indemnity function Y , this indemnification cost is a proportional cost of the form ρ Y , for a given loading factor ρ > 0 specified exogenously and a priori. Consequently, starting from an initial wealth W0Ins , the total state-continent wealth for the insurer after receiving the premium Π from the DM and committing to the indemnity schedule Y ∈ B (Σ ) is the random variable W Ins ∈ B (Σ ) given by W Ins := W0Ins + Π − (1 + ρ) Y − C . 2.4. Pareto-optimality

Definition 2.4. An indemnity function Y ∗ ∈ B (Σ ) is Paretooptimal (P.O.) if 0 ≤ Y ∗ ≤ X , Q -a.s., and there is no Y ∈ B (Σ ) such that 0 ≤ Y ≤ X , Q -a.s., and (1) u (W0 − Π − X + Y ) dP ≥ u (W0 − Π − X + Y ∗ ) dP, ∫ ( ∫ ( Ins ) (2) v W0Ins + Π − (1 + ρ) Y − C dQ ≥ v W0 + Π − (1 + ρ) Y ∗ − C ) dQ ,

∫

(x) ≤ 0.

Note that the limit conditions in Assumption 2.2 are milder than the usual Inada-type conditions typically used in the literature.1 ( ′ )−1

The DM has initial wealth W0 > Π and his total statecontingent wealth is the random variable W ∈ B (Σ ) defined by

We are interested in Pareto-optimal indemnity schedules. These are defined as follows.

As in Arrow’s (1971) framework, the DM is risk averse, such that his utility function u satisfies the following assumption.

( ′ )−1

81

(x) = +∞ and lim

x→+∞

0. These are stronger requirements than those in Assumption 2.2.

( ′ )−1 u

(x) =

∫

with at least one of the two inequalities above being strict. Clearly, by assumption of risk-neutrality of the insurer, condition (2) in Definition 2.4 is equivalent to

∫

∫ YdQ ≤

Y ∗ dQ .

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M. Ghossoub / Insurance: Mathematics and Economics 89 (2019) 79–91

Consider now the following class of budget-constrained optimal insurance problems, which is parameterized by κ ∈ [ ∫ ] 0, XdQ . Problem 2.5.

(Pκ )

{∫ sup

Y ∈B(Σ )

∫ Q -a.s.;

(

u W0 − Π − X + Y

)

dP : 0 ≤ Y ≤ X ,

Y dQ ≤ κ .

• P = Pac + Ps . • Pac ≪ Q (Pac is absolutely continuous with respect to Q ). That is, for all B ∈ Σ , Pac (B) = 0 whenever Q (B) = 0. • Ps ⊥ Q (Ps and Q are mutually singular). That is, there exists some A ∈ Σ such that Q (S \ A) = Ps (A) = 0, which then implies that Pac (S \ A) = 0 and Q (A) = Q (S ) = 1. Therefore, by the Radon–Nikodým Theorem (Aliprantis and Border, 2006, Theorem 13.20) there exists a Q -a.s. unique h ∈ ∫ L1 (S , Σ , Q ) such that h : S → [0, +∞) and Pac (C ) = C h dQ , for all C ∈ Σ . Moreover, note that h can be interpreted as a likelihood ratio: dPac h= , dQ and h > 0, Q -a.s., if and only if Q ≪ Pac (e.g., Bogachev, 2007, p. 179). Furthermore, for all Z ∈ B (Σ ),

∫ Z dP =

∫ Z dPac +

∫A A

∫ Z dPs =

∫ S \A Zh dQ +

=

∫ Zh dQ +

∫A Z dP =

S \A

Z dPs

∫ S \A Zh dQ +

Z dP ,

S \A

equality follows from the fact that ∫where the second-to-last ∫ Z dPs = S \A Z dP, since Pac (S \ A) = 0, and the last equality S \A follows from the fact that Q (S \ A) = 0. Consequently, for all Z ∈ B (Σ ), ∫ ∫ ∫ Z dP = Zh dQ = Zh dQ , A

3. Optimal indemnification 3.1. The DM’s demand problem

}

Remark 2.6. By Lebesgue’s Decomposition Theorem (Aliprantis and Border, 2006, Theorem 10.61) there exists a unique pair (Pac , Ps ) of (nonnegative) finite measures on (S , Σ ) such that:

∫

is equivalent to finding all P.O. insurance indemnity schedules. However, if P ⊥ Q (and so Q ≪ Pac does not hold), then there might exist solutions to Problem 2.5 that are not Pareto optimal. In this paper, we will examine a budget-constrained optimal insurance design problem in the class (Pκ ).

A

In what follows, the set A and the function h are fixed all throughout. The following result examines the relationship between P.O. indemnity schedules and solutions to budget-constrained optimal insurance problems of the form (Pκ ). Its proof is given in Appendix B. Boonen and Ghossoub (2019) provide a similar result in the context of bilateral risk-sharing with heterogeneous beliefs. Lemma 2.7. (1) If Y ∗ is ∫ P.O., then it is optimal for Problem 2.5 with parameter κ := Y ∗ dQ . [ ∫ ] (2) If Q ≪ Pac , then for any κ ∈ 0, XdQ , if Y ∗ solves ∗ Problem 2.5 with parameter κ , then[Y ∫is P.O.] (3) If Q ≪ Pac , then for any κ ∈ ∫0, XdQ , if Y ∗ solves Problem 2.5 with parameter κ , then Y ∗ dQ = κ . Note that the condition Q ≪ Pac , which is equivalent to h > 0, Q a.s., and to the mutual absolute continuity of Q and Pac (denoted by Q ∼ Pac ), is weaker than the condition Q ≪ P, and it still allows for singularity between P and Q . If Q ≪ Pac , [Lemma ∫ 2.7 ] implies that finding solutions to problem (Pκ ), for κ ∈ 0, XdQ ,

The DM’s problem is that of finding an indemnity function that maximizes his subjective expected utility of terminal wealth, subject to the premium constraint and the constraint that the indemnity does not exceed the total loss (indemnity constraint). Specifically, the DM’s problem is given by the following. Problem 3.1.

{∫ sup

Y ∈B(Σ )

(

u W0 − Π − X + Y

∫ Q -a.s.;

˜ := YdQ ≤ Π

)

dP : 0 ≤ Y ≤ X ,

Π−

∫

CdQ

1+ρ

}

.

˜ can also be interpreted in Note that the constraint YdQ ≤ Π terms of a premium principle. Specifically, if one assumes that the insurer uses the modified expected value premium principle ∫

Ξ (Y ) = (1 + ρ)

∫

∫ YdQ +

CdQ , ∀Y ∈ B (Σ ) ,

which accounts for the expected state verification cost, then a

∫constraint of the form Ξ (Y ) ≤ Π is equivalent to the constraint ˜ . Hence, the existence of the SVC modifies the YdQ ≤ Π premium principle in the problem formulation, and ultimately shifts the cost of verifying the state to the insured. Given that this problem is formulated as a budget-constrained optimal insurance problem, this means that, compared with the related literature, the SVC reduces the DM’s total insurance budget from 1Π to +ρ

˜= Π

∫ Π − CdP . 1+ρ

Letting R := X − Y be the retention random variable, the problem can now be restated as: Problem 3.2.

{∫ sup

R∈B(Σ )

(

)

u W0 − Π − R dP : 0 ≤ R ≤ X ,

∫ Q -a.s.;

∫ R dQ ≥ R0 :=

}

˜ . XdQ − Π

Clearly, R∗ is optimal for Problem 3.2 if and only if Y ∗ := X − R∗ is optimal for Problem 3.1. Therefore, we focus on solving Problem 3.2. ˜ ≤ we will assume that 0 ≤ Π ∫ To rule out trivial situations, ∫ XdQ (that is, 0 ≤ R0 ≤ XdQ ): Assumption 3.3.

∫

CdQ ≤ Π ≤

∫

CdQ + (1 + ρ)

∫

XdQ .

Indeed, if Π < CdQ , then no indemnity schedule ∫ would satisfy the CdQ + ∫ premium constraint. Moreover, if Π > (1 + ρ) XdQ , then it is easy to see that full insurance is always optimal for the DM, at that level of premium, since the DM’s utility function is strictly increasing.

∫

Assumption 3.4. The function h is a continuous random variable2 on the probability space (S , Σ , Q ) (i.e., Q ◦ h−1 is nonatomic). 2 This assumption can be dropped, but one would have to use the Distributional Transform approach of Rüschendorf (2009) (see also Föllmer and Schied, 2016, Exercise A.3.2). All the results of this paper would still hold, with adequate modifications.

M. Ghossoub / Insurance: Mathematics and Economics 89 (2019) 79–91

3.2. Optimal indemnities In this section, Theorem 3.6 provides a closed-form characterization of the optimal indemnification, for any level of belief divergence. We also examine the two extreme cases that correspond to maximal and minimal belief singularity of P with respect to Q : (i) Proposition 3.5 considers the case where P and Q are mutually singular (i.e., h ≡ 0); and, (ii) Proposition 3.8 examines the case where P is absolutely continuous with respect to Q . Subsequently, Proposition 3.13 examines the special case where P = Q (i.e., h ≡ 1), and Proposition 3.12 studies the case of a constant likelihood ratio. Proposition 3.5. Let A and h = dPac /dQ be as in Remark 2.6. If Assumptions 2.2, 2.3, and 3.3 hold, and if h ≡ 0, then P ∫⊥ Q , and for any R∗ ∈ B (Σ ) that satisfies 0 ≤ R∗ ≤ X , Q -a.s., and RdQ ≥ R0 , the indemnity function Y ∗ := X − R∗ 1A + X 1S \A

(

)

is optimal for Problem 3.1. In particular, P [Y ∗ = X ] = 1, and hence the optimal indemnity is full insurance, P-a.s. Hence, when the beliefs are mutually singular, an optimal indemnity for the DM provides full insurance over an event to which the DM assigns full probability but the insurer assigns zero probability. The proof of Proposition 3.5 is given in Appendix C. The following result gives a closed-form characterization of the optimal indemnification, for any level of belief divergence.

Remark 3.7. It is important to note that, on a conceptual level, our main interest in this paper is the divergence in beliefs and the singularity between the beliefs. Consequently, it is precisely the shape of optimal indemnities as a function of the likelihood ratio h and the singular event A that are this paper’s main focus. Theorem 3.6, which is this paper’s main result, characterizes the shape of the optimal indemnity as a function of h and A. While on the event S \ A, the optimal indemnity provides full insurance, on the event A the optimal indemnity provides a partial insurance coverage that is increasing in the likelihood ratio h. Of course, when the likelihood ratio itself displays some monotonicity with respect to the loss X , this translates into monotonicity of the optimal indemnity as a function of the loss X , as examined in Section 3.4. However, this monotonicity with respect to X (which is the main concern of Ghossoub (2017) and Chi (2019)) is not our primary focus in this paper; and we believe that we can allow ourselves this shift in focus precisely because of the insurer’s ability to verify the state of the world. As an immediate consequence of Theorem 3.6, Proposition 3.8 examines the case where P is absolutely continuous with respect to Q , an extreme case in which P displays no singularity with Q . Proposition 3.8. Suppose that Assumptions 2.2, 2.3, and 3.3 hold, dP and that P ≪ Q with ζ := dQ . Define the function φζ on [0, 1] by

φζ (t ) =

t

∫ 0

Theorem 3.6. Let A and h = dPac /dQ be as in Remark 2.6, and define the function φ on [0, 1] by

φ (t ) =

t

∫

Fh−,Q1

0

(1 − x) dx.

Fζ−,Q1 (1 − x) dx.

If X := FX−,1Q 1 − Fζ ,Q (ζ ) ≤ X , Q -a.s.,

)

(

R∗ := min X , d∗ (ζ )

]

[

) ( ˜ X := FX−,1Q 1 − Fh,Q (h) ≤ X , Q -a.s.,

(3.1)

is optimal for Problem 3.2, and hence the indemnity function

then the retention function

Y ∗ := X − R∗ = max X − d∗ (ζ ) , X − X

R∗ := min ˜ X , d∗ (h) 1A

is optimal for Problem 3.1, where:

[

]

is optimal for Problem 3.2, and hence the indemnity function Y ∗ := X − R∗ = max X − d∗ (h) , X − ˜ X 1A + X 1S \A

[

]

is optimal for Problem 3.1, where:

{

( ′ )−1 (

• d (h) := max 0, W0 − Π − u ∗

(3.2)

then the retention function

If Assumptions 2.2, 2.3, 3.3, and 3.4 hold, and if

[

83

} ( ( ))) λ v φ 1 − Fh,Q (h) , ∗ ′

a state-contingent deductible that depends on the state of the world only through the likelihood ratio h; • For all t ∈ [0, φ (1)], v (∫t ) = φ −1 (t ); ∫ ˜. • λ∗ is chosen such that R∗ dQ = R0 , that is, Y ∗ dQ = Π

{

If, moreover, Q ≪ Pac (i.e., h > 0, Q -a.s.), then d∗ = max 0, W0 −

} ( )−1 ( λ∗ ) Π − u′ and Y ∗ is P.O. h ∗ Note ∫that since ∫Q (S \[A) = 0, it follows ] that λ is such that ˜ = Y ∗ dQ = max X − d∗ (h) , X − ˜ Π X dQ . Note also that the optimal indemnity provides full insurance over the event S \ A. However, the insurer assigns zero probability to this event. The DM, on the other hand, assigns a nonzero measure to this event. Moreover, Theorem 3.6 does not require X to be a continuous random variable. Hence, the distribution of X can have mass points, in particular at 0. The proof of Theorem 3.6 is given in Appendix D.

]

} { ))) ( )−1 ( ∗ ′ ( ( , • d∗ (ζ ) := max 0, W0 −Π − u′ λ v φ 1 − Fζ ,Q (ζ ) a state-contingent deductible that depends on the state of the world only through the likelihood ratio ζ ; • For all t ∈ [0, 1], v (t ) ∫= φ −1 (t ); ∫ ˜. • λ∗ is chosen such that R∗ dQ = R0 , that is, Y ∗ dQ = Π

{

If, moreover, Q ≪ P (i.e., ζ > 0, Q -a.s.), then d∗ (ζ ) = max 0, W0 −

} ( ′ )−1 ( λ∗ )

Π− u

ζ

and Y ∗ is P.O.

Proposition 3.8 is a direct consequence of Theorem 3.6, and its proof is therefore omitted. Condition (3.2) is a special case of Condition (3.1). 3.3. Condition (3.1) Condition (3.1) is a consistency requirement between the beliefs P and Q that Proposition 3.9 shows to be weaker than a MLR condition, which is extensively used in economic theory. Indeed, since the seminal contribution of Milgrom (1981), it is well-known that the MLR property plays a significant role in the existence (and monotonicity) of solutions to various problems in economic theory, with information asymmetry.

84

M. Ghossoub / Insurance: Mathematics and Economics 89 (2019) 79–91

Specifically, since h = dPac /dQ is Σ -measurable, and Σ =

σ {X }, it follows that h = Ψ ◦ X,

(3.3)

for some Borel-measurable and Q ◦ X −1 -integrable map Ψ : X (S ) → [0, +∞). The following result states that when this likelihood ratio is monotone, then Condition (3.1) is satisfied. Proposition 3.9. If the function Ψ in Eq. (3.3) is decreasing (h is anti-comonotonic with X ), then Condition (3.1) is satisfied.

probability to smaller values of the loss. Condition (3.1) could also be interpreted in this light. Indeed, for any s0 ∈ S, Pac [h ≥ h (s0 )] = EQ h1[h≥h(s0 )]

[

= Q [h ≥ h (s0 )] EQ [h|h ≥ h (s0 )] . Consequently, Condition (3.1) implies that, for any s0 ∈ S, Q [X ≤ X (s0 )] ≥

= FX−,1Q

) (h) be the nonincreasing

(

Proof. := 1 − Fh,Q Q -equimeasurable rearrangement of X with respect to h (Appendix A.1). By Q -a.s. uniqueness of the rearrangement (Proposition A.1), since X and h are anti-comonotonic, it follows that X = ˜ X , Q -a.s. □ Let ˜ X

An important special case, in which h is anti-comonotonic with X , is the case of an Esscher premium principle, where the insurer’s probability measure Q is an Esscher transformation of the DM’s probability measure P, and the two probability measures are equivalent (e.g., Goovaerts and Laeven (2008)). Although no singularity between the measures exists in this case, this is an important special case, since the Esscher transform is a time-honoured tool that has a rich history in risk measurement (Denuit et al., 2006; Gerber and Goovaerts, 1981; Goovaerts and Laeven, 2008) and financial economics (Bühlmann et al., 1998; Embrechts, 2000; Gerber and Shiu, 1994, 1996), with a solid economic foundation (Buhlmann, 1980). Moreover, with ˜ X as in Condition (3.1), it follows that ˜ X and X are identically distributed under the probability measure Q . Hence, by Shaked and Shanthikumar (2007, Theorem 1.A.1), there exist random variables Xˆ 1 and Xˆ 2 , defined on the same probability space, such that Xˆ 1 , Xˆ 2 , and X (and ˜ X ) are identically distributed under the probability measure Q , and Xˆ 1 = Xˆ 2 , Q -a.s. In this sense, Condition (3.1) is relatively mild. Recall that for two random variables Z1 and Z2 , with distribution functions F1 and F2 , respectively, Z1 is larger than Z2 is the sense of first-order stochastic dominance, written as Z1 ≽ Z2 , FSD

if P [X ≤ t] ≤ Q [X ≤ t], for all t. Then one should note that the MLR and MHR conditions used by Ghossoub (2017) and Chi (2019), respectively, imply a strong form of first-order stochastic dominance. In fact, the following result follows from Shaked and Shanthikumar (2007, Eq. 1.B.7 & Theorem 1.C.5). Proposition 3.10. For any Borel set B ∈ B (R) and any probability measure µ on (S , Σ ), let [X |X ∈ B]µ denote the conditional distribution of X under µ, conditioning on the event [X ∈ B]. (1) If P and Q satisfy the MHR condition of Chi (2019) (that is, if Q [X >t] is decreasing in t), then [X |X > t]P ≽ [X |X > t]Q , for P[X >t] any t. In particular, X P ≽ X Q .

FSD

FSD

(2) If P and Q have respective densities f and g that satisfy the g(t) MLR condition of Ghossoub (2017) (that is, if f (t) is decreasing in t), then [X |a ≤ X ≤ b]P ≽ [X |a ≤ X ≤ b]Q , for all a ≤ b. In particular, X P ≽ X Q .

FSD

FSD

Hence, the assumptions used by Chi (2019) or Ghossoub (2017) imply that the insurer is more optimistic than the DM about the realizations of the loss, in the sense of assigning a higher

Pac [h ≥ h (s0 )]

EQ [h|h ≥ h (s0 )] P [h ≥ h (s0 )] − Ps [h ≥ h (s0 )] EQ [h|h ≥ h (s0 )]

.

(3.4)

In other words, Condition (3.1) implies Eq. (3.4), which gives a lower bound on the insurer’s assessment of the likelihood of the event [X ≤ X (s0 )], for any state of the world s0 ∈ S. 3.4. Monotonicity of optimal indemnities Nothing can be inferred in general from Theorem 3.6 about the monotonicity of R∗ , and hence of Y ∗ , with respect to X . However, as noted, the insurer can verify the loss by incurring a state-verification cost. Nevertheless, in a situation of monotone likelihood ratio, we can obtain a monotonicity property of the optimal indemnity. We refer to Ghossoub (2017) and Chi (2019) for necessary conditions for optimal indemnities to be nondecreasing in the loss. Ghossoub (2017) showed that a monotone likelihood ratio yields optimality of a variable deductible indemnity schedule, in which the deductible depends on the state of the world only through the likelihood ratio, hence recovering Arrow’s theorem as a special case. However, Ghossoub’s (2017) setting does not allow for possible singularity between the measures (i.e., no disagreement about zero-probability events). Here, we further extend this result to allow for such singularities and disagreements about zero-probability events. Corollary 3.11 (MLR). Let A and h = dPac /dQ be as in Remark 2.6, and define the function φ on [0, 1] by

φ (t ) =

t

∫ 0

FSD

if F1 (t ) ≤ F2 (t ), for all t. Similarly, we will say that X P ≽ X Q

]

Fh−,Q1 (1 − x) dx.

If Assumptions 2.2, 2.3, 3.3, and 3.4 hold, and if it is further assumed that h is anti-comonotonic with X (i.e., the likelihood ratio Ψ in Eq. (3.3) is a nonincreasing function), then the function

[

]

Y ∗ := max X − d∗ (h), 0 1A + X 1S \A is optimal for Problem 3.1, where:

} ( ′ )−1 ( ∗ ′ ( ( ))) • d (h) := max 0, W0 − Π − u λ v φ 1 − Fh,Q (h) , ∗

{

a state-contingent deductible that depends on the state of the world only through the likelihood ratio h; • For all t ∈ [0, φ (1)], v (∫t ) = φ −1 (t ); ∫ ˜. • λ∗ is chosen such that R∗ dQ = R0 , that is, Y ∗ dQ = Π

≪ Pac (i.e., h} > 0, Q -a.s.), then d∗ (h) := [ ( )−1 ( λ∗ ) . In particular, Y ∗ = max X − max 0, W0 − Π − u′ h ] d∗ (h) , 0 , Q -a.s., a variable deductible contract, and Y ∗ is P.O. If, moreover, Q {

Proof. Follows from ) Theorem 3.6, since in this case X = ( ˜ X = FX−,1Q 1 − Fh,Q (h) , Q -a.s., by the Q -a.s. uniqueness of the

nonincreasing rearrangement of X with respect to h (Proposition A.1). □

M. Ghossoub / Insurance: Mathematics and Economics 89 (2019) 79–91

As a special case of MLR, we obtain the following result, which characterizes the optimal indemnity in case of a constant likelihood ratio (CLR), hence not requiring nonatomicity of Q ◦h−1 . The proof of Proposition 3.12 is given in Appendix E. Proposition 3.12 (CLR). Let A and h = dPac /dQ be as in Remark 2.6. If Assumptions 2.2, 2.3, and 3.3 hold, and if h is a constant other than 0, then the function Y ∗ := max X − d∗ , 0 1A + X 1S \A

(

)

(3.5)

is optimal for Problem 3.1, where d∗ ∈ R+ is chosen such that

∫

˜= Y ∗ dQ = Π

∫

max X − d∗ , 0 dQ .

[

]

In particular, Y ∗ = max (X − d∗ , 0) , Q -a.s., a linear deductible contract, and Y ∗ is P.O. As a consequence of Proposition 3.12, we obtain the following result, the proof of which is omitted. This result states that, even in the presence of a state-verification cost, if the beliefs of the two parties coincide, then a linear deductible is optimal, as in Arrow’s Theorem (Arrow, 1971). Proposition 3.13. Let A and h = dPac /dQ be as in Remark 2.6. If Assumptions 2.2, 2.3, and 3.3 hold, and if h ≡ 1, then P = Q , and a linear deductible indemnity function of the form Y ∗ := max ∫(X − d∗ , 0) is optimal for Problem 3.1, where d∗ is chosen such ˜ . Moreover, Y ∗ is P.O. that Y ∗ dQ = Π 4. Conclusion In this paper, we introduced belief heterogeneity in Arrow’s classical model of demand for insurance indemnification. Unlike parts of the literature, we did not impose the no sabotage condition on admissible indemnities, but assumed instead that the insurer can incur a state-verification cost in order to verify the loss severity, thereby ruling out ex post moral hazard issues that could otherwise arise from possible misreporting of the loss by the insured. Under mild a consistency requirement between the probability measures of the DM and the insurer that is weaker than the Monotone Likelihood Ratio (MLR) condition, we characterized the optimal indemnity and we showed that it has a simple two-part structure, consisting of (i) full insurance on an event to which the insurer assigns zero probability; and, (ii) a variable deductible on the complement of this event. We also examined the special cases of a MLR and a Constant Likelihood Ratio (CLR), and we gave a closed-form characterization of the optimal indemnity in each case. Appendix A. Equimeasurable rearrangements and supermodularity The classical theory of monotone equimeasurable rearrangements of Borel-measurable functions on R dates back to the work of Hardy, Littlewood and Pólya, who gave the first integral inequalities involving functions and their rearrangements. Here, the idea of an equimeasurable rearrangement of a random variable with respect to an other random variable is discussed. All of the results in this Appendix are taken from Ghossoub (2015) and references therein, to which we refer for proofs, additional results, and additional references.

85

A.1. The rearrangement Let (S , G , µ) be a probability space and let V ∈ L∞ (S , G , µ) be a continuous random variable (i.e., µ ◦ V −1 is nonatomic) with range V (S ) ⊂ R+ . ( ) For each Z ∈ L∞ (S , G , µ), let FZ ,µ (t ) = µ {s ∈ S : Z (s) ≤ t } denote the cumulative distribution function of Z with respect to 1 the probability measure µ, and let FZ−,µ (t ) be the left-continuous inverse of the distribution function FZ ,µ (that is, the quantile function of Z w.r.t.µ). Proposition A.1. For any Y ∈ L∞ (S , G , µ), define ˜ Yµ and Y µ as follows: 1 Y µ = FY−,µ FV ,µ (V )

(

)

and

( ) 1 ˜ Yµ = FY−,µ 1 − FV ,µ (V ) .

Then, (i) (ii) (iii) (iv)

Y, ˜ Yµ , and Y µ have the same distribution under µ. Y µ is comonotonic with V . ˜ Yµ is anti-comonotonic with V . For each X ∈ L∞((S , G , µ), If )0 ≤ Y ≤ X , µ-a.s., then 1 0 ≤ ˜ Yµ ≤ FX−,µ 1 − FV ,µ (V ) , µ-a.s., and 0 ≤ Y µ ≤

1 FX−,µ FV ,µ (V ) , µ-a.s. In particular, for any N ∈ R+ , if 0 ≤ Y ≤ N , µ-a.s. then 0 ≤˜ Yµ ≤ N , µ.a.s., and 0 ≤ Y µ ≤ N , µ.a.s. (v) If Z ∗ is any other element of L∞ (S , G , µ) that has the same distribution as Y under µ and that is comonotonic with V , then Z ∗ = Y µ , µ-a.s. (vi) If Z ∗∗ is any other element of L∞ (S , G , µ) that has the same distribution as Y under µ and that is anti-comonotonic with V , then Z ∗∗ = ˜ Yµ , µ-a.s.

(

)

˜ Yµ is called the nonincreasing µ-rearrangement of Y with respect to V , and Y µ is called the nondecreasing µ-rearrangement of Y with respect to V . Since µ ◦ V −1 is nonatomic, it follows that FV ,µ (V ) has a uniform distribution over (0, 1) (Föllmer and Schied, 2016, Lemma A.25). Letting U := FV ,µ (V ), it follows that U is a random variable on the probability space (S , Σ , µ) with a uniform distribution on 1 (0, 1) and that V = FV−,µ (U ) , µ-a.s., that is, V µ = V , µ-a.s. A.2. Supermodularity and Hardy–Littlewood–Pólya Inequalities A partially ordered set (poset) is a pair (T , ≽) where ≽ is a reflexive, transitive and antisymmetric binary relation on T . For any x, y ∈ S denote by x ∨ y (resp. x ∧ y) the least upper bound, or supremum (resp. greatest lower bound, or infimum) of the set {x, y}. A poset (T , ≽) is called a lattice when x ∨ y, x ∧ y ∈ T , for each x, y ∈ T . For instance, the Euclidian space Rn is a lattice for the partial order ≽ defined as follows: for x = (x1 , . . . , xn ) ∈ Rn and y = (y1 , . . . , yn ) ∈ Rn , write x ≽ y when xi ≥ yi , for each i = 1, . . . , n. It is then easy to see that x ∨ y = (max (x1 , y1 ) , . . . , max (xn , yn )) and x ∧ y = (min (x1 , y1 ) , . . . , min (xn , yn )). Definition A.2. Let (T , ≽) be a lattice. A function L : T → R is said to be supermodular if for each x, y ∈ T , L (x ∨ y) + L (x ∧ y) ≥ L (x) + L (y) .

(A.1)

2

In particular, a function L : R → R is supermodular if for any x1 , x2 , y1 , y2 ∈ R with x1 ≤ x2 and y1 ≤ y2 , one has L (x2 , y2 ) + L (x1 , y1 ) ≥ L (x1 , y2 ) + L (x2 , y1 ) .

(A.2)

Eq. (A.2) then implies that a function L : R2 → R is supermodular if and only if the function η (y) := L (x + h, y) − L (x, y) is nondecreasing on R, for any x ∈ R and h ≥ 0.

86

M. Ghossoub / Insurance: Mathematics and Economics 89 (2019) 79–91

( ) ¯ ε . Then Y¯ ≤ ˜ ˜ ∫min X , Y + ∫ ∫ Y∗, 0 ≤ Y ≤ X , Q -a.s., and ¯ ˜ Y dQ ≤ Y dQ + ε = Y dQ . Hence, ˜ Y is feasible for Problem 2.5 with parameter ∫ κ. ∫ ∫ If Y¯ = X , Q -a.s., then( Y¯ dQ) = XdQ ≥ ( Y ∗ dQ), a 3 ¯ ˜ contradiction. Hence, Q) Y¯ < X ( > 0 )and so Q Y < Y > ( ¯ ¯ ˜ ˜ 0. Therefore, P Y < Y ≥ Pac Y < Y > 0, since Q ≪ Pac .

Example A.3. The following are supermodular functions: (1) If g : R → R is concave and a ∈ R, then the function L1 : R2 → R defined by L1 (x, y) = g (a − x + y) is supermodular. Moreover, if g is strictly concave, then L1 is strictly supermodular. (2) If h : R → R is concave, a ∈ R, and ψ : R → R is increasing and nonnegative, then the function L2 : R2 → R defined by L2 (x, y) = h (a − x + y) ψ (x) is supermodular. (3) If ψ, φ : R → R are both nonincreasing or both nondecreasing functions, then the function L3 : R2 → R defined by L3 (x, y) = φ (x) ψ (y) is supermodular.

Hence,

∫

u W0 − Π − X + ˜ Y dP

(

∫ [˜ Y =Y¯ ]

∫

>

Lemma A.5 (Hardy–Littlewood–Pólya Inequality). Let Y ∈ L∞ (S , G , µ), let ˜ Yµ be the nonincreasing µ-rearrangement of Y with respect to V . If L is supermodular then L V,˜ Yµ

(

)

dµ ≤

∫

(

L V, Y

)

dµ ≤

∫

L V, Yµ

)

∫

(1) Suppose that Y ∗ is P.O. Then, in particular, it is feasible for ∫ Problem 2.5 with parameter κ := Y ∗ dQ . Suppose that Y ∗ is not optimal for Problem 2.5 with parameter κ . Let Y¯ ∈ B (Σ ) be optimal for Problem 2.5 with parameter κ . Then 0 ≤ Y¯ ≤ X , Q -a.s., and u W0 − Π − X + Y¯ dP >

(

)

∫

u W0 − Π − X + Y ∗ dP .

(

Moreover, since Y¯ dQ ≤ κ = identity function, it follows that

∫

∫

)

∫

∫

(

) u W0 − Π − X + Y¯ dP = (

∫

∗

u W0 − Π − X + Y ∗ dP .

(

)

(

)

u W0 − Π − X + Y¯ dP

(

)

u W0 − Π − X + Y¯ dP

(

∫

with at least one of the two inequalities above being strict. In particular, since v is the identity function, (b) implies ∫ ∫ that Y¯ dQ ≤ Y ∗ dQ , and so Y¯ is feasible for Problem 2.5 with parameter κ . If the inequality in (a) is strict, then this contradicts the optimality of Y ∗ for Problem 2.5 with parameter κ . Hence

)

∫ that the inequality in (b) is strict. Then Y¯ dQ < ∫Assume ∫ ∫ Y ∗ dQ . Let ε := Y ∗ dQ − Y¯ dQ > 0, and ˜ Y :=

)

(

[Y ∗ =Y¯ ]

>

+ ∫

[Y ∗
u W0 − Π − X + Y¯ dP

)

u W0 − Π − X + Y ∗ dP

(

∫

=

)

(

[Y ∗ =Y¯ ]

+ ∫

∫

)

u W0 − Π − X + Y¯ dP

u W0 − Π − X + Y¯ dP

=

u W0 − Π − X + Y¯ dP ≥ u (W0 − Π − X ∗ + Y dP, ) ∫ ( ∫ ( Ins ) (b) v W0Ins + Π − (1 + ρ) Y¯ − C dQ ≥ v W0 ∗ +Π − (1 + ρ) Y − C ) dQ , (a)

∫

)

(

[˜ Y >Y¯ ]

∫ ( ) v W0Ins + Π − (1 + ρ) Y¯ − C dQ ∫ ( ) ≥ v W0Ins + Π − (1 + ρ) Y ∗ − C dQ ,

contradicting the Pareto-optimality of Y . Hence, Y is ∫ optimal for Problem 2.5 with parameter κ = Y ∗ dQ . [ ∫ ] (2) Suppose that Q ≪ Pac . Fix κ ∈ 0, XdQ , and suppose that Y ∗ solves Problem 2.5 with parameter κ . Suppose that Y ∗ is not P.O. Then there exists Y¯ ∈ B (Σ ) such that 0 ≤ Y¯ ≤ X , Q -a.s., and

∫

(

contradicting the optimality of Y ∗ for Problem 2.5 with parameter κ . Hence Y ∗ is P.O. [ ∫ ] (3) Suppose that Q ≪ Pac . Fix κ ∈ 0, XdQ , and suppose that ∫Y ∗ solves Problem 2.5 with parameter ∫ κ . Suppose that Y ∗ dQ < κ , and let ε := κ − Y ∗ dQ > 0. ∗ Let Y¯ := min . Then Y ∗ ≤ Y¯ , 0 ≤ Y¯ ≤ X , ∫ (X , Y ∫+ ε) ∗ ¯ Q -a.s., and Y dQ ≤ Y dQ + ε = κ . Hence Y¯ is feasible ∗ for Problem ∫ ∗ 2.5 ∫with parameter∫ κ . ∗If Y = X , Q -a.s., then Y dQ = XdQ ≥ κ > ( Y dQ ,) a contradiction. ∗ ∗ ¯ Hence ( ∗ Q ()Y < X )( >∗ 0 and ) so Q Y < Y > 0. Therefore, P Y < Y¯ ≥ Pac Y < Y¯ > 0, since Q ≪ Pac . Hence,

Y ∗ dQ and v is the

∗

)

u W0 − Π − X + Y ∗ dP ,

=

Appendix B. Proof of Lemma 2.7

u W0 − Π − X + ˜ Y dP

u W0 − Π − X + Y¯ dP

∫

dµ,

)

(

[˜ Y >Y¯ ]

[˜ Y =Y¯ ]

+ ∫

provided the integrals exist.

∫

+ ∫

=

(

u W0 − Π − X + ˜ Y dP

(

=

Remark A.4. Note that, as observed by Carlier and Dana (2005), a function L ∈ C 2 (R × R, R) is supermodular if and only if ∂ 2 L(x,y) ≥ 0. ∂x ∂y

∫

)

)

u W0 − Π − X + Y ∗ dP

(

[Y ∗
)

u W0 − Π − X + Y ∗ dP ,

(

)

contradicting the optimality of Y ∗ for Problem 2.5 with ∫ parameter κ . Hence, Y ∗ dP = κ . Appendix C. Proof of Proposition 3.5 Denote by FSB the feasibility set for Problem 3.2:

{ FSB =

R ∈ B (Σ ) : 0 ≤ R ≤ X , Q -a.s., and

∫

} RdQ ≥ R0 , (C.1)

3 It is immediate to verify that {s ∈ S : Y¯ (s) = X (s)} ⊆ {s ∈ S : Y¯ (s) = ˜ Y (s)}. Now, if s ∈ S is such that Y¯ (s) = ˜ Y (s) but Y¯ (s) < X (s), then X (s) > ˜ Y (s) = min{X (s), Y¯ (s) + ε}, and hence ˜ Y (s) = Y¯ (s) + ε > Y¯ (s), a contradiction. Consequently, {s ∈ S : Y¯ (s) = ˜ Y (s)} ⊆ {s ∈ S : Y¯ (s) = X (s)}, and hence {s ∈ S : Y¯ (s) = ˜ Y (s)} = {s ∈ S : Y¯ (s) = X (s)}. Therefore, {s ∈ S : Y¯ (s) < ˜ Y (s)} = {s ∈ S : Y¯ (s) < X (s)}.

M. Ghossoub / Insurance: Mathematics and Economics 89 (2019) 79–91 = and let FSB be defined as

{ =

FSB =

R ∈ B (Σ ) : 0 ≤ R ≤ X , Q -a.s., and

To show optimality of R∗ for Problem 3.2, let Z be any other feasible function for Problem 3.2. Then Z is feasible for Problem C.2, and hence the optimality of R∗1 for Problem C.2 yields

}

∫

RdQ = R0 .

∫

u W0 − Π − R∗ dP =

)

(

Z ∈ B (Σ ) is such that Z = ˜ Z , Q -a.s., then (If )˜ ˜ Z dP, for any function f such that the integrals

u W0 − Π − R∗1 dP

)

(

≥

=

Assumption 3.3 then implies that FSB ̸ = ∅ and hence FSB ̸ = ∅. Let A and h be as in Remark 2.6. Then the following result follows immediately.

∫ ∫A

A

(C.2)

Proposition C.1. ∫ ∫ f Z dP = f ( ) A A are defined.

87

u W0 − Π − Z dP .

(

)

A

Moreover, 0 ≤ ∫ Z 1S \A ≤ X 1S \A , Q -a.s., since 0 ≤ Z ≤ X , Q -a.s.. Additionally, S \A Z dQ = 0, since Q (A) = 1. Hence, Z is feasible for Problem C.3. Thus, by Lemma C.5,

(

)

u W0 − Π P (S \ A) =

≥

Now, consider the following two problems:

∫

(

u W0 − Π

∫S \A (

)

dP

u W0 − Π − Z

)

dP .

S \A

Problem C.2.

{∫

∫

)

u W0 − Π − R dP : 0 ≤ R ≤ X ,

sup

R∈B(Σ )

Therefore,

(

u W0 − Π − R

∗

(

)

+ ∫

R dQ ≥ R0 .

{∫

=

u W0 − Π − R dP : 0 ≤ R1S \A ≤ X 1S \A ,

}

∫

R dQ = 0 .

Q -a.s.; S \A

Remark C.4. Since the function u is continuous (Assumption 2.2), it is bounded on any closed and bounded subset of R. Therefore, since the range of X is closed and bounded, the supremum of each of the above two problems is finite when their feasibility sets are nonempty. Now, the constant function R = 0 is feasible for Problem C.3, and so Problem C.3 has a nonempty feasibility set. Moreover, the constant function R0 is feasible for Problem C.2, by Assumption 3.3, and so Problem C.2 has a nonempty feasibility set. Lemma C.5. The constant function R∗ := 0 is optimal for Problem C.3. Proof. The feasibility of R∗ := 0 for Problem C.3 is clear. To show optimality, let R be any feasible solution for Problem C.3. Then for Q -a.a. s ∈ S \ A, 0 ≤ R (s) ≤ X (s). Therefore, since u is increasing, we have for each s ∈ S \ A, u (W0 − Π − R (s)) ≤ u (W0 − Π ) = u (W0 − Π − R∗ (s)). Thus, u (W0 − Π − R) dP ≤

∫

S \A

u W0 − Π − R

(

∗

)

dP

S \A

= u (W0 − Π ) P (S \ A) . □ Lemma C.6. If R∗1 is optimal for Problem C.2 then R∗ := R∗1 1A is optimal for Problem 3.2. Proof. By the feasibility of R∗1 for Problem C.2, we have 0 ≤ R∗1 ≤ ∫ ∗ X , Q -a.s., and R1 dP ≥ R0 . Therefore, 0 ≤ R∗ ≤ X , Q -a.s., and since Q (A) = 1, we have R dQ =

(

)

S \A

u W0 − Π − R∗1 dP

(

)

( ) + u W0 − Π P (S \ A) ∫ ( ) ≥ u W0 − Π − Z dP A ∫ ( ) + u W0 − Π − Z dP ∫ S(\A ) = u W0 − Π − Z dP .

)

S \A

∫

u W0 − Π − R∗ dP

A

(

sup

∗

)

(

∫

Problem C.3.

∫

u W0 − Π − R∗ dP

A

}

Q -a.s.;

∫

dP =

A

∫

R∈B(Σ )

∫

∗

∫

R1 1A dQ =

∗

R1 dQ = A

Hence, R∗ is feasible for Problem 3.2.

∫

Consequently, R∗ is optimal for Problem 3.2.

□

Now, if h ≡ 0 then Pac ≡ 0, and so P = Ps ⊥ Q . In this case, P (A) = 0 and Q (A) = 1. Hence, any feasible solution for Problem C.2 is optimal for Problem C.2. Choose any R∗ ∈ FSB . By Lemma C.6, the function R∗ 1A is optimal for Problem 3.2, and hence the function Y ∗ := X − R∗ 1A = X − R∗ 1A + X 1S \A

(

)

is optimal for Problem 3.1. This concludes the proof of Proposition 3.5. □ Appendix D. Proof of Theorem 3.6 This proof builds upon the proof of Proposition 3.5. By assumption, P and Q are not mutually singular, and therefore h is not the constant function equal to 0. Hence, it follows form Remark 2.6 that we can rewrite Problem C.2 as the following problem. Problem D.1.

{∫ sup

R∈B(Σ )

(

∫ Q -a.s.;

)

u W0 − Π − R h dQ : 0 ≤ R ≤ X ,

}

R dQ ≥ R0 .

Now, for each Z ∈ B+ (Σ ), let FZ ,Q (t ) = Q {s ∈ S : Z (s) ≤ t } denote the cumulative distribution function of Z with respect to the probability measure Q , and let FZ−,Q1 (t ) denote the quantile function of Z w.r.t. Q .

(

R1 dQ ≥ R0 . ∗

)

88

M. Ghossoub / Insurance: Mathematics and Economics 89 (2019) 79–91

Lemma D.2. For each R ∈ FSB , let ˜ R := FR−,Q1 1 − Fh,Q (h) . Then ˜ ˜ R ∈ FSB , R is anti-comonotonic with h, F˜ R = FR , and

(

∫

(

)

u W0 − Π − ˜ R h dQ ≥

∫

(

)

)

u W0 − Π − R h dQ .

∫

∫

(

)

u W0 − Π − ˜ R h dQ =

∫

L h, ˜ R dQ ≤

(

∫ =−

)

∫

∫

∫ ∫

Q = f ∈ Q : 0 ≤ f (t ) ≤ ∗

FX−,1Q

}

(t ) , for each 0 < t < 1 .

(D.1)

In light of Lemma D.2, we can focus on looking for solutions to Problem D.1 of the form f (1 − U ), where f ∈ Q∗ and U = Fh,Q (h) (since 0 ≤ R ≤ X , Q -a.s., whenever 0 ≤ FR−,Q1 (t ) ≤ FX−,1Q (t ), for 0 < t < 1). Now, let R ∈ B (Σ ) be of the form R = f (1 − U ), where U = Fh,Q (h) and f ∈ Q∗ . Then f = FR−,Q1 and 0 ≤ R ≤ X , Q -a.s., by Eq. (3.1). Since U is uniformly distributed and h = Fh−,Q1 (U ) , Q -a.s., we have

∫

(

)

u W0 − Π − R dP =

∫

(

1

= 0

∫

Moreover, RdQ = following problem:

∫1 0

(

1

u W0 − Π − f (t )

sup f ∈Q∗

(

0

1

∫

u (W0 − Π − f (t )) dφ (t ) .

= 0

Letting v (t ) := φ −1 (t ), for t ∈ [0, φ (1)], and using the change of variable z = v −1 (t ) = φ (t), we obtain 1

∫

u (W0 − Π − f (t )) Fh−,Q1 (1 − t ) dt

0

1

∫

u (W0 − Π − f (t )) dv −1 (t )

= 0

φ (1)

=

) u W0 − Π − f (t ) Fh−,Q1 (1 − t ) dt . 0

f (t ) dt. Consider the

u (W0 − Π − f (v (z ))) dz

0

φ (1)

∫

u (W0 − Π − q (t )) dt ,

0

where the function q is defined by q (t ) = f (v (t )) . Moreover, 1

∫

f (t ) dt = 0

φ (1)

∫

f (v (z )) dv (z ) =

0

φ (1)

∫

q (t ) v ′ (t ) dt .

0

Now, define the set Q∗∗ by

Problem D.3.

{∫

u (W0 − Π − f (t )) Fh−,Q1 (1 − t ) dt

0

∫

(

∫1

)

1

)

FR−,Q1 (t ) dt =

)

u W0 − Π − R∗ dP ,

)

u W0 − Π − f (1 − t ) Fh−,Q1 (t ) dt

0

∫

(

=

1

=

u W0 − Π − f ∗ (t ) Fh−,Q1 (1 − t ) dt

∫t

) u W0 − Π − f (1 − U ) Fh−,Q1 (U ) dQ

∫

1

∫0 (

(

=

)

Fix f ∈ Q∗ and define the function φ by φ (t ) = 0 Fh−,Q1 (1 − x) ∫ dx. Then φ (0) = 0 and φ (1) = hdQ = Pac (S) ≤ 1. Hence,

u W0 − Π − R h dQ

∫

(

where the first inequality follows from an argument similar to the one used in the proof of Lemma D.2, and the second inequality follows from the optimality of f ∗ for Problem D.3. Therefore, R∗ is optimal or Problem D.1. □

∫ {

u W0 − Π − FR−,Q1 (t ) Fh−,Q1 (1 − t ) dt

≤ =

)

1

∫

(

)

(

0

)

⏐ } ⏐ Q = f : (0, 1) → R ⏐ f is nondecreasing and left-continuous ,

f ∗ (t ) dt ≥ R0 ,

u W0 − Π − FR−,Q1 (1 − U ) Fh−,Q1 (U ) dQ

=

u W0 − Π − R h dQ ,

{

1

∫ 0

(

=

Let Q denote the collection of all quantile functions under the measure Q , and let Q∗ denote the collection of all quantile functions f that satisfy 0 ≤ f (t ) ≤ FX−,1Q (t ), for all t ∈ (0, 1). Then

f (U ) dQ = ∗

u W0 − Π − FR−,Q1 (1 − U ) h dQ

≤

∫

∫ ( ) ) u W0 − Π − ˜ R h dQ ≥ u W0 − Π − R h dQ . □

(U ) dQ =

∫

u (W0 − Π − R) h dQ

L (h, R) dQ

(

FR−∗1,Q

where the last inequality follows from the feasibility of f ∗ for Problem D.3. Hence, R∗ is feasible for Problem D.1. To show optimality of R∗ for Problem D.1, let R by any other feasible solution for Problem D.1 and FR−,Q1 its quantile function. Then FR−,Q1 is feasible for Problem D.3, and hence

s that is,

∫

∫

∗

R dQ =

Proof. Since Q ◦ h−1 is nonatomic (by Assumption 3.4), it follows that Fh,Q (h) has a uniform distribution ) ( over (0, 1) (Föllmer and Schied, 2016, Lemma A.25), that is, Q {s ∈ S : Fh,Q (h) (s) ≤ t } = t for each t ∈ (0, 1). Letting U := Fh,Q (h), it follows that U is a random variable on the probability space (S , Σ , Q ) with a uniform distribution on (0, 1) and that h = Fh−,Q1 (U ) , Q -a.s. Fix R ∈ FSB , and let ˜ R := FR−,Q1 (1 − U ) be the non-increasing rearrangement of R with respect to h. Then ˜ R ∈ FSB , by Proposition A.1 and Eq. (3.1). Moreover, since the function u is increasing, it follows that the map L : R × R → R defined by L (x, y) := −u (W0 − Π − y) x is supermodular (see Example A.3 and Remark A.4). Consequently, by Lemma A.5, it follows that

−

Proof. Let f ∗ be optimal for Problem D.3 and R∗ := f ∗ (1 − U ), with U = Fh,Q (h). Then, since f ∗ ∈ Q∗ , it follows from Eq. (3.1) that 0 ≤ R∗ ≤ X , Q -a.s. Moreover,

)

Fh−,Q1

(1 − t ) dt :

1

∫

}

f (t ) dt ≥ R0 . 0

Lemma)D.4. If f ∗ is optimal for Problem D.3, then R∗ := f ∗ (1− Fh,Q (h) is optimal for Problem D.1 and anti-comonotonic with h.

⏐ ⏐

{

Q∗∗ := q : [0, φ (1)] → R ⏐ g

is nondecreasing and left-continuous, 0 ≤ q (t ) ≤

FX−,1Q

(D.2)

}

(v (t )) , for each t ∈ [0, φ (1)] ,

where v = φ −1 , and consider the following problem:

M. Ghossoub / Insurance: Mathematics and Economics 89 (2019) 79–91

Problem D.5. φ (1)

{∫ sup q∈Q∗∗

(

)

u W0 − Π − q (t ) dt :

0

φ (1)

∫

}

q (t ) v ′ (t ) dt ≥ R0 .

0

Lemma D.6. If q∗ is optimal for Problem D.5, then f ∗ := q∗ ◦ φ is optimal for Problem D.3.

Remark D.7. For all t ∈ [0, 1], φ (t ) = 0 Fh−,Q1 (1 − x) dx. Therefore, for all t ∈ [0, 1], φ ′ (t ) = Fh−,Q1 (1 − t ) ≥ 0, and so φ is increasing and concave. Moreover, since Fh−,Q1 is strictly increasing by Assumption 3.4, the inverse function theorem implies that v = φ −1 is convex and increasing. Consequently, v ′ is positive and increasing on [0, φ (1)]. Lemma D.8. If q∗ ∈ Q∗∗ satisfies:

Proof. Suppose q∗ is optimal for Problem D.5, and let f ∗ := q∗ ◦φ . Since q∗ ∈ Q∗∗ , it follows that f ∗ ∈ Q∗ . Moreover, using the change of variable z = v −1 (t ), where v = φ −1 and φ are as defined above, we have

∫

1

f ∗ (t ) dt =

φ (1)

∫

φ (1)

∫ =

φ (1)

=

then q∗ is optimal for Problem D.5. q (φ (v (t ))) v (t ) dt q∗ (t ) v ′ (t ) dt ≥ R0 ,

where the last inequality follows from the feasibility of q∗ for Problem D.5. Therefore, f ∗ is feasible for Problem D.3. To show optimality of f ∗ for Problem D.3, let f be any other feasible solution for Problem D.3. Then, with v (t ) = φ −1 (t ) as above, and using the change of variable z = v −1 (t ), we have 1 0

∫

(t )) Fh−,Q1

φ (1)

=

)

(

Hence, φ (1)

(

)

u W0 − Π − q∗ (t ) dt −

φ (1)

=

φ (1)

∫

)

0

φ (1)

[∫

v (t ) q (t ) dt − ′

φ (1)

∫

v ′ (t ) q∗ (t ) dt

]

0

0

u (W0 − Π − q (t )) dt

(

u W0 − Π − q (t ) dt

≥ λ [R0 − R0 ] = 0. ∫ φ (1) Therefore, 0 u (W0 − Π − q∗ (t )) dt q (t )) dt. □

u (W0 − Π − f (v (z ))) dz

)

] [ ≥ λ v ′ (t ) q (t ) − v ′ (t ) q∗ (t ) .

≥λ

0

∫

(

u W0 − Π − q∗ (t ) − u W0 − Π − q (t )

0

u (W0 − Π − f (t )) dv −1 (t )

0

∫

Proof. Let q∗ ∈ Q∗∗ be such that the two conditions above are satisfied. Then q∗ is feasible for Problem D.5. To show optimality, let q ∈ Q∗∗ be any feasible solution for Problem D.5. Then, by definition of q∗ , it follows that for each t ∈ [0, φ (1)],

∫

(1 − t ) dt

1

=

}

u (W0 − Π − y) + λyv ′ (t ) ,

argmax

′

∗

0

u (W0 − Π − f

{

q∗ (t ) =

0≤y≤FX−,1Q (v(t ))

0

∫

∫ φ (1)

(1) 0 v ′ (t ) q∗ (t ) dt = R0 ; and, (2) There exists some λ ≥ 0 such that for all t ∈ [0, φ (1)],

f ∗ (v (z )) dv (z )

0

0

∫

89

∫t

≥

∫ φ (1) 0

u (W0 − Π −

0

φ (1)

∫

Remark D.9. Assumption 2.2 implies that u is continuously ′ differentiable ( ′ )−1 and that u is strictly decreasing. This then implies is continuous and strictly decreasing, by the Inverse that u Function Theorem. Moreover, the continuity of u implies that u is bounded on every closed and bounded subset of R.

u W0 − Π − q∗ (t ) dt

)

(

≤ 0

∫

φ (1)

u W0 − Π − f ∗ (v (z )) dz

)

(

= 0

∫

1

u W0 − Π − f ∗ (t ) dv −1 (t )

(

=

)

Lemma D.10. For each λ ≥ 0, define the function q∗λ on [0, φ (1)] by

0

∫

1

u W0 − Π − f ∗ (t ) φ ′ (t ) dt

)

(

=

[

0

∫ = 0

q∗λ (t ) := min

1

u W0 − Π − f ∗ (t ) Fh−,Q1 (1 − t ) dt ,

(

)

where q := f ◦ v . Therefore, to show optimality of f ∗ for Problem D.3, it remains to show that q is feasible for Problem D.5. Since f is feasible for Problem D.3, it is nondecreasing, leftcontinuous, and satisfies, for all t ∈ (0, 1), 0 ≤ f (t ) ≤ FX−,1Q (t ). Therefore, since v is increasing and continuous (by the inverse function theorem), q is nondecreasing, left-continuous, and satisfies, for all t ∈ [0, φ (1)], 0 ≤ q (t ) = f (v (t )) ≤ FX−,1Q (v (t )). Therefore, q ∈ Q∗∗ . Furthermore, φ (1)

∫ 0

q (t ) v ′ (t ) dt =

φ (1)

∫ 0

f (v (z )) dv (z ) =

1

∫

f (t ) dt ≥ R0 , 0

FX−,1Q

}] ( ′ )−1 ( ′ ) λv (t ) . (v (t )) , max 0, W0 − Π − u {

(D.3)

∫ φ (1)

Then the function Γ : λ ↦ → 0 nondecreasing in λ. Moreover, lim Γ (λ) = 0 and

λ→0

q∗λ (t ) v ′ (t ) dt is continuous and

lim Γ (λ) =

λ→+∞

∫

XdQ .

( )−1

Proof. As v ′ ≥ 0 and u′ (·) is continuous and decreasing, q∗λ (t ) is continuous and increasing in λ for any given t ∈ [0, φ (1)]. Therefore, Γ is continuous and nondecreasing in λ. The rest follows from Assumptions 2.2 and 2.3. □

where the last inequality follows from the feasibility of f for Problem D.3. Thus, q is feasible for Problem D.5, which concludes the proof that f ∗ is optimal for Problem D.3. □

Lemma D.11. For each λ ≥ 0, let the function q∗λ on [0, φ (1)] be defined as in Eq. (D.3). Then:

In light of Lemma D.6, we turn our attention to solving Problem D.5.

(1) For each λ ≥ 0, q∗λ ∈ Q∗∗ ; ∫ φ (1) (2) There exists λ∗ ≥ 0 such that 0 v ′ (t ) q∗λ∗ (t ) dt = R0 ; and

90

M. Ghossoub / Insurance: Mathematics and Economics 89 (2019) 79–91

(3) For all t ∈ [0, φ (1)], q∗λ (t ) =

Now, if Y ∗ denotes the optimal indemnity, then R∗ := X − Y ∗ solves the following problem.

{ argmax

} u (W0 − Π − y) + λyv ′ (t ) .

Problem E.1.

0≤y≤FX−,1Q (v(t ))

{∫ Proof. (1) and (3) follow from Remark D.9, and from the monotonicity and continuity properties of v ′ (Remark D.7). (2) follows from Lemma D.10 and Assumption 3.3, by the Intermediate Value Theorem. □ Therefore, Lemmata D.8 and D.11 imply that for λ∗ such that ∫ φ (1) v ′ (t ) q∗λ∗ (t ) dt = R0 , and for all q ∈ Q∗∗ , 0 ∫ φ (1) [ ( ] ) u W0 − Π − q (t ) + λ∗ q (t ) v ′ (t ) dt 0 ∫ φ (1) [ ( ] ) ≤ u W0 − Π − q∗λ∗ (t ) + λq∗λ∗ (t ) v ′ (t ) dt , 0

where is as in Eq. (D.3). Hence, the optimal solution to Problem D.1 is given by q∗λ∗ . Thus, by Lemmata D.4, D.6, D.8, and D.11, the function R∗ := ∗ qλ∗ (φ (1 − U )) is optimal for Problem D.1 (and hence also for Problem C.2) and anti-comonotonic with h, where:

• U = Fh,Q (h) is uniformly distributed on (0, 1) for the measure Q , by Föllmer and Schied (2016, Lemma A.25);

• For all t ∈ [0, φ (1)], [ q∗λ∗ (t ) := min FX−,1Q

{ }] ( ′ )−1 ( ∗ ′ ) λ v (t ) ; (v (t )) , max 0, W0 − Π − u

• For all t ∈ [0, 1], v (t ) = φ∫ −1 (t ); t • For all t ∈ [0, 1], φ (t ) = 0 Fh−,Q1 (1 − x) dx; and, ∫ φ (1) • λ∗ is chosen such that 0 q∗λ∗ (t ) v ′ (t ) dt = R0 . Consequently, by Lemma C.6, R∗ 1A is optimal for Problem 3.2, and hence Y ∗ := X − R∗ 1A is optimal for Problem 3.1. Moreover, R∗ is anti-comonotonic with h, and ˜ X := FX−,1Q (1 − U ) is the nonincreasing Q -rearrangement of X with respect to h. Then, by Proposition A.1 and Eq. (3.1), 0 ≤ ˜ X ≤ X , Q -a.s., and FX ,Q ≡ F˜ X ,Q . Now, suppose that Q ≪ Pac . Then h > 0, and hence since v = φ −1 and φ ′ (t ) = Fh−,Q1 (1 − t ), it follows that R∗

[ FX−,1Q

[ = min

R∈B(Σ )

Q -a.s.,

(

FX−,1Q

{

(

(1 − U ) , max 0, W0 − Π − u

{

λ∗

( ′ )−1

( ′ )−1

(1 − U ) , max 0, W0 − Π − u

)} ]

Fh−,Q1 (U )

(

λ∗ h

)}] ,

Q –a.s., where the second equality follows from Assumption 3.4, by Föllmer and Schied (2016, Lemma A.25). Finally, when Q ≪ ∗ ˜ ∈ P [ ac ,∫Lemma ] 2.7 implies that Y is Pareto-optimal, since Π 0, XdQ by Assumption 3.3. This concludes the proof of Theorem 3.6. □ Appendix E. Proof of Proposition 3.12 ac Let h = dP := x ∈ R+ . Then x ∈ [0, 1], since 0 ≤ Pac (S ) ≤ dQ Q (S ) = 1. By assumption, we have x ̸ = 0 and so x ∈ (0, 1]. Recall that there exists A ∈ Σ such that Q (S \ A) = Ps (A) = 0, which then implies that Pac (S \ A) = 0 and Q (A) = Q (S ). Therefore, by the Radon–Nikodým Theorem (Aliprantis 2006, ∫ and Border, ∫ ∫Theorem 13.20), ∫ we have for all Z ∈ B (Σ ), A Z dP = A Z xdQ = Z xdQ = x ZdQ .

)

u W0 − Π − R dP : 0 ≤ R ≤ X ,

∫

}

R dQ ≥ R0 .

However, as in the proof of Proposition 3.5, if R∗ solves Problem E.2, then R∗ 1A solves Problem E.1. Problem E.2.

{∫ sup

R∈B(Σ )

(

)

u W0 − Π − R x dQ : 0 ≤ R ≤ X ,

∫

q∗λ

= min

sup

Q -a.s.;

}

R dQ ≥ R0 .

Since x ∈ (0, 1], it suffices to solve Problem E.3.

{∫ sup

R∈B(Σ )

Q -a.s.,

(

)

u W0 − Π − R dQ : 0 ≤ R ≤ X ,

∫

}

R dQ ≥ R0 .

Hence, we are back in the classical setup of belief homogeneity, in which it is known that a solution to Problem E.3 is given ∗ ∗ + by R∗ = min (X , d∗ ) = X∫ − max (X − ∫ d ∗, 0), where ∫ ∗d ∈ R is ∗ chosen such that R = R dQ = R dQ = R 1 dQ , that 0 A ∫ ∫ A ˜ = max (X − d∗ , 0) dQ = (X − R∗ ) dQ . Finally, since is, Π h > 0, then Q ≪ Pac , and[ hence 2.7 implies that Y ∗ is ∫ Lemma ] ˜ ∈ 0, XdQ by Assumption 3.3. □ Pareto-optimal, since Π References Aliprantis, C.D., Border, K.C., 2006. Infinite Dimensional Analysis, third ed. Springer-Verlag. Amarante, M., Ghossoub, M., 2016. Optimal insurance for a minimal expected retention: The case of an ambiguity-seeking insurer. Risks 4 (1), 8. Amarante, M., Ghossoub, M., Phelps, E.S., 2015. Ambiguity on the insurer’s side: The demand for insurance. J. Math. Econom. 58, 61–78. Arrow, K.J., 1971. Essays in the theory of risk-bearing. Markham Publishing Company, Chicago. Assa, H., 2015. On optimal reinsurance policy with distortion risk measures and premiums. Insurance Math. Econom. 61, 70–75. Balbás, A., Balbás, B., Heras, A., 2009. Optimal reinsurance with general risk measures. Insurance Math. Econom. 44 (3), 374–384. Bernard, C., He, X., Yan, J.A., Zhou, X.Y., 2015. Optimal insurance design under rank-dependent expected utility. Math. Finance 25 (1), 154–186. Bogachev, V.I., 2007. Measure Theory, Vol. 1. Springer-Verlag. Boonen, T.J., 2016. Optimal reinsurance with heterogeneous reference probabilities. Risks 4 (3), 26. Boonen, T.J., Ghossoub, M., 2019. Bilateral Risk Sharing with Heterogeneous Beliefs and Exposure Constraints. Mimeo. Buhlmann, H., 1980. An economic premium principle. Astin Bull. 11 (1), 52–60. Bühlmann, H., Delbaen, F., Embrechts, P., Shiryaev, A.N., 1998. On esscher transforms in discrete finance models. Astin Bull. 28 (2), 171–186. Cai, J., Liu, H., Wang, R., 2017. Pareto-optimal reinsurance arrangements under general model settings. Insurance Math. Econom. 77, 24–37. Carlier, G., Dana, R.A., 2005. Rearrangement inequalities in non-convex insurance models. J. Math. Econom. 41 (4–5), 483–503. Cheung, K.C., Chong, W.F., Yam, S.C.P., 2015. The optimal insurance under disappointment theories. Insurance Math. Econom. 64, 77–90. Cheung, K.C., Lo, A., 2017. Characterizations of optimal reinsurance treaties: A cost-benefit approach. Scand. Actuar. J. 2017 (1), 1–28. Cheung, K.C., Sung, K.C.J., Yam, S.C.P., Yung, S.P., 2011. Optimal reinsurance under general law-invariant risk measures. Scand. Actuar. J. 2014 (1), 72–91. Chi, Y., 2019. On the optimality of a straight deductible under belief heterogeneity. Astin Bull. 49 (1), 243–262.

M. Ghossoub / Insurance: Mathematics and Economics 89 (2019) 79–91 Cui, W., Yang, J., Wu, L., 2013. Optimal reinsurance minimizing the distortion risk measure under general reinsurance premium principles. Insurance Math. Econom. 53 (1), 74–85. Denuit, M., Dhaene, J., Goovaerts, M.J, Kaas, R., Laeven, R.J.A., 2006. Risk measurement with equivalent utility principles. Statist. Decisions 24 (1), 1–25. Embrechts, P., 2000. Actuarial versus financial pricing of insurance. J. Risk Financ. 1 (4), 17–26. Föllmer, H., Schied, A., 2016. Stochastic Finance: An Introduction in Discrete Time, fourth ed. Walter de Gruyter. Gerber, H.U., Goovaerts, M.J, 1981. On the representation of additive principles of premium calculation. Scand. Actuar. J. 4, 221–227. Gerber, H.U., Shiu, E.S.W., 1994. Option pricing by esscher transforms. Trans. Soc. Actuar. 46, 99–191. Gerber, H.U., Shiu, E.S.W., 1996. Actuarial bridges to dynamic hedging and option pricing. Insurance Math. Econom. 18 (3), 183–218. Ghossoub, M., 2015. Equimeasurable rearrangements with capacities. Math. Oper. Res. 40 (2), 429–445. Ghossoub, M., 2016. Optimal insurance with heterogeneous beliefs and disagreement about zero-probability events. Risks 4 (3), 29.

91

Ghossoub, M., 2017. Arrow’s theorem of the deducible with heterogeneous beliefs. N. Am. Actuar. J. 21 (1), 15–35. Goovaerts, M.J, Laeven, R.J.A., 2008. Actuarial risk measures for financial derivative pricing. Insurance Math. Econom. 42 (2), 540–547. Lo, A., 2017. A neyman-pearson perspective on optimal reinsurance with constraints. Astin Bull. 47 (2), 467–499. Milgrom, P., 1981. Good news and bad news: Representation theorems and applications. Bell J. Econ. 12 (2), 380–391. Morris, S., 1995. The common prior assumption in economic theory. Econ. Phil. 11 (2), 227–253. Rüschendorf, L., 2009. On the distributional transform, sklar’s theorem, and the empirical copula process. J. Statist. Plann. Inference 139 (11), 3921–3927. Schmeidler, D., 1989. Subjective probability and expected utility without additivity. Econometrica 57 (3), 571–587. Shaked, M., Shanthikumar, J.G., 2007. Stochastic Orders. Springer. Xu, Z.Q., Zhou, X.Y., Zhuang, S.C., 2018. Optimal insurance under rank-dependent utility and incentive compatibility. Math. Finance 21 (4), 503–534. Zhuang, S.C., Weng, C., Tan, K.S., Assa, H., 2016. Marginal indemnification function formulation for optimal reinsurance. Insurance Math. Econom. 67, 65–76.