The Schmeidler expected utility model: Theory, implications, and an application to European call options

The Schmeidler Expected Utility Model

Theory, Implications, and an Application to European Call Options

Simon Edvinsson

Spring 2016

Thesis Project, 30ECTS

Degree of Master of Science in Mathematics, 120ECTS

Department of Mathematics and Mathematical Statistics

Abstract

In this essay we present a self-contained proof of the Schmeidler expected utility theorem. Furthermore, we give an overview of the implications of today with focus on the optimal investment with non-additive expected utility. Finally, an applica- tion to pricing of European call options is conducted where ambiguity in the market is shown to be present, and forecasted using ARIMA models.

Sammanfattning

Denna uppsats ger ett fullständigt bevis av Schmeidlers nyttofunktionssats. Vi

presenterar en översikt av dagens implikationer av denna sats med särskilt fokus

på optimala investeringar och dess följder. Slutligen applicerar vi en optionspris-

sättningsmodell på europeiska köp-optioner som bygger på Schmeidlers sats, där

osäkerhetsaversion påvisas och modelleras med ARIMA-modeller.

Contents

Introduction i

Acknowledgements v

Theory 1

Chapter 1. Set Functions and the Choquet Integral 3

1.1. Set Functions 3

1.2. The Asymmetric Choquet Integral 11

1.3. The Subadditivity Theorem 19

Chapter 2. Binary Relations and the Herstein–Milnor Theorem 25

2.1. Relations and Representation 25

2.2. The Herstein–Milnor Theorem 29

Chapter 3. The Schmeidler Expected Utility Theorem 35

3.1. An Integral Representation Theorem 35

3.2. Theorem and Proof 39

3.3. The Gilboa–Schmeidler Maxmin Theorem 43

Implications 47

Chapter 4. An Overview 49

4.1. Ambiguity Theory in Financial Markets 49

4.2. Contractual Agreements 50

4.3. Game Theory 51

4.4. Miscellaneous 51

Chapter 5. Optimal Investments with Choquet Expected Utility 53

An Application to European Call Options 59

Chapter 6. Economic Overview 61

6.1. Option Contracts 61

6.2. Risk Neutral Option Pricing 62

Chapter 7. Ambiguous Black-Scholes Derivation 65

7.1. Continuous Choquet Random Walk 65

6 CONTENTS

7.2. Ambiguous Option Price 66

7.3. Unit Root Non-stationary Models 70

Chapter 8. Pricing under Ambiguity 73

8.1. Option Pricing Formula Evaluation 73

8.2. Empirical Evaluation 78

Chapter 9. Conclusions 85

Appendix A. Data 87

A.1. Banco Santander 87

A.2. Deutsche Bank 93

A.3. HSBC 97

A.4. Royal Bank of Scotland 103

Appendix B. Code 109

Bibliography 115

List of Figures 119

Index 123

Introduction

The Merton–Margrabe–Black–Scholes model, in short the MMBS–model, for option pricing is not used in the derivatives market by professional investors and traders, and never has been (see Haug and Taleb (2011)). Its distributional assump- tions are not compatible with the real world dynamics of the derivatives market, since it only takes into account the probability of known events to which it is possi- ble to assign known probabilities. The MMBS-model is only found in the academy where it is taught to those studying financial mathematics, nearly exclusively with- out mentioning the shortcomings that arises when this model is applied on the real world markets.

This attitude can have two possible explanations: lack of knowledge in the the- ory of set functions, or non-existent connection with the actual derivatives market.

Black and Scholes themselves, two years after receiving the Nobel prize in eco- nomics, underestimated the risk of the derivatives market as the hedge-fund they managed experienced fatal blow-ups (see Edwards (1999)). In this essay we shall look at a model that incorporates real world uncertainty aversion, the so called Schmeidler’s non-additive subjective probability expected utility model, also known as Choquet’s expected utility theorem (in short the CEU–model). We shall use this theory to the pricing of options under uncertainty.

The mathematical highlight of this essay is to present a self-contained proof of the Schmeidler expected utility model, which is stated as follows:

The Schmeidler Expected Utility Model (Theorem 3.6). Let Ω be a set and let A be an algebra of subsets on Ω. Let L ₀ denote the set of all A-measurable finite valued functions from Ω to the set Y of simple probability measures as defined in Definition 1.8. Let  be a binary relation defined on L 0 . Consider the following conditions for :

A1: negative transitivity, A2: asymmetry,

A3: independence, A4: continuity, A5: monotonicity,

A6: comonotonic additivity, and

i

ii INTRODUCTION

A7: non degeneracy.

The following conditions are then equivalent:

1) the binary relation  satisfies assumptions A1–A7 for L 0 ,

2) there exists an affine function u : Y → R that is unique up to a positive linear transformation, and a unique monotone, submodular set function v defined on A such that

f < g if and only if Z

Ω

(u ◦ f ) dv ≥ Z

Ω

(u ◦ g) dv.

Here the integral is the asymmetric Choquet integral as defined in Chap- ter 1, Section 2.

Theorem 3.6 was formulated in context of choice theory, and aimed at solving the Ellsberg paradox. The paradox states that a utility maximizing agent is indif- ferent to choose between unknown outcomes with known probability and unknown outcomes with unknown probability. The Ellsberg paradox shows the shortcom- ing of the utility maximizing homo economicus that has no preference structure with respect to uncertainty, which is a common assumption in economics and fi- nance (Ellsberg, 1961).

As noted by Keynes (1921), market volatility breaks the uniform behaviour of the investors, and their actions reflect heterogeneous individual preferences with respect to uncertainty. Theorem 3.6 can therefore be used to explain behaviour of uncertainty averse investors on volatile financial markets, which was partly done by Dow and da Costa Werlang (1992). They showed that trade inertia, the phe- nomena where a decrease in price of an risky asset is not offset by an increase in demand, was compatible with the CEU–model. This result serves as a possible ex- planation to how volatile markets with fluctuating prices are compatible with utility maximizing investor inertia.

The second part of the essay is devoted to bridge the gap between theory and the real world. An overview of the implications that Theorem 3.6 have had on various research areas that handles uncertainty is presented. The results of Dow and da Costa Werlang (1992) is studied in closer detail. This is because Dow and da Costa Werlang (1992) in a convenient manner show the effects of heterogeneous preference structures of investors with respect to risky asset acquisition on volatile markets, which is our final main topic.

The last part of the essay is devoted to a new application of Theorem 3.6 with regards to pricing of options. An uncertainty adjusted Wiener process that is dynamically consistent with the axiomatic structure of Theorem 3.12 can be constructed and employed within a new model framework for pricing options and estimating market uncertainty (Driouchi et al. (2015); Kast and Lapied (2010); Kast et al. (2014)).

Following Knight (1921), we define risk to be events with unknown outcome with

known probability. Uncertainty is then defined as events with unknown outcome to

which it can not be assigned a probability. Keynes (1921) was, as already mentioned,

INTRODUCTION iii

an early contributor in examining the impact on investment decision with respect to uncertainty averse investors.

In the risk neutral pricing framework, it is assumed that every future financial claim can be replicated by a trading strategy. The market is effectively complete by the fact that every claim can be assigned a perfect hedge. The theoretical consequences are clear in a financial aspect. Firstly, there are unique risk neutral prices for every asset on the market. Secondly, pricing of the claims are independent of any preference structure of the market agents, including uncertainty (Björk, 2004).

If no perfect hedge exist and the market is incomplete, then there are no unique replicating strategies. Uncertainty is then infused in the market, and the preference structure of the investors can have an impact on prices. The MMBS-model has been criticized for not creating perfect hedging options regarding the aversion of irrational agents on the market, or extremal events that has not before been encountered in the economy (see e.g. Haug and Taleb (2011); Kahneman and Tversky (1979)).

These shortcomings of the MMBS-model regarding the market impact of un- certainty averse investors have often been explained through cognitive science. In volatile markets the participants marginal utility is heterogeneous as a consequence of their individual preferences towards uncertainty (Smith and McCardle, 1998).

Since the preference structure in an imperfect market creates a mispricing with the MMBS-model, we are forced to revert to utility theory to account for the impact of the investors preference structure with respect to uncertainty.

By allowing for varying marginal utility with respect to uncertainty, Driouchi et al. (2015) constructs a new option pricing model using ambiguity adjusted price dynamics by a Choquet Wiener process, dynamically consistent with the axiomatic structure of Theorem 3.12. This model enables pricing of options with respect to a nonuniform uncertainty preference structure of the investors. In the application part of the essay, a pricing model for European call options is derived in a similar manner as by Driouchi et al. (2015), who shows the model for European exchange and put options.

The uncertainty adjusted pricing model is evaluated by pricing options for vari- ous levels of uncertainty preferences. By numerical backsolving, the optimal market ambiguity for various European call options is estimated during the eurozone crisis with beginning in late 2009 continuing through early 2011. Finally, an ARIMA model is fitted to the obtained estimates and used to forecast the market ambiguity for the various considered options.

In conclusion, the aim of the essay is to give a comprehensive theoretical back- ground as well as the proof of the Choquet expected utility model. In addition, implications of Theorem 3.6 are examined, with a special focus on trade inertia as described by Dow and da Costa Werlang (Theorem 5.4). Finally, it is demonstrated that Theorem 3.12 can be used in the modeling of an uncertainty adjusted option pricing formula, which can be used to empirically estimate the preference structure of the market agents.

The essay is divided into three parts: theory, implications, and application. In

the first part the necessary theory that is needed to state and prove Theorem 3.6 is

iv INTRODUCTION

developed. In Chapter 1 nonadditive set functions, the asymmetric integral, and the subadditivity theorem are presented. Chapter 2 covers relations and the Herstein–

Milnor theorem (Theorem 2.16). Theorem 3.6 is stated and proven in Chapter 3, which also contains the extension Theorem 3.12.

In the first chapter of the implication part of the essay a comprehensive overview of the various areas of research where Theorem 3.6 has had an impact is presented.

The second chapter covers the optimal investment choices and presence of trade inertia in the context of Theorem 5.4.

The final part examines option pricing with respect to heterogenous uncertainty

preferences. An uncertainty adjusted pricing model is presented and applied on

observed real world market prices. Finally, the uncertainty estimates are fitted and

forecasted by an ARIMA model. The essay is concluded with a final discussion and

necessary appendices.

Acknowledgements

First of all I would like to thank my advisor Per Åhag at Umeå university, who has contributed with his expertise, patience and encouragement. It takes a certain level of devotion to not only see the beauty of mathematics, but also to be able to reveal this inherent beauty to others. In this art form, there are none equal to him. But foremost for teaching me how to fit Bugs Bunny-motion with the ARIMA model, also known as the Arima Bunny. Secondly, I want to thank my examinator Lisa Hed for her indispensable advice which greatly improved this essay.

I am thankful for insightful comments and sound advice, generously contributed by Anton Vernersson, Tim Larsson, and André Berglund. Lastly, I would like to thank Robin Törnkvist and Sarah Pontén. The former for introducing me to mathematics in general and analysis in particular. The latter for unyielding support, and for her enthusiasm to share a world outside of mathematics with me.

v

Theory

CHAPTER 1

Set Functions and the Choquet Integral

The following chapter is devoted to the theory of set functions, the asymmetric Choquet integral, and the subadditivity theorem. A solid foundation of the theory used in the essay is presented and motivated. It will be evident that the subadditiv- ity of the asymmetric Choquet integral implies submodularity of the set function.

The chapter is therefore concluded with the subadditivity theorem, which shows that the converse relation holds: submodularity of the set function is a necessary condition for subadditivity of the integral. For further information and historical references, see Denneberg (1994).

1.1. Set Functions

This section is devoted to set functions and their elementary properties, with focus on nonadditivity. The definition of distribution functions, measurability and the concept of comonotonicity follows by the introduction of upper set systems for nonadditive set functions. Denote the basic nonempty set Ω and let 2 ^Ω be the power set of Ω, conventionally defined as the class of all subsets of Ω. For additional notations see the list of notations.

Definition 1.1. A set system S is any S ⊆ 2 ^Ω such that ∅ ∈ S.

Definition 1.2. The set system S is an algebra if it is closed under the forma- tion of finite unions, as well as closed under complementation.

A σ-algebra is an algebra such that it is closed under the formation of countable union. Set systems that are algebras are referred to as a paving or pavage. Con- sider the definition of set functions and its elementary properties, which are used throughout the essay.

Definition 1.3. A set function µ is a mapping from a set system S to the non-negative extended real line, µ : S → R ⁺ such that µ(∅) = 0.

Definition 1.4. A set function µ is

monotone: if A, B ∈ S, A ⊂ B implies µ(A) ≤ µ(B);

finite: if µ(A) < ∞ for all A ∈ S;

submodular : if A, B ∈ S such that A ∪ B, A ∩ B ∈ S implies µ(A ∪ B) + µ(A ∩ B) ≤ µ(A) + µ(B);

3

4 1. SET FUNCTIONS AND THE CHOQUET INTEGRAL

supermodular : if A, B ∈ S such that A ∪ B, A ∩ B ∈ S implies µ(A ∪ B) + µ(A ∩ B) ≥ µ(A) + µ(B);

subadditive: if A, B ∈ S such that A ∪ B ∈ S, A ∩ B = ∅ implies µ(A ∪ B) ≤ µ(A) + µ(B);

superadditive: if A, B ∈ S such that A ∪ B ∈ S, A ∩ B = ∅ implies µ(A ∪ B) ≥ µ(A) + µ(B);

A set function with any of these properties will be referred to with suitable corresponding adjective. Submodularity is also referred to as strong subadditivity or concavity. A set function that is both submodular and supermodular is said to be modular, and a set function that is subadditive and superadditive is correspondingly referred to as additive.

Definition 1.5. The set function µ is σ-additive if µ

∞

[

n=1

A

!

=

∞

X

n=1

µ(A

)

for sequences A

∈ S of pairwise disjoint sets such that ∪ ^∞

A

∈ S.

Definition 1.6. A set function µ on a set system S is continuous from below if for A

∈ S, A

⊂ A

for n ∈ N

A =

∞

[

n=1

A

∈ S, implies

n→∞

lim µ(A

) = µ(A).

Definition 1.7. A set function µ on a set system S is continuous from above if for A

∈ S, A

⊃ A

for n ∈ N

A =

∞

\

n=1

A

∈ S, implies

n→∞

lim µ(A

) = µ(A).

If S is an algebra, then µ is additive if and only if it is modular. If S is a σ- algebra, then µ is σ-additive if and only if it is continuous from below and additive.

Such a set function µ is referred to as a measure. For an algebra S a probability

measure is an finitely additive µ such that µ(Ω) = 1. Note that a probability

measure not necessarily satisfies σ-additivity.

1.1. SET FUNCTIONS 5

Definition 1.8. Let Ω be a set and let S be an algebra of subsets on Ω. Let µ : S → R ⁺ be a probability measure. Then µ is a simple probability measure if the set {ω ∈ Ω : µ(ω) 6= 0} is finite and P

ω∈Ω

µ(ω) = 1 holds.

Definition 1.9. Let µ be a set function µ : S → R + where S is any set system S ⊆ 2 ^Ω . Define the outer set function of µ as the set function µ ^∗ : 2 ^Ω → R + such that

µ ^∗ (A) = inf{µ(B) : A ⊆ B ∈ S}

for A, B ∈ 2 ^Ω .

Definition 1.10. Let µ be a set function µ : S → R + where S is any set system such that S ⊆ 2 ^Ω . We define the inner set function of µ as the set function µ ∗ : 2 ^Ω → R + such that

µ _∗ (A) = sup{µ(C) : C ∈ S, C ⊆ A}

for A ∈ 2 ^Ω .

The inner and outer set functions are respectively the smallest and largest extension of the set function µ defined on S onto the power set 2 ^Ω . Some of the properties of µ are inherited by the outer set function µ ^∗ .

Proposition 1.11. Let µ be a monotone set function on the set system S ⊆ 2 ^Ω , where S is closed under union and intersection. Then

i) if µ is submodular, then µ ^∗ is submodular,

ii) if µ is subadditive and S is an algebra, then µ ^∗ is subadditive.

Proof. Begin with i). Let µ be a monotone and submodular set function, and let A, B ∈ 2 ^Ω . If µ ^∗ (A) = ∞ or µ ^∗ (B) = ∞, the proof is done. For the other case, for ε > 0 there exist A 1 , B 1 ∈ S, A ⊆ A 1 , B ⊆ B 1 where

µ(A ₁ ) ≤ µ ^∗ (A) + ε 2 , and

µ(B ₁ ) ≤ µ ^∗ (B) + ε 2 .

By the monotonicity of µ ^∗ , the submodularity of µ and A 1 ∪ B 1 , A 1 ∩ B 1 ∈ S, we get

µ ^∗ (A ∪ B) + µ ^∗ (A ∩ B) ≤ µ ^∗ (A 1 ∪ B 1 ) + µ ^∗ (A 1 ∩ B 1 )

= µ(A ₁ ∪ B ₁ ) + µ(A ₁ ∩ B ₁ ) ≤ µ(A ₁ ) + µ(B ₁ ) ≤ µ ^∗ (A) + µ ^∗ (B) + ε.

By ε → 0, the proof is complete.

To show ii), let S be an algebra. Define the sets A 1 , B 2 as in i). For any ε > 0, subadditivity of µ, and A 1 , B 1 \A 1 ∈ S we get

µ ^∗ (A ∪ B) ≤ µ(A 1 ∪ B 1 ) ≤ µ(A 1 ) + µ(B 1 \A 1 )

≤ µ(A 1 ) + µ(B 1 ) ≤ µ ^∗ (A) + µ ^∗ (B) + ε.

Let ε → 0, and the proof is done. 

6 1. SET FUNCTIONS AND THE CHOQUET INTEGRAL

Further properties inherited from µ by the outer measure µ ^∗ is continuity from below. Define the closure from below S of S ⊆ 2 ^Ω by

S = (

∈ 2 ^Ω : A =

∞

X

n=1

A

)

for some increasing sequence A

∈ S. Then S is said to be closed from below if S = S.

Proposition 1.12. Let µ be a monotone set function that is continuous from below on S ∈ 2 ^Ω . Let S be closed under union and intersection and closed from below. Then µ ^∗ is continuous from below.

Proof. Let A

be an increasing sequence on 2 ^Ω , where A = ∪ ^∞

A

. Since µ ^∗ is monotone, it holds that lim

µ ^∗ (A

) ≤ µ ^∗ (A). It suffices to show the validity of the converse inequality, i.e. for any ε > 0

µ ^∗ (A) ≤ lim

n→∞

µ ^∗ (A

) + ε.

Fix an arbitrary ε > 0. By definition of µ ^∗ there exists for each n ∈ N a set B

∈ S such that

µ(B

) ≤ µ ^∗ (A

) + ε 2

for A

⊆ B

. Then C

= ∪

B

∈ S and it will be shown by induction that µ(C

) ≤ µ ^∗ (A

) + ε

n

X

k=1

1

2

. (1.1)

For n = 1, nothing has to be proved. Assume that (1.1) is true. By (1.1), mono- tonicity of A

⊆ C

∩ B

and submodularity, it follows that

µ(C

) ≤ µ(C

) + µ(B

) − µ(C

∩ B

)

≤ µ ^∗ (A

) + ε

n

X

k=1

1

2

+ µ ^∗ (A

) + ε

2

− µ ^∗ (A

)

= µ ^∗ (A

) + ε

n+1

X

k=1

1 2

. The induction is done. Since C

∈ S is an increasing sequence and C = ∪ ^∞

C

∈ S, by the continuity of µ and (1.1), we have

µ ^∗ (A) ≤ µ(C) = lim

n→∞

µ(C

) ≤ lim

n→∞

µ ^∗ (A

) + ε

n

X

k=1

1 2

!

= lim

n→∞

µ ^∗ (A

) + ε.

 Define a chain of sets in 2 ^Ω to be a completely ordered set system M. That is, if A ⊂ B or B ⊂ A for A, B ∈ M, then M is a chain. Let X : Ω → R + . Then the strict upper level set of X for the level x ∈ R is defined as

{X > x} = {ω ∈ Ω : X(ω) > x}, (1.2)

1.1. SET FUNCTIONS 7

whereas the corresponding weak upper level set of X for the corresponding level x is defined by exchanging the strict inequality in (1.2) with ≥. The upper set system of X is defined as the union of all upper level set systems. The upper set system is a chain since it is completely ordered.

Definition 1.13. Let µ : 2 ^Ω → R + be a monotone set function and let X be an arbitrary function on Ω such that X : Ω → R. The (decreasing) distribution function of X with respect to µ is defined by

G

(x) = G

(x) = µ(X > x).

Here, µ(X > x) = µ({X > x}) for convenience. If µ is continuous from below, then G

(x) is right continuous. The pseudo-inverse of G

(x) is the quantile function. To be able to define the quantile function, the concept of pseudo-inverse in context of decreasing functions needs to be made clear.

Definition 1.14. Define the decreasing function f : I → R where I ⊂ R is on the x-axis. Define the interval J = [inf

f (x), sup

f (x)] where J ⊂ R is on the y-axis. We say that the function ˇ f : J → I is the pseudo-inverse of f if for a = inf I a ∨ sup{x : f (x) > y} ≤ ˇ f (y) ≤ a ∨ sup{x : f (x) ≥ y}. (1.3) Observe that we define sup ∅ = −∞ and inf ∅ = ∞. The pseudo-inverse ˇ f (y) is uniquely determined by the decreasing function f if its pseudo-inverse is continuous at y, since for any ε < 0 it holds that

f (y + ε) ≤ a ∨ sup{x : f (x) ≥ y + ε} ≤ a ∨ sup{x : f (x) > y} ˇ

≤ ˇ f (y) ≤ a ∨ sup{x : f (x) ≥ y} ≤ a ∨ sup{x : f (x) > y − ε} ≤ ˇ f (y − ε).

Let then ε → 0 and uniqueness ensues.

The pseudo-inverse of a decreasing function preserves order. In other words, the pseudo-inverse of decreasing functions f, g ordered f ≤ g, implies ˇ f ≤ ˇ g. This is verified by observing that for a point of common continuity, we have

f (y) = a ∨ sup{x : f (x) > y} ≤ a ∨ sup{x : g(x) > y} = ˇ ˇ g(y).

Proposition 1.15. For a decreasing function f : I → R it holds that ( ˇ f )ˇ= f

except on an at most countable set.

Proof. Let x be a point of continuity of ( ˇ f )ˇ. Then

( ˇ f )ˇ(x) = sup{y : ˇ f (y) > x} = sup{y : ˇ f (y) ≥ x}.

We begin by showing that ( ˇ f )ˇ(x) ≤ f (x). By definition of ˇ f , the inequality ˇ f (y) > x implies y ≤ f (x). Hence,

( ˇ f )ˇ(x) = sup{y : ˇ f (y) > x} ≤ sup{y : y ≤ f (x)} = f (x).

The reverse inequality is proven conversely by replacing {y : ˇ f (y) > x} with {y :

f (y) ≥ x}, along with using the fact that y < f (x) implies ˇ ˇ f (y) ≥ x. 

8 1. SET FUNCTIONS AND THE CHOQUET INTEGRAL

Denote the pseudo-inverse of the distribution function, i.e. the quantile function of X with respect to µ defined on [0, µ(Ω)) with ˇ G

. The order of functions X, Y is preserved through transgression to the distribution function and the quantile function. The order

X ≤ Y implies

G

≤ G

,

which follows from monotonicity of µ and {X > a} ⊂ {Y > a} for a level a ∈ R.

Furthermore, the order

G

≤ G

, if and only if G ˇ

≤ ˇ G

holds by Definition 1.14. The distribution function is, as we have seen, defined for set functions on 2 ^Ω . Let µ be a monotone set function on and arbitrary S ⊆ 2 ^Ω , with extensions onto the power set including µ ^∗ and µ _∗ .

Definition 1.16. Let µ be a monotone set function on the set system S ⊆ 2 ^Ω . The function X : Ω → R is upper µ-measurable if

G

= G

. which is unique except on an at most countable set.

Denote the function in Definition 1.16 by G

and call it the decreasing dis- tribution function of X with respect to µ on S. Conversely, a function X is lower µ-measurable if it is upper µ-measurable for −X. A real valued function X on Ω is said to be µ-measurable if it is both lower- and upper µ-measurable.

A function X is lower or upper S-measurable if it is lower or upper µ-measurable for a monotone set function µ on S. The function X is strongly S-measurable if

M

, M _−X ⊆ S.

The function X is upper S-measurable if M

⊆ S. If S is an algebra, then the condition M

⊆ S is a sufficient condition for X to be strongly S-measurable.

Proposition 1.17. Let S be a set system on 2 ^Ω . A function X : Ω → R is upper S-measurable if and only if for every pair a, b ∈ R, a < b, there exists a set S ∈ S so that

{X > b} ⊆ S ⊆ {X > a}.

Proof. Begin by showing the implication from left to right. Let µ be a mono- tone set function on S and let x ∈ R be a point where G

is continuous. It suffices to show that G

(x) = G

(x). If x < b, it holds that

G

(b) = inf{µ(A) : {X > b} ⊆ A ∈ S}

≤ sup{µ(B) : B ⊆ {X > x}, B ∈ S} = G

(x).

Let b → x, and continuity of G

at x implies G

(x) ≤ G

(x).

The reversed inequality is shown by using µ ∗ ≤ µ ^∗ .

1.1. SET FUNCTIONS 9

For the reversed implication, let a < b. Define a suitable monotone set function µ on S which allows us to assure that the desired set S ∈ S exists. Define the monotone set function v on M

by

v(X > x) = x ∧ b − x ∨ a, v(X ≥ x) = v(X > x).

Now define

µ(A) = inf{v(M ) : A ⊆ M, M ∈ M

}

for A ∈ S. Since M

is a chain, v(M 1 ) < v(M 2 ) implies M 1 ⊂ M 2 and for µ it can be concluded that

µ(A) < v(X > x)

implies A ⊂ {X > x}. By assumption, X is upper µ-measurable. There is therefore a real number x, a < x < b with

G

(x) = G

(x).

Writing out this equation we get

v(X > x) ≥ sup{µ(A) : A ∈ S, A ⊆ {X > x}}

= inf{µ(B) : B ∈ S, {X > x} ⊆ B} ≥ v(X > x).

Hence, equality holds and there is a set B ∈ S with {X > x} ⊆ B and µ(B) arbitrarily close to v(X > x). Since c(X > x) < v(X > a), by construction of µ, the set B can be chosen so that

v(X > x) ≤ µ(B) < v(X > a).

It can be concluded that

{X > b} ⊆ {X > x} ⊆ B ⊆ {X > a}.

 Real valued functions X : Ω → R are simple if the image X(Ω) ⊂ R is a finite set. If X is strongly S-measurable, then the simple function X : Ω → R can be written as X = P

i=1

d

χ

with S

∈ S, S 1 ⊂ · · · ⊂ S

and d

> 0.

Consider the following examples of measurability.

Example 1.18. Let X be a S-measurable simple function, then X is strongly S-measurable. It suffices to show {X ≥ x}, {X > x} ∈ S, x ∈ X(Ω) for upper S-measurable simple X. Since X is simple there exists for x ∈ X(Ω) an a ∈ R, a < x, so that [a, x) ∩ X(Ω) = ∅. Then

{X ≥ a} = {X ≥ x}

and by Proposition 1.17 this set is contained in S. A similar argument shows that

{X > x} ∈ S, x ∈ X(Ω). 

Example 1.19. Let S ⊆ 2 ^Ω be closed under intersection and X an upper S- measurable function. If A ∈ S is minimal with respect to set inclusion in S\{∅}

then X is constant on A. Assume the contrary, that there exists ω 1 , ω 2 ∈ A so that

x 1 = X(ω 1 ) < X(ω 2 ) = x 2 . Fix x 0 with x 1 < x 0 < x 2 . Then by Proposition 1.17

there exists a set S ∈ S with {X > x 0 } ⊆ S ⊆ {X > x 1 }. Now S ∩ A ∈ S and not

10 1. SET FUNCTIONS AND THE CHOQUET INTEGRAL

empty since it contains ω 2 and it is not A since ω 1 ∈ {X > x / 1 }. Hence we achieved

∅ 6= S ∩ A ⊂ A, contradicting the minimal property of A.  The section is concluded with a discussion regarding comonotonic functions.

The term comonotonic derives from common monontonicity. Consider the following definition.

Definition 1.20. Two functions X, Y : Ω → R are comonotonic if there is no pair ω ₁ , ω ₂ ∈ Ω such that X(ω ₁ ) < X(ω ₂ ) and Y (ω ₁ ) > Y (ω ₂ ).

Comonotonicity of the functions X, Y : Ω → R are equivalent with their re- spective upper level sets being a chain. Assume A ∈ M

and B ∈ M

such that A 6⊂ B and B 6⊂ A. Choose ω ₁ ∈ A\B and ω 2 ∈ B\A. Then

X(ω 1 ) > a ≥ X(ω 2 ) when A = {X > a}, and

X(ω 1 ) ≥ a > X(ω 2 )

for A={X ≥ a}. It is then the case that X(ω 1 ) > X(ω 2 ) and Y (ω 2 ) > Y (ω 1 ) which contradicts comonotonicity. Therefore, a class C of functions X : Ω → R are comonotonic if

[

X∈C

M

.

Comonotonicity implies additivity of the quantile functions.

Corollary 1.21. Assume a monotone set function µ on 2 ^Ω , and let X, Y : Ω → R be comonotonic set functions. Then we have that

G ˇ

= ˇ G

+ ˇ G

(1.4) except for an at most countable set.

Proof. Let Z = X + Y and let u, v : R → R be increasing and continuous functions with u + v = id. We have that

X = u(Z), Y = v(Z).

From the discussion regarding equation (1.12) in the coming chapter, we get the implication that

G ˇ

= u ◦ ˇ G

, G ˇ

= v ◦ ˇ G

and therefore

G ˇ

+ ˇ G

= (u + v) ◦ ˇ G

= ˇ G

.



1.2. THE ASYMMETRIC CHOQUET INTEGRAL 11

1.2. The Asymmetric Choquet Integral

By the means of the integral of decreasing functions and the quantile function, this section defines an nonadditive integral with respect to the monotone and non- additive set function µ. This asymmetric integral was first introduced by Choquet (1954). Various equivalent definitions of the integral will be presented, allowing for its representation with regards to both the quantile function as well as distribution function. The outset for the section is the integration of decreasing functions.

Consider a decreasing function f : I → R where I ⊆ R. Let d : Z → I such that d is a subdivision of the interval I with d

≤ d

for n ∈ Z. Furthermore, it holds that sup

d

= sup I and inf

d

= inf I. Then the lower sum is defined as

S(f, d) =

∞

X

n=−∞

f (d

)(d

− d

) (1.5) Excluding the cases where the summands sums to ∞ and −∞, then S(f, d) ∈ R.

The integral on I ⊂ R is defined as Z

I

f (x) dx = sup

d

S(f, d). (1.6)

Consider some elementary properties of the integral (1.6), which are verified by standard methods as for the Riemann integral. Let f, g be decreasing functions, then the following properties of (1.6) hold:

i) R

I

(f + g) dx = R

I

f dx + R

I

g dx.

ii) f ≤ g on I implies R

I

f (x) dx ≤ R

I

g(x) dx.

iii) R

a

f (x) dx = R

a

f (x) dx + R

b

f (x) dx for a ≤ b ≤ c.

iv) R

a+c

f (y − c) dy = R

a

f (x) dx where c ∈ R.

v) − R −a

−b (−f (−y)) dy = R

a

f (x) dx.

Property iv) show the translation invariance of the integral, and property v) the invariance under reflection. For completeness, consider the monotone convergence theorem for the integral of decreasing functions.

Proposition 1.22. Assume an interval I ⊆ R, and let f

be a sequence of decreasing functions f

: I → R ⁺ on the interval I. Assume that 0 ≤ f

≤ f

. Then

n→∞

lim Z

I

f

(x) dx = Z

I

n→∞

lim f

(x) dx.

Proof. By the assumption that R f

(x) dx increases, the interchangeability of supremums, and (1.6) it holds that

n→∞

lim Z

I

f

(x) dx = sup

n

Z

I

f

(x) dx = sup

n

sup

d

S(f

, d) = sup

d

sup

n

S(f

, d), (1.7)

12 1. SET FUNCTIONS AND THE CHOQUET INTEGRAL

since for increasing sequences it holds that the limit is equal to the supremum. By observing the inner supremum on the right hand side of (1.7), it holds that

sup

n

S(f

, d) = sup

n

sup

k k

X

i=−k

f

(d

)(d

− d

) = sup

k

sup

n k

X

i=−k

f

(d

)(d

− d

)

= sup

k

lim

X

i=−k

f

(d

)(d

− d

) = sup

k k

X

i=−k

lim

f

(d

)(d

− d

)

=

∞

X

i=−∞

lim

f

(d

)(d

− d

) = S(lim

n

f

, d).

From this it follows that

n→∞

lim Z

I

f

(x) dx = sup

d

sup

n

S(f

, d) = sup

d

S( lim

n→∞

f

, d) = Z

I

n→∞

lim f

(x) dx.

 Having defined the integral for decreasing functions, we can make the following definition by the fact that the quantile function is a decreasing function.

Definition 1.23. Let µ be a nonnegative real valued monotone set function on S ⊆ 2 ^Ω . Given an upper µ-measurable function X : Ω → R with distribution function G

and quantile function ˇ G

the asymmetric integral of X with respect to µ is defined as

Z

Xdµ =

µ(Ω)

Z

0

G ˇ

(t) dt (1.8)

This integral is similarly referred to as the Choquet integral, sometimes denoted (C) for distinction. By Definition 1.14 the pseudo-inverse of the quantile function is the distribution function. To be able to define the Choquet integral in terms of the distribution function, it has to be shown that the integration of a decreasing function f is equivalent to the integration of its pseudo-inverse ˇ f .

Lemma 1.24. Consider the decreasing functions f, g : I → R where I ⊆ R and f = g on a dense subset of I. Then

Z

I

f (x) dx = Z

I

g(x) dx

Proof. Assume that for any ε > 0 and any subdivision of I, denoted d, there is a subdivision e = e(d) of I, such that

S(f, d) − ε ≤ S(g, e). (1.9)

Assume that (1.9) holds. By taking supremum over d in (1.9), it holds that Z

I

f (x) dx − ε ≤ sup

d

S(g, e(d)) ≤ Z

I

g(y) dy

1.2. THE ASYMMETRIC CHOQUET INTEGRAL 13

which becomes

Z

I

f (x) dx ≤ Z

I

g(y) dy

since ε was arbitrary. To show the reverse implication, interchange f and g and perform the same operations. It remains to show the validity of assumption (1.9).

It is possible that g(d

) ≤ f (d

) for a given subdivision d on I ⊆ R. In other words, S(f, d) is not a lower sum for g. Select a new point d ⁰

for each n ∈ Z on the interval d

< d

, d

< d ⁰

< d

such that

(f (d

) − g(d

))(d

− d ⁰

) < ε 3 × 2 ^|n|

Consider the subdivision as given by the points d

, d ⁰

for n ∈ Z, and let this subdivision be e. On a dense subset of I the functions f and g agree. It can be derived from d ⁰

< d

that

f (d

) ≤ g(d ⁰

).

Omitting the summands where d

− d

= 0, we get

S(f, d) − S(g, e) =

∞

X

n=−∞

(f (d

)(d

− d ⁰

) + f (d

)(d ⁰

− d

))

−

∞

X

n=−∞

(g(d

)(d

− d ⁰

) + g(d

)(d ⁰

− d

)) ≤ X

n∈

Z ε 3 × 2 ^|n|

= ε 3 1 + 2

∞

X

n=1

1 2

!

= ε.

 Proposition 1.25. For any pseudo-inverse ˇ f of a decreasing function f : R + → R + , it holds that

Z ∞ 0

f (y) dy = ˇ Z ∞

0

f (x) dx.

Proof. By Lemma 1.24 it can be assumed that ˇ f is left continuous. In other words, ˇ f assumes the largest value of

f (y) = 0 ∧ sup{x : f (x) ≥ y} ˇ for each y. We want to show that

Z ∞ 0

f (y) dy ≥ ˇ Z ∞

0

f (x) dx (1.10)

which suffices since the inverse inequality is proven in a converse manner.

Let d : Z → R + be any subdivision on the x-axis such that d

≤ d

, lim

d

= 0, and lim

d

= ∞ for n ∈ Z. Define the subdivision e = e(d) on the y-axis such that e(d) : Z → R ⁺ with e : n = f (d

) for n ∈ Z. Since the function f is decreasing, it holds that e

≤ e

, and lim

e

= 0 if R ∞

0 f (x) dx < ∞.

14 1. SET FUNCTIONS AND THE CHOQUET INTEGRAL

Assume that R ∞

0 f (x) dx < ∞. Define b = lim

e

which gives that ˇ f (y) = 0 for y > b. It follows that

Z ∞ 0

f (y) dy = ˇ Z

0

f (y) dy ˇ

where e is the subdivision of [0, b]. To show that inequality (1.10) holds, it has to be shown that

S( ˇ f , e) ≥ S(f, d) (1.11)

is true. First, note that

f (f (x)) = sup{t : f (t) ≥ f (x)} ≥ x ˇ

from which it follows that ˇ f (e

) ≥ d _−n . If we use that lim

d

= 0 and lim

f (d

) = lim

e _−n = 0, we get

S( ˇ f , e) = X

n

f (e ˇ

)(e

− e

) ≥ X

n

d −n (f (d −n ) − f (d _−(n−1) ))

= X

n

d

(f (d

) − f (d

)) = X

n

X

k≤n

(d

− d

)(f (d

) − f (d

)

= X

n

X

n≥k

(f (d

) − f (d

))(d

− d

) = X

k

f (d

)(d

− d

) = S(f, d).

Therefore inequality (1.11) holds, and it can be derived from (1.10) that Z

0

f (y) dy ≥ sup ˇ

d

S( ˇ f , e(d)) ≥ sup

d

S(f, d) = Z ∞

0

f (x) dx.

 By Proposition 1.25 the Choquet integral (1.8) can be expressed by the distri- bution function G instead of its pseudo-inverse ˇ G, such as

Z

X dµ = Z ∞

0

G

(x) dx

if X > 0 or ˇ G

(x) ≥ 0. To be able to express the Choquet integral for arbi- trary upper µ-measurable functions X, consider the following Corollary to Propo- sition 1.25.

Corollary 1.26. For a decreasing function f : [0, b] → R where b ∈ (0, ∞), we have

Z

0

f (x) dx = Z 0

−∞

( ˇ f (y) − b) dy + Z ∞

0

f (y) dy. ˇ

Proof. Since f is decreasing there is a point a ∈ [0, b] such that f (x) ≥ 0 for x < a and f (x) ≤ 0 for x > a. Split the integral of f at a

Z

0

f (x) dx = Z

0

f (x) dx + Z

a

f (x) dx.

1.2. THE ASYMMETRIC CHOQUET INTEGRAL 15

Apply Proposition 1.25 twice. First, let g = f on [0, a) and g = 0 on [a, ∞]. Then Z

0

f (x) dx = Z

0

g(x) dx = Z

0

g(x) dx + Z ∞

a

g(x) dx

= Z ∞

0

g(x) dx = Z ∞

0

ˇ

g(y) dy = Z ∞

0

f (y) dy. ˇ Secondly, define the function h(t) by

h(t) =

( 0 if t ≤ a − b, f (t + b) if a − b < t < 0.

Proposition 1.25 is valid for functions f : R − → R − , it holds by property iv) of the integral (1.6) that

Z

f (x) dx = Z 0

−∞

h(t) dt = Z 0

−∞

ˇ h(y) dy = Z 0

−∞

( ˇ f (y) − b) dy.

 By Proposition 1.15 and Proposition 1.26 it holds that

Z

X dµ = Z 0

−∞

(G

(x) − µ(Ω)) dx + Z ∞

0

G

(x) dx

if µ(Ω) < ∞ for arbitrary upper µ-measurable X. If µ is a σ-additive measure, the integral coincides with the usual except for infinite measures. In the case of infinite measures, the integral is the usual one for functions X ≥ 0.

Example 1.27. Let A ⊆ 2 ^Ω be a finite algebra, and let µ be a monotone set function on A with µ(Ω) = 1. Denote the minimal elements of A\{∅} which in addition are disjoint and unite to Ω by A ₁ , . . . , A

. Then an upper A-measurable function X on Ω is constant on the sets A

and the quantile function ˇ G

is a step function.

If the A

are enumerated so that x

= X(A

) are in descending order, i.e.

x ₁ > x ₂ > · · · > x

, then Z

X dµ = Z 1

0

G ˇ

(t) dt =

n

X

i=1

x

(µ(S

) − µ(S

)) =

n

X

i=1

(x

− x

)µ(S

) where S

= A 1 ∪· · ·∪A

for i = 1, . . . , n, S 0 = ∅, and x

= 0. Notice that the two sums correspond to the following two representations of X as linear combination of indicator functions

X =

n

X

i=1

x

χ

=

n

X

i=1

(x

− X

)1

where the last has the advantage that the summands form a class of comonotonic

functions.

16 1. SET FUNCTIONS AND THE CHOQUET INTEGRAL

If the set function µ is additive, i.e. a probability measure that can be denoted p, then p(S

) − p(S

) = p(A

) and the integral has the well known representation, know as expected probability,

Z

X dp =

n

X

i=1

X(A

)p(A

).

 Assume that the monotone set function µ is defined on 2 ^Ω for simplicity. For a set function µ defined on a arbitrary set system S ⊂ 2 ^Ω , the inner and outer set function defined on 2 ^Ω assures

Z

X dµ = Z

X dµ ^∗ = Z

X dµ ∗

for upper µ-measurable functions X. This follows by definition of upper µ-measurability, by

G

= G

= G

.

Note that a monotone increasing transformation of the real axis is compatible with performing the quantile function. Consider the monotone set function µ on 2 ^Ω and X : Ω → R. For an increasing function u : R → R and the distribution function G

it holds that

G ˇ

(x) = u ◦ ˇ G

, (1.12) which can be seen to be true for u(x) = cx for a constant c ∈ R ⁺ .

Proposition 1.28. Let µ denote a monotone set function on 2 ^Ω , and assume functions X, Y : Ω → R. Then:

i) R χ

dµ = µ(A), for A ∈ P (Ω).

ii) X ≤ Y implies R X dµ ≤ R Y dµ.

iii) R (X + c) dµ = R X dµ + cµ(Ω), for c ∈ R.

iv) If X, Y are comonotonic and real value then R (X + Y ) dµ = R X dµ + R Y dµ.

Proof. Let the set up be that of the proposition.

i) The proof is trivial.

ii) Since X ≤ Y can be relaxed to G

≤ G

, which is equivalent to G ˇ

≤ ˇ G

, the property follows readily from property ii) of (1.6).

iii) Follows from Proposition 1.12 with ˇ G

= ˇ G

+ c for c ∈ R.

iv) From Corollary 1.21 we have ˇ G

= ˇ G

+ ˇ G

, and the proof follows from property i) of (1.6).



1.2. THE ASYMMETRIC CHOQUET INTEGRAL 17

Consider an upper S-measurable function X : Ω → R and a monotone set function µ on S ⊂ 2 ^Ω . For a constant c > 0 the multiple cµ is then a monotone set function and it can be seen that

G

= cG

,

and Z

X d(cµ) = c Z

X dµ.

1.3. THE SUBADDITIVITY THEOREM 19

1.3. The Subadditivity Theorem

If the asymmetric Choquet integral is subadditive for a monotone set function µ, i.e.

Z

(X + Y ) dµ ≤ Z

X dµ + Z

Y dµ

then µ is submodular. This section is devoted to show the reverse implication, that a submodular µ is a sufficient condition for subadditivity of the asymmetric Choquet integral. This is known as the subadditivity theorem. By Proposition 1.11 the outer set function of a submodular set function µ is submodular. Without loss of generality, µ can therefore be defined on the power set of Ω, and no measurability conditions need to be imposed on the functions. Before the theorem is considered, several lemma are stated that will be of use in the proof of the theorem. The first lemma requires a interval continuity proposition for the integral (1.6).

Proposition 1.29. Let f be a decreasing function on the interval [a, b] ⊆ R.

Then it holds that

lim

x→b

Z

f (t) dt = Z

a

f (t) dt.

Proof. By property (iii) of the integral for decreasing functions (1.6), assumes that f ≤ 0, a ∈ R and by property (iv) that a = 0. Consider the increasing sequence x

tending to b, and define f

= f |[0, x

]. Then ˇ f

= x

∨ ˇ f and ˇ f

− x

≤ 0. ˇ f

is a decreasing sequence with

n→∞

lim ( ˇ f

(y) − x

) = ˇ f (y) − b

where y ≤ 0. By Corollary 1.26 and property vi) of the integral (1.6) it follows that

n→∞

lim Z

0

f (t) dt = lim

n→∞

Z 0

−∞

( ˇ f

(y) − x

) dy = Z 0

−∞

( ˇ f (y) − b) dy = Z

0

f (t) dt.

 Lemma 1.30. Let µ be a monotone set function on 2 ^Ω . For any q ∈ [0, 2 ^Ω ], define

µ

(A) = q ∧ µ(A)

for A ∈ 2 ^Ω . µ

is monotone and for an arbitrary function X : Ω → R lim

q→µ(Ω)

Z

X dµ

= Z

X dµ.

Proof. The distribution functions with respect to µ

and µ coincide if the values are below q,

G

(x) = µ

(X > x) = q ∧ µ(X > x) = q ∧ G

(x).

20 1. SET FUNCTIONS AND THE CHOQUET INTEGRAL

In other words, the quantile functions with respect to µ and µ

coincide on [0, q).

By Proposition 1.29 it follows that Z

X dµ

= Z

0

G ˇ

(t) dt

= Z

0

G ˇ

(t) dt −→

q→µ(Ω)

Z

0

G ˇ

(t) dt = Z

X dµ.

 The following lemma is used extensively to show the validity of the Subaddi- tivity theorem.

Lemma 1.31. Let Ω be the disjoint union of the sets A 1 , . . . , A

, and let A be the algebra generated by A ₁ , . . . , A

. Consider also the monotone set function µ defined as µ : A → [0, 1] such that µ(∅) = 0 and µ(Ω) = 1. For any permutation π of (1, . . . , n) define

S

=

i

[

j=1

A

for i = 1, . . . , n, and S ₀

= ∅. Define the probability measure p

on A through p

(A

) = µ(S

) − µ(S

)

for i = 1, . . . , n. Let X : Ω → R be upper A-measurable, that is constant on the sets A

. If the set function µ is submodular, then

Z

X dµ ≥ Z

X dp

,

and the equality holds if X(A

) ≥ X(A

) ≥ · · · ≥ X(A

).

Proof. We only need to prove the case π = id. Assume that for some i, we have x

= X(A

) and for i + 1 it holds x

= X(A

). Furthermore, assume that x

< x

. Denote the permutation that interchanges i and i + 1 with ϕ. Given these assumptions, the objective is to show that

Z

X dp

≥ Z

X dp

. (1.13)

Given that inequality (1.13) holds, construct a permutation π such that X(A

) ≥ X(A

) ≥ · · · ≥ X(A

)

from the permutations of type ϕ. Given the permutation above and by Example 1.27 it holds that

Z

X dp

≥ Z

X dp

whose left hand side is R X dµ. The inequality (1.13) needs to be proven for the case where the summands are not identical. For simplification, denote p = p

, S

= S

. We want to show that

x

p

(A

) + x

p

(A

) ≥ x

p(A

) + x

p(A

)