THE SET OF PRIORS IN THE REPRESENTATION OF CHOQUET EXPECTATION WHEN A CAPACITY IS SUBMODULAR
THE SET OF PRIORS IN THE REPRESENTATION OF CHOQUET EXPECTATION WHEN A CAPACITY IS SUBMODULAR
The Pure and Applied Mathematics. 2015. Nov, 22(4): 333-342
• Received : August 22, 2015
• Accepted : November 16, 2015
• Published : November 30, 2015 PDF e-PUB PubReader PPT Export by style
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JU HONG, KIM

Abstract
We show that the set of priors in the representation of Choquet expectation is the one of equivalent martingale measures under some conditions, when given capacity is submodular. It is proven via Peng’s g -expectation and related topics.
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1. INTRODUCTION
A starting point for a mathematical definition of risk is simply as standard deviation. The more risk we take, the more we stand to lose or gain. Standard deviation (or volatility) is a kind of simple risk measure. Different families of risk measures have been proposed in literature like coherent, convex, spectral risk measures, conditional value-at-risk etc. and discussed to measure or quantify the market risks in theoretical and practical perspectives. Risk measures are also linked to insurance premiums.
Markowitz  used the standard deviation to measure the market risk in his portfolio theory but his method doesn’t tell the difference between the positive and the negative deviation. Artzer et al. [1 , 2] proposed a coherent risk measure in an axiomatic approach, and formulated the representation theorems. Fritelli  proposed sublinear risk measure to weaken coherent axioms. Heath  firstly studied the convex risk measures and Föllmer & Schied [8 , 9 , 10] and Frittelli & Rosazza Gianin  extended them to general probability spaces. They had weakened the conditions of positive homogeneity and subadditivity by replacing them with convexity.
There exist stochastic phenomena like Allais paradox and Ellsberg paradox which can not be dealt with linear mathematical expectation in economics. So Choquet  introduced a nonlinear expectation called Choquet expectation which applied to many areas such as statistics, economics and finance. Choquet expectation is equivalent to the convex(or coherent) risk measure if given capacity is submodular. But Choquet expectation has a difficulty in defining a conditional expectation. Peng  introduced a nonlinear expectation, g -expectation which is a solution of a nonlinear backward stochastic differential equation. It’s easy to define conditional expectation with Peng’s g -expectation (see papers [5 , 13 , 15 , 17 , 20 , 22] for related topics).
In this paper, we show that the set of priors in the representation of Choquet expectation is the one of equivalent martingale measures under some conditions, when the distortion is submodular. That is, if a capacity c is submodular, then we have the representation PPT Slide
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where Qc := { Q M 1,f : Q [ A ] ≤ c ( A ) ∀ A FT }. There is no specific explanation in the literature for the structure of the set Qc . It is worthy of examining it. By using g -expectation and related topics, we’ll show that PPT Slide
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for some density generator set Θ g .
This paper consists of as follows. Introduction is given in section 1. Definitions of Choquet expectation( or integral) and risk measures are stated in section 2. Definition of Peng’s g -expectation and related topics are given in section 3. The set of priors in the representation of Choquet expectation is discussed and the main Theorem 4.4 is given in section 4.
2. DEFINITIONS OF CHOQUET EXPECTATION( OR INTEGRAL) AND RISK MEASURES
In this section, we give definitions of Choquet expectation( or integral) and coherent( or convex) risk measures. Let (Ω,( Ft ) t∈[0,T] , P ) be the given filtered probability space.
Definition 2.1. A set function c : F → [0, 1] is called monotone if PPT Slide
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and normalized if PPT Slide
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The monotone and normalized set function is called a capacity . A monotone set function is called submodular or 2- alternating if PPT Slide
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Two real functions X and Y defined on Ω are called comonotonic if PPT Slide
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A class of function X is said to be comonotonic if for every pair ( X , Y ) ∈ X × X , X and Y are comonotonic.
Definition 2.2. Let ψ : [0, 1] → [0, 1] be increasing function with ψ (0) = 0 and ψ (1) = 1. The set function PPT Slide
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is called distortion of P with respect to the distortion function ψ .
The cψ defined in Definition 2.2 becomes normalized monotone function. The notion of integral with respect to a capacity is due to Choquet  .
Definition 2.3. Let c : F → [0, 1] be monotone and normalized set function. The Choquet integral or concave distortion risk measure of X L 2 ( FT ) with respect to c is defined as PPT Slide
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The following is the definition of coherent risk measure of which concept is borrowed from one of norm.
Definition 2.4. A coherent risk measure ρ : X → ℝ is a mapping satisfying for X , Y X
(1) ρ ( X ) ≥ ρ ( Y ) if X Y (monotonicity),
(2) ρ ( X + m ) = ρ ( X ) − m for m ∈ ℝ (translation invariance),
(3) ρ ( X + Y ) ≤ ρ ( X ) + ρ ( Y ) (subadditivity),
(4) ρ ( λX ) = λρ ( X ) for λ ≥ 0 (positive homogeneity).
The subadditivity and the positive homogeneity can be relaxed to a weaker quantity, i.e. convexity PPT Slide
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which means diversification should not increase the risk.
A convex risk measure ρ :→ ℝ is a functional satisfying monotonicity, translation invariance and convexity.
Definition 2.5. Choquet integral of the loss is defined as PPT Slide
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where c is a capacity.
Choquet integral of the loss ρ : X → ℝ satisfies monotonicity, cash invariance, positive homogeneity and the others.
(1) λdc = λ for constant λ (constant preserving).
(2) If X Y , then (− X ) dc (− Y ) dc (monotonicity).
(3) For λ ≥ 0, λ (− X )dc = λ (− X ) dc (positive homogeneity).
(4) If X and Y are comonotone functions, then PPT Slide
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.
(5) If c is submodular or concave function, then PPT Slide
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3. PENG’S G-EXPECTATION
In this section, the definition of g-expectation is given. Let g : Ω×[0, T ]×ℝ×ℝ n → ℝ be a function that g g ( t , y , z ) is measurable for each ( y , z ) ∈ ℝ×ℝ n and satisfy the following conditions PPT Slide
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Lager Image PPT Slide
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Theorem 3.1 (  ) . For every terminal condition ξ L 2 ( FT ) := L 2 (Ω, FT , P ) the following backward stochastic differential equation PPT Slide
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has a unique solution PPT Slide
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Definition 3.2. For each ξ L 2 ( FT ) and for each t ∈ [0, T ] g −expectation of X and the conditional g −expectation of X under Ft is respectively defined by PPT Slide
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where yt is the solution of the BSDE (3.2).
3.1. Two sets of probability measures, PPT Slide
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and PPT Slide
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Let g be independent of y and g ( t , y , 0) = 0. We define two sets of probability measures on the measurable space (Ω, FT ), PPT Slide
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where t ∈ [0, T ] and Θ g is defined as PPT Slide
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Let’s see properties of set of priors, Qc as in (1.1). Set PPT Slide
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Then PPT Slide
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is a P -martingale since PPT Slide
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. Also PPT Slide
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is a P -density on FT since PPT Slide
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. A probability measure Qθ on (Ω, F ) is equivalent to P , where Qθ is defined as PPT Slide
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We can easily see that Qc is convex and weakly compact in L 1 (Ω, F , P ). For every deterministic τ ∈ [0, T ] and every B Fτ , PPT Slide
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where PPT Slide
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denotes the restriction of Q 3 to Fτ (See the paper  for details).
If θ ∈ Θ g , i.e. θt · z g ( t , z ), then we have θt · z ≤ | g ( t , z )| ≤ K | z | and so | θt | ≤ K by taking z = θt . The Girsanov transformation implies that there exists a probability measure Qθ on the space (Ω, Ft ) such that PPT Slide
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and PPT Slide
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, t ∈ [0, T ] is a Qθ -Brownian motion.
The two prior sets, PPT Slide
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and PPT Slide
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are the same set under some conditions.
Theorem 3.3 (  ) . Let g be independent of y and satisfy the conditions (3.1a) and (3.1c). Then PPT Slide
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Definition 3.4. Let g be independent of y and satisfy the conditions (3.1a) and (3.1c). The generator g is said to be sublinear with respect to z if for a ≥ 0, z 1 , z 2 ∈ ℝ d PPT Slide
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Theorem 3.5 (  ) . Let g be independent of y and satisfy the conditions (3.1a) and (3.1c). Then PPT Slide
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ξ L 2 (Ω, Ft , P ) if and only if g is sublinear with respect to z .
4. THE SET OF PRIORS IN THE REPRESENTATION OF CHOQUET EXPECTATION
The following theorem is about the equivalent properties on the Choquet integral with respect to a capacity c .
Theorem 4.1 (  ) . For the Choquet integral with respect to a capacity c, the followings are equivalent .
(1) ρ ( X ) := (− X ) dc is a convex risk measure on L 2 ( FT ).
(2) ρ ( X ) := (− X ) dc is a coherent risk measure on L 2 ( FT ).
(3) For Qc := { Q M 1,f : Q [ A ] ≤ c ( A ) ∀ A FT }, PPT Slide
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where M 1,f := M 1,f (Ω, F ) is the set of all finitely additive normalized set functions Q : F → [0, 1].
(4) The set function c is submodular . In this case , Qc = Qmax , where Qmax := { Q P : αmin ( Q ) = 0} is in the representation of the convex risk measure PPT Slide
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Peng’s g-expectation provides various features. We will use the properties of g-expectation to investigate the set of prior, Qc . The classical mathematical expectation can be represented by the Choquet expectation if g is linear function of z . The following theorem deals with the one-dimensional Brownian motion case, and y , z ∈ ℝ.
Theorem 4.2 (  ) . Suppose that g satisfies the conditions (3.1a), (3.1b) and (3.1c) . Then there exists a Choquet expectation whose restriction to L 2 (Ω, F , P ) is equal to a g-expectation if and only if g is independent of y and is linear in z, i.e. there exists a continuous function νt such that PPT Slide
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Set g ( y , z , t ) = νtz through the last of this paper. Then Choquet expectation is equal to g -expectation by Theorem 4.2. I.e., there exist a capacity cg such that PPT Slide
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If we take ξ = IA for A F in (4.2), then the capacity cg satisfies PPT Slide
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Then we can prove that cg is submodular.
Theorem 4.3. The capacity cg in (4.2) is submodular .
Proof . By Dellacherie’s theorem in Dellacherie  , Choquet expectation on L 2 (Ω, F , P ) is comonotonic additive. That is, if Ɛg is Choquet expectation, then we have Ɛg [ ξ + η ] = Ɛg [ ξ ] + Ɛg [ η ] whenever ξ and η are comonotonic.
Note that I AB and I AB is a pair of comonotone functions for all A , B F . Hence comonotonicity and subadditivity of Ɛg imply PPT Slide
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So the proof is done.    ☐
Since cg is submodular, by Theorem 4.1 we have the representation PPT Slide
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where Qcg := { Q M 1,f : Q [ A ] ≤ cg ( A ) ∀ A FT }.
The following is the main theorem.
Theorem 4.4. PPT Slide
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where Qcg is the prior set in the representation (4.3) .
Notice that PPT Slide
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is defined as PPT Slide
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where t ∈ [0, T ] and Θ g is defined as PPT Slide
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Proof . Since PPT Slide
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has the same expression as PPT Slide
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becomes Qcg . Since g ( y , z , t ) = νtz is independent of y and satisfy the conditions (3.1a) and (3.1c), PPT Slide
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= PPT Slide
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by Theorem 3.3. Therefore, we have PPT Slide
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.    ☐
In fact, for the linear function g ( t , y , z ) = νtz , let us consider the BSDE PPT Slide
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The above differential equation (4.4) is reduced to PPT Slide
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By Girsanov’s Theorem, PPT Slide
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-Brownian motion under Qν defined as PPT Slide
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Therefore we have the relations PPT Slide
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which means g -expectation is a classical mathematical expectation.
Acknowledgements
This work was supported by the research grant of Sungshin Women’s University in 2015.
References