Advanced
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
Journal of the Korean Society of Mathematical Education Series B The Pure and Applied Mathematics. 2015. Nov, 22(4): 333-342
Copyright © 2015, Korean Society of Mathematical Education
  • Received : August 22, 2015
  • Accepted : November 16, 2015
  • Published : November 30, 2015
Download
PDF
e-PUB
PubReader
PPT
Export by style
Share
Article
Author
Metrics
Cited by
TagCloud
About the Authors
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.
Keywords
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 [18] 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 [11] proposed sublinear risk measure to weaken coherent axioms. Heath [14] firstly studied the convex risk measures and Föllmer & Schied [8 , 9 , 10] and Frittelli & Rosazza Gianin [12] 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 [4] 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 [21] 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
Lager Image
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
Lager Image
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
Lager Image
and normalized if
PPT Slide
Lager Image
The monotone and normalized set function is called a capacity . A monotone set function is called submodular or 2- alternating if
PPT Slide
Lager Image
Two real functions X and Y defined on Ω are called comonotonic if
PPT Slide
Lager Image
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
Lager Image
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 [4] .
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
Lager Image
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
Lager Image
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
Lager Image
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
Lager Image
.
(5) If c is submodular or concave function, then
PPT Slide
Lager Image
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
Lager Image
PPT Slide
Lager Image
PPT Slide
Lager Image
Theorem 3.1 ( [21] ) . For every terminal condition ξ L 2 ( FT ) := L 2 (Ω, FT , P ) the following backward stochastic differential equation
PPT Slide
Lager Image
PPT Slide
Lager Image
has a unique solution
PPT Slide
Lager Image
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
Lager Image
where yt is the solution of the BSDE (3.2).
3.1. Two sets of probability measures,
PPT Slide
Lager Image
and
PPT Slide
Lager Image
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
Lager Image
where t ∈ [0, T ] and Θ g is defined as
PPT Slide
Lager Image
Let’s see properties of set of priors, Qc as in (1.1). Set
PPT Slide
Lager Image
Then
PPT Slide
Lager Image
is a P -martingale since
PPT Slide
Lager Image
. Also
PPT Slide
Lager Image
is a P -density on FT since
PPT Slide
Lager Image
. A probability measure Qθ on (Ω, F ) is equivalent to P , where Qθ is defined as
PPT Slide
Lager Image
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
Lager Image
where
PPT Slide
Lager Image
denotes the restriction of Q 3 to Fτ (See the paper [23] 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
Lager Image
and
PPT Slide
Lager Image
, t ∈ [0, T ] is a Qθ -Brownian motion.
The two prior sets,
PPT Slide
Lager Image
and
PPT Slide
Lager Image
are the same set under some conditions.
Theorem 3.3 ( [16] ) . Let g be independent of y and satisfy the conditions (3.1a) and (3.1c). Then
PPT Slide
Lager Image
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
Lager Image
Theorem 3.5 ( [16] ) . Let g be independent of y and satisfy the conditions (3.1a) and (3.1c). Then
PPT Slide
Lager Image
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 ( [7] ) . 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
Lager Image
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
Lager Image
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 ( [3] ) . 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
Lager Image
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
Lager Image
If we take ξ = IA for A F in (4.2), then the capacity cg satisfies
PPT Slide
Lager Image
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 [6] , 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
Lager Image
So the proof is done.    ☐
Since cg is submodular, by Theorem 4.1 we have the representation
PPT Slide
Lager Image
where Qcg := { Q M 1,f : Q [ A ] ≤ cg ( A ) ∀ A FT }.
The following is the main theorem.
Theorem 4.4.
PPT Slide
Lager Image
where Qcg is the prior set in the representation (4.3) .
Notice that
PPT Slide
Lager Image
is defined as
PPT Slide
Lager Image
where t ∈ [0, T ] and Θ g is defined as
PPT Slide
Lager Image
Proof . Since
PPT Slide
Lager Image
has the same expression as
PPT Slide
Lager Image
PPT Slide
Lager Image
becomes Qcg . Since g ( y , z , t ) = νtz is independent of y and satisfy the conditions (3.1a) and (3.1c),
PPT Slide
Lager Image
=
PPT Slide
Lager Image
by Theorem 3.3. Therefore, we have
PPT Slide
Lager Image
.    ☐
In fact, for the linear function g ( t , y , z ) = νtz , let us consider the BSDE
PPT Slide
Lager Image
The above differential equation (4.4) is reduced to
PPT Slide
Lager Image
By Girsanov’s Theorem,
PPT Slide
Lager Image
-Brownian motion under Qν defined as
PPT Slide
Lager Image
Therefore we have the relations
PPT Slide
Lager Image
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
Artzner P. , Delbaen F. , Eber J.-M. , Heath D. (1989) Heath: Thinking coherently Risk 10 68 - 71
Artzner P. , Delbaen F. , Eber J.-M. , Heath D. (1999) Heath: Coherent measures of risk Mathematical Finance 9 203 - 223    DOI : 10.1111/1467-9965.00068
Chen Z. , Chen T. , Davison M. (2005) Choquet expectation and Peng’s g-expectation The Annals of Probability 33 1179 - 1199    DOI : 10.1214/009117904000001053
Choquet G. (1953) Theory of capacities Ann. Inst. Fourier (Grenoble) 5 131 - 195
Coquet F. , Hu Y. , Mémin J. , Peng S. (2002) Filtration consistent nonlinear expectations and related g-expectations Probability Theory and Related Fields 123 1 - 27    DOI : 10.1007/s004400100172
Dellacherie C. 1970 Quelques commentaires sur les prolongements de capacités. Séminaire de Probabilités V. Strasbourg Springer Lecture Notes in Math. 191 77 - 81
Föllmer H. , Schied A. 2002 Stochastic Finance: An Introduction in Discrete Time Springer-Verlag New York
Föllmer H. , Schied A. (2002) Convex measures of risk and trading constraints Finance & Stochastics 6 429 - 447    DOI : 10.1007/s007800200072
Föllmer H. , Schied A. , Sandmann K. , Schönbucher P.J. Robust preferences and convex measures of risk Springer-Verlag Advances in Finance and Stochastics
Föllmer H. , Schied A. 2004 Stochastic Finance: An introduction in discrete time Walter de Gruyter Berlin
Frittelli M. 2000 Representing sublinear risk measures and pricing rules http://www.mat.unimi.it/users/frittelli/pdf/Sublinear2000.pdf
Frittelli M. , Rosazza Gianin E. (2002) Putting order in risk measures Journal of Banking & Finance 26 1473 - 1486    DOI : 10.1016/S0378-4266(02)00270-4
He K. , Hu M. , Chen Z. (2009) The relationship between risk measures and Choquet expectations in the framework of g-expectations Statistics and Probability Letters 79 508 - 512    DOI : 10.1016/j.spl.2008.09.025
Heath D. 2000 Back to the future. Plenary lecture at the First World Congress of the Bachelier Society Paris
Jiang L. (2006) Convexity, translation invariance and subadditivity for g-expectation and related risk measures Annals of Applied Provability 18 245 - 258
Jiang L. (2009) A necessary and sufficient condition for probability measures dominated by g-expectation Statistics and Probability Letters 79 196 - 201    DOI : 10.1016/j.spl.2008.07.037
El Karoui N. , Peng S.G. , Quenez M.-C. (1997) Quenez: Backward stochastic differential equations in finance Math. Finance 7 1 - 71    DOI : 10.1111/1467-9965.00022
Markowitz H. (1952) Portfolio selection The Journal of Finance 26 1443 - 1471
Peng S. (1997) Backward SDE and related g-expectation, backward stochastic DEs Pitman 364 141 - 159
Pardoux E. , Peng S.G. (1990) Adapted solution of a backward stochastic differential equation Systems and Control Letters 14 55 - 61    DOI : 10.1016/0167-6911(90)90082-6
Peng S. , El Karoui N. , Mazliak L. (1997) Backward SDE and related g-expectations, in: Backward stochastic differential equations Pitman Res. Notes Math. Ser. 364 141 - 159
Gianin E. Rosazza (2006) Some examples of risk measures via g-expectations Insurance: Mathematics and Economics 39 19 - 34    DOI : 10.1016/j.insmatheco.2006.01.002
Chen Zengjing , Epstein Larry (2002) Ambiguity, risk and asset returns in continuous time Econometrica 70 1403 - 1443    DOI : 10.1111/1468-0262.00337