THE SET OF PRIORS IN THE REPRESENTATION OF CHOQUET EXPECTATION WHEN A CAPACITY IS SUBMODULAR

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.

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

where Qc := {Q ∈ M1,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

for some density generator set Θg.

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

and normalized if

The monotone and normalized set function is called a capacity. A monotone set function is called submodular or 2-alternating if

Two real functions X and Y defined on Ω are called comonotonic if

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

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 ∈ L2(FT ) with respect to c is defined as

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

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

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

.

(5) If c is submodular or concave function, then

 

3. PENG’S G-EXPECTATION

In this section, the definition of g-expectation is given. Let g : Ω×[0, T]×ℝ×ℝn → ℝ be a function that g ↦ g(t, y, z) is measurable for each (y, z) ∈ ℝ×ℝn and satisfy the following conditions

Theorem 3.1 ([21]). For every terminal condition ξ ∈ L2(FT ) := L2(Ω, FT , P) the following backward stochastic differential equation

has a unique solution

Definition 3.2. For each ξ ∈ L2(FT ) and for each t ∈ [0, T] g−expectation of X and the conditional g−expectation of X under Ft is respectively defined by

where yt is the solution of the BSDE (3.2).

3.1. Two sets of probability measures, and 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 ),

where t ∈ [0, T] and Θg is defined as

Let’s see properties of set of priors, Qc as in (1.1). Set

Then is a P-martingale since . Also is a P-density on FT since . A probability measure Qθ on (Ω, F) is equivalent to P, where Qθ is defined as

We can easily see that Qc is convex and weakly compact in L1(Ω, F, P). For every deterministic τ ∈ [0, T] and every B ∈ Fτ ,

where denotes the restriction of Q3 to Fτ (See the paper [23] for details).

If θ ∈ Θg, 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

and

, t ∈ [0, T] is a Qθ-Brownian motion.

The two prior sets, and 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

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, z1, z2 ∈ ℝd

Theorem 3.5 ([16]). Let g be independent of y and satisfy the conditions (3.1a) and (3.1c). Then

∀ξ ∈ L2(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 L2(FT ).

(2) ρ(X) := ∫(−X) dc is a coherent risk measure on L2(FT ).

(3) For Qc := {Q ∈ M1,f : Q[A] ≤ c(A) ∀A ∈ FT },

where M1,f := M1,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

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 L2(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

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

If we take ξ = IA for A ∈ F in (4.2), then the capacity cg satisfies

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 L2(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 IA∪B and IA∩B is a pair of comonotone functions for all A, B ∈ F. Hence comonotonicity and subadditivity of Ɛg imply

So the proof is done.    ☐

Since cg is submodular, by Theorem 4.1 we have the representation

where Qcg := {Q ∈ M1,f : Q[A] ≤ cg(A) ∀A ∈ FT }.

The following is the main theorem.

Theorem 4.4. where Qcg is the prior set in the representation (4.3).

Notice that is defined as

where t ∈ [0, T] and Θg is defined as

Proof. Since has the same expression as

becomes Qcg. Since g(y, z, t) = νtz is independent of y and satisfy the conditions (3.1a) and (3.1c), = by Theorem 3.3. Therefore, we have .    ☐

In fact, for the linear function g(t, y, z) = νtz, let us consider the BSDE

The above differential equation (4.4) is reduced to

By Girsanov’s Theorem, -Brownian motion under Qν defined as

Therefore we have the relations

which means g-expectation is a classical mathematical expectation.