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 valueatrisk 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
Q_{c}
:= {
Q
∈
M
_{1,f}
:
Q
[
A
] ≤
c
(
A
) ∀
A
∈
F_{T}
}. There is no specific explanation in the literature for the structure of the set
Q_{c}
. It is worthy of examining it. By using
g
expectation and related topics, we’ll show that
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,(
F_{t}
)
_{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
c_{ψ}
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}
(
F_{T}
) 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 GEXPECTATION
In this section, the definition of gexpectation 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
Theorem 3.1
(
[21]
)
.
For every terminal condition
ξ
∈
L
^{2}
(
F_{T}
) :=
L
^{2}
(Ω,
F_{T}
,
P
)
the following backward stochastic differential equation
has a unique solution
Definition 3.2.
For each
ξ
∈
L
^{2}
(
F_{T}
) and for each
t
∈ [0,
T
]
g
−expectation of
X
and the conditional
g
−expectation of
X
under
F_{t}
is respectively defined by
where
y_{t}
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,
F_{T}
),
where
t
∈ [0,
T
] and Θ
^{g}
is defined as
Let’s see properties of set of priors,
Q_{c}
as in (1.1). Set
Then
is a
P
martingale since
. Also
is a
P
density on
F_{T}
since
. A probability measure
Q^{θ}
on (Ω,
F
) is equivalent to
P
, where
Q^{θ}
is defined as
We can easily see that
Q_{c}
is convex and weakly compact in
L
^{1}
(Ω,
F
,
P
). For every deterministic
τ
∈ [0,
T
] and every
B
∈
F_{τ}
,
where
denotes the restriction of
Q
^{3}
to
F_{τ}
(See the paper
[23]
for details).
If
θ
∈ Θ
^{g}
, i.e.
θ_{t}
·
z
≤
g
(
t
,
z
), then we have
θ_{t}
·
z
≤ 
g
(
t
,
z
) ≤
K

z
 and so 
θ_{t}
 ≤
K
by taking
z
=
θ_{t}
. The Girsanov transformation implies that there exists a probability measure
Q^{θ}
on the space (Ω,
F_{t}
) such that
and
,
t
∈ [0,
T
] is a
Q^{θ}
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,
z
_{1}
,
z
_{2}
∈ ℝ
^{d}
Theorem 3.5
(
[16]
)
.
Let g be independent of y and satisfy the conditions (3.1a) and (3.1c). Then
∀
ξ
∈
L
^{2}
(Ω,
F_{t}
,
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}
(
F_{T}
).
(2)
ρ
(
X
) :=
∫
(−
X
)
dc is a coherent risk measure on
L
^{2}
(
F_{T}
).
(3)
For
Q_{c}
:= {
Q
∈
M
_{1,f}
:
Q
[
A
] ≤
c
(
A
) ∀
A
∈
F_{T}
},
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
,
Q_{c}
=
Q_{max}
,
where
Q_{max}
:= {
Q
≪
P
:
α_{min}
(
Q
) = 0}
is in the representation of the convex risk measure
Peng’s gexpectation provides various features. We will use the properties of gexpectation to investigate the set of prior,
Q_{c}
. The classical mathematical expectation can be represented by the Choquet expectation if
g
is linear function of
z
. The following theorem deals with the onedimensional 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 gexpectation if and only if g is independent of y and is linear in z, i.e. there exists a continuous function
ν_{t}
such that
Set
g
(
y
,
z
,
t
) =
ν_{t}z
through the last of this paper. Then Choquet expectation is equal to
g
expectation by Theorem 4.2. I.e., there exist a capacity
c_{g}
such that
If we take
ξ
=
I_{A}
for
A
∈
F
in (4.2), then the capacity
c_{g}
satisfies
Then we can prove that
c_{g}
is submodular.
Theorem 4.3.
The capacity c_{g} 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
Ɛ_{g}
is Choquet expectation, then we have
Ɛ_{g}
[
ξ
+
η
] =
Ɛ_{g}
[
ξ
] +
Ɛ_{g}
[
η
] whenever
ξ
and
η
are comonotonic.
Note that
I
_{A∪B}
and
I
_{A∩B}
is a pair of comonotone functions for all
A
,
B
∈
F
. Hence comonotonicity and subadditivity of
Ɛ_{g}
imply
So the proof is done. ☐
Since
c_{g}
is submodular, by Theorem 4.1 we have the representation
where
Q_{cg}
:= {
Q
∈
M
_{1,f}
:
Q
[
A
] ≤
c_{g}
(
A
) ∀
A
∈
F_{T}
}.
The following is the main theorem.
Theorem 4.4.
where
Q_{cg}
is the prior set in the representation (4.3)
.
Notice that
is defined as
where
t
∈ [0,
T
] and Θ
^{g}
is defined as
Proof
. Since
has the same expression as
becomes
Q_{cg}
. Since
g
(
y
,
z
,
t
) =
ν_{t}z
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
) =
ν_{t}z
, let us consider the BSDE
The above differential equation (4.4) is reduced to
By Girsanov’s Theorem,
Brownian motion under
Q^{ν}
defined as
Therefore we have the relations
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.
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