A standard deviation has been a starting point for a mathematical definition of risk. As a remedy for drawbacks such as subadditivity property discouraging the diversification, coherent and convex risk measures are introduced in an axiomatic approach. Choquet expectation and 𝑔expectations, which generalize mathematical expectations, are widely used in hedging and pricing contingent claims in incomplete markets. The each risk measure or expectation give rise to its own pricing rules. In this paper we investigate relationships among dynamic risk measures, Choquet expectation and dynamic 𝑔expectations in the framework of the continuoustime asset pricing.
1. INTRODUCTION
Various kinds of risk measures have been proposed and discussed to measure or quantify the market risks in theoretical and practical perspectives. A starting point for a mathematical definition of risk is simply as standard deviation. Markowitz
[19]
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. Artzner et al.
[2
,
3]
proposed a coherent risk measure in an axiomatic approach, and formulated the representation theorems. Frittelli
[12]
proposed sublinear risk measures to weaken coherent axioms. Heath
[16]
firstly studied the convex risk measures and Föllmer & Schied
[9
,
10
,
11]
and Frittelli & Rosazza Gianin
[13]
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
[1]
and Ellsberg paradox
[8]
which can not be dealt with linear mathematical expectation in economics. Choquet
[6]
introduced a nonlinear expectation called Choquet expectation which applied to many areas such as statistics, economics and finance. But Choquet expectation has a difficulty in defining a conditional expectation. Peng
[21]
introduced a nonlinear expectation, 𝑔expectation which is a solution of a nonlinear backward stochastic differential equation. It’s easy to define conditional expectation with Peng’s 𝑔expectation. In this paper, we show that Choquet expectation is equal to 𝑔expectation under some conditions via {
Ƒ_{t}
}
_{t∈[0, T]}
consistent expectation
Ɛ
satisfying
Ɛ
^{μ}
domination and translability condition.
The coherent (or convex) risk measure which is a static risk measures is defined in section 2. Peng’s 𝑔expectation, Choquet expectation and dynamic risk measure are introduced in section 3. The relationships between Choquet expectation and 𝑔expectation are given as in the literature in section 4. It is shown that Choquet expectation is equal to 𝑔expectation under some conditions via {
Ƒ_{t}
}
_{t∈[0, T]}
consistent expectation
Ɛ
in section 5.
2. STATIC RISK MEASURES
Risk measures are introduced to measure or quantify investors’ risky positions such as financial contracts or contingent claims. Let ( Ω,
Ƒ
,
P
) be a probability space and
T
be a fixed horizon time. Assume that 𝒳 =
L^{p}
( Ω,
Ƒ
,
P
), with 1 ≤
p
≤ + ∞ is the space of financial positions to be quantified or measured.
L^{p}
( Ω,
Ƒ
,
P
) is endowed with its norm topology for
p
∈ (1, + ∞) and with the weak topology
σ
(
L
^{∞}
, +
L
^{1}
) for
p
= + ∞.
Definition 2.1.
A
coherent risk measure
ρ
:
𝒳
→ ℝ is a mapping satisfying for
X
,
Y
∈
𝒳

(1)ρ(X) ≥ρ(Y) ifX≤Y(monotonicity),

(2)ρ(X+m) =ρ(X) −mform∈ 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
ρ(λX + (1 − λ)Y) ≤ λρ(X) + (1 − λ)ρ(Y) ∀λ ∈ [0,1],
which means diversification should not increase the risk.
3. PENG’S 𝑔EXPECTATION AND CHOQUET EXPECTATION
Let (
W_{t}
)
_{t≥0}
a standard d−dimensional Brownian motion and (
Ƒ_{t}
)
_{t≥ 0}
the augmented filtration associated with the one generated by (
W_{t}
)
_{t≥0}
. Let
be the space of the adapted processes (
ξ_{t}
)
_{t∈[0, T]}
such that
where ∥·∥ represents the Euclidean norm on ℝ
^{n}
.
Suppose that for
t
∈ [0,
T
],
L
^{2}
(
Ƒ_{t}
) :=
L
^{2}
(Ω,
Ƒ_{t}
,
P
) is the space of realvalued,
Ƒ_{t}
measurable and square integrable random variables endowed with the
L
^{2}
norm ∥·∥
_{2}
topology.
Let 𝑔 : Ω × [0,
T
] × ℝ × ℝ
^{n}
→ ℝ 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
(
[20]
)
.
For every terminal condition X
∈
L
^{2}
(
Ƒ_{T}
)
the following backward stochastic differential equation
has a unique solution
Definition 3.2.
For each
X
∈
L
^{2}
(
Ƒ_{T}
) and for each
t
∈ [0,
T
] 𝑔 expectation of
X
and the conditional 𝑔expectation of
X
under
Ƒ_{t}
is respectively defined by
Ɛ_{𝑔}[X] := y_{0}, Ɛ_{𝑔}[XƑ_{t}] := y_{t},
where
y_{t}
is the solution of the BSDE (3.2).
Since 𝑔expectation and conditional 𝑔expectation can be considered as the extension of classic mathematical expectation and conditional mathematical expectation, they preserve most properties of classic mathematical expectation and conditional mathematical expectation except the linearity.
Definition 3.3.
A realvalued set function
c
:
Ƒ
→ [0, 1] is called
capacity
if it satisfies (1)
c
(
A
) ≤
c
(
B
) for
A
⊂
B
, (2)
c
(∅) = 0 and
c
(Ω) = 1.
Definition 3.4.
A capacity is called
submodular
or 2
alternating
if
c(A∪B) + c(A∩B) ≤ c(A) + c(B).
Definition 3.5.
Two measurable functions
X
and
Y
on (Ω,
Ƒ
) are called
comonotone
if there exists a measurable function
Z
on (Ω,
Ƒ
) and increasing functions
f
and 𝑔 on ℝ such that
X = f(Z) and Y = 𝑔(Z).
A risk measure
ρ
on
L^{p}
(
Ƒ_{T}
) is called
comonotonic
if
ρ(X + Y) = ρ(X) + ρ(Y)
whenever
X
and
Y
are comonotonic.
Define the Choquet integral of the loss as
Then
ρ
: 𝒳 → ℝ satisfies monotonicity, translation invariance and positive homogeneity, and other properties according to the given conditions.

(1) (Constant preserving)∫λdc= λ for constant λ.

(2) (Monotonicity) IfX≤Y, then∫(−X)dc≥∫(−Y)dc.

(3) (Positive homogeneity) For λ ≥ 0,∫λ(−X)dc= λ∫(−X)dc.

(4) (Translation invariance)∫(−X+m)dc=∫(−X)dc+m,m∈ ℝ.

(5) (Comonotone additivity) IfXandYare comonotone functions, then

(6) (Subadditivity) Ifcis submodular or concave function, then
The static risk measures do not account for payoffs or new information according to the time evolution(refer to
[25
,
26]
).
Definition 3.6.
A dynamic risk measures are defined as the mappings (
ρ_{t}
)
_{t∈[0, T]}
satisfying

(1)ρt:Lp(ƑT) →L0(Ω,Ƒt,P), for allt∈ [0,T],

(2)ρ0is a static risk measure,

(3)ρT(X) = −X P−a.s., for allX∈Lp(ƑT).
4. NONLINEAR EXPECTATIONS AND NONLINEAR PRICING
To quantify riskiness of financial positions, coherent (or convex) risk measures, Choquet expectation and 𝑔expectation are widely used. It depends on practitioner’s appropriate choices. The paper
[5]
shows that the pricing with the coherent risk measure is less than one with the Choquet expectation.
Denote the Choquet expectation 𝒞(·) as 𝒞
_{𝑔}
(·) with respect to the capacity
V
𝑔 defined as
V_{𝑔}(A) := Ɛ_{𝑔}[I_{A}] ∀A ∈ Ƒ_{T}.
Theorem 4.1
(
[5]
).
If Ɛ
_{𝑔}
[·]
is a coherent risk measure, then
Ɛ_{𝑔}
[·]
is bounded by the Choquet expectation
𝒞_{𝑔}
(·),
that is
Ɛ_{𝑔}[X] ≤ 𝒞_{𝑔}(X), X ∈ L^{2}(Ω, Ƒ, P)
But if
Ɛ_{𝑔}
[·]
is a convex risk measure, then the above inequality does not hold generally.
Theorem 4.2
(
[15]
).
Let 𝑔 be convex function with respect to z, independent of y and deterministic. Let 𝑔 also satisfy (3.1). Then ρ^{𝑔}
(
X
) ≤
𝒞_{𝑔}
[−
X
]
for
X
∈
L
^{2}
(
Ƒ_{T}
)
if and only if ρ^{𝑔} is a coherent risk measure. Here ρ^{𝑔}
(
X
)
is defined as ρ^{𝑔}
(
X
) :=
Ɛ_{𝑔}
[−
X
]
for
X
∈
L
^{2}
(
Ƒ_{T}
).
Note that
ρ
^{𝑔}
:
L
^{2}
(
Ƒ_{T}
) ⟼ ℝ is a coherent (or convex) risk measure if and only if 𝑔 is independent of
y
and is positively homogeneous and subadditive (or convex) with respect to
z
(see
[23
,
14
,
22]
).
The positive homogeneity and comonotonic additivity hold in the Choquet expectation. The time consistency holds in the 𝑔expectation.
E[ξ + η] = E[ξ] + E[η] ∀ξ, η ∈ L^{2}(Ω, Ƒ, P).
The above equality holds for the Choquet expectation if
ξ
and
η
are comonotonic. But if 𝑔 is nonlinear, the above equality does not hold for the 𝑔expectation even if
ξ
and
η
are comonotonic. These facts means that 𝑔expectation is more nonlinear than the Choquet expectation on
L
^{2}
(Ω,
Ƒ
,
P
)
[15]
.
The following Lemmas (4.3) and (4.6), Proposition (4.4), and Theorem (4.5) are from the paper
[5]
.
Lemma 4.3.
For any X
∈
L
^{2}
(Ω,
Ƒ
_{T}
,
P
),
there exists unique
η
∈
L
^{2}
(Ω,
Ƒ_{t}
,
P
)
such that
Ɛ_{𝑔}[I_{A}X] = Ɛ_{𝑔}[I_{A}η] ∀A ∈ Ƒ_{t}.
The η is called the conditional 𝑔expectation of X and it is written as
Ɛ_{𝑔}
[
X

Ƒ
_{t}
].
This Ɛ_{𝑔}
[
X

Ƒ
_{t}
] is exactly the
y_{t} which is the solution of BSDE (3.2).
Proposition 4.4.
Let μ
= {
μ_{t}
}
_{t∈[0, T]}
be a continuous functions. Suppose that g
(
t
,
y
,
z
) =
μ_{t}

z_{t}

and the process
(
z_{t}
}
_{t∈[0, T)}
is one dimensional. Then for any
ξ
∈
L
^{2}
(Ω,
Ƒ
,
P
),
the conditional 𝑔expectation satisfies
Ɛ_{𝑔}[ξƑ_{t}] = ess sup_{Q∈Q}E_{Q}[ξƑ_{t}] for μ > 0
where Q is a set of probability measures defined as
Theorem 4.5
(
[5]
).
Suppose that 𝑔 satisfies the given Hypotheses. Then there exists a Choquet expectation whose restriction to
L
^{2}
(Ω,
Ƒ
,
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 v
(
t
)
such that
𝑔(y, z, t) = v(t)z.
Lemma 4.6.
Suppose that 𝑔 is a convex (or concave) function. If
Ɛ_{𝑔}
[·]
is comonotonic additive on
,
then
Ɛ_{𝑔}
[·
Ƒ
_{t}
]
is also comonotonic additive on
for any
t
∈ [0,
T
).
Corollary 4.7.
Suppose that 𝑔 is a convex (or concave) function. If
Ɛ_{𝑔}
[·]
is a Choquet expectation on
then Ɛ_{𝑔}
[·
Ƒ
_{t}
]
is also a Choquet expectation on
for any t
∈ [0;
T
).
5.ƑtCONSISTENT EXPECTATION
In this section, an {
Ƒ
_{t}
}
_{t∈[0, T]}
consistent expectation
Ɛ
is defined as a nonlinear functional on
L
^{2}
(
Ƒ
_{T}
). We’ll show that Choquet expectation is an {
Ƒ
_{t}
}
_{t∈[0, T]}
consistent expectation
Ɛ
under some conditions.
Definition 5.1.
A
nonlinear expectation
is defined as a functional
Ɛ
:
L
^{2}
(
Ƒ
_{T}
) → ℝ satisfying

(1) (Monotonicity) IfX≥YPa.s., thenƐ(X) ¸Ɛ(Y). Moreover, under the inequality X ≥ Y ,Ɛ(X) =Ɛ(Y) if and only ifX=YPa.s..

(2) (Constancy)Ɛ(c) =c∀c∈ ℝ.
Definition 5.2.
An {
Ƒ
_{t}
}
_{t∈[0, T]}

consistent expectation
is defined as the nonlinear expectation
Ɛ
such that if for any
X
∈
L
^{2}
(
Ƒ_{T}
) and any
t
∈ [0;
T
] there exists
η
∈
L
^{2}
(
Ƒ_{t}
) satisfying
The
η
satisfying (5.1) is called
conditional
{
Ƒ
_{t}
}
_{t∈[0, T]}

consistent
expectation of
X
under
Ƒ
_{t}
and denoted by
Ɛ
[
X

Ƒ_{t}
].
Definition 5.3.
It is called that {
Ƒ
_{t}
}
_{t∈[0, T]}
consistent expectation
Ɛ
is
dominated
by
Ɛ^{u}
(
u
> 0) if
Ɛ[X + Y ] − Ɛ[X] ≤ Ɛ^{u}[Y] ∀X, Y ∈ L^{2}(Ƒ_{T} )
where
Ɛ^{u}
is 𝑔expectation with 𝑔(
t
,
y
,
z
) =
u

z
.
An {
Ƒ
_{t}
}
_{t∈[0, T]}
consistent expectation
Ɛ
is called to satisfy the
translability condition
if
The following theorem tells us the relationships between conditional 𝑔expectation and {
Ƒ
_{t}
}
_{t∈[0, T]}
consistent expectation.
Theorem 5.4
(
[7]
).
Let
Ɛ
:
L
^{2}
(
Ƒ_{T}
) → ℝ
be a
{
Ƒ
_{t}
}
_{t∈[0, T]}

consistent expectation
.
If
Ɛ
is
Ɛ^{u}

dominated for some
u
> 0
and if it satisfies translability condition (5.2), then there exists a unique 𝑔 which is independent of y, satisfies the assumptions (3.1) and
𝑔(
t
,
z
) ≤
u

z
 such that
Ɛ[X] = Ɛ_{𝑔}[X] and Ɛ[XƑ_{t}] = Ɛ_{𝑔}[XƑ_{t}] ∀X ∈ L^{2}(Ƒ_{T} ).
Theorem 5.5
(
[11]
).
For the Choquet integral with respect to a capacity c, the following are equivalent.

(1)ρ0(X) := ∫(−X)dc is a convex risk measure onL2(ƑT).

(2)ρ0(X) := ∫(−X)dc is a coherent risk measure onL2(ƑT).

(3)ForQc:={Q∈M1,fQ[A] ≤c(A) ∀A∈ƑT},

(4)The set function c is submodular. In this case,Qc=Qmax.
The set ℳ
_{1,f}
= ℳ
_{1,f}
(Ω,
Ƒ
) in Theorem (5.3) is the one of all finitely additive set functions
Q
:
Ƒ
→ [0; 1] which is normalized to
Q
[Ω] = 1. The
Q_{max}
is defined as
where
A_{ρ}
is defined as
A_{ρ} := {X ∈ L^{2}(Ƒ_{T}) ρ(X) ≤ 0 }.
From the viewpoint of Proposition (4.4) and Theorem (4.5), the set
Q_{c}
of (5.3) is unnecessarily too large so that it could be reduced to a suitable set of probability measures for consistency, i.e.
It can be shown that
Q_{c}
is indeed the set of equivalent martingale measures by the following Proposition (5.6).
Proposition 5.6
(
[11]
).
If
Q
<<
P on Ƒ
,
then Q is equivalent to P if and only if
Pa.s.
Assume that the capacity
c
is submodular. Under the new set
Q_{c}
as in (5.4), we define a nonlinear expectation
as
We will show that the above
Ɛ
[
X
] satisfies all the assumptions of Theorem (5.4). It is easy to show that
Ɛ
[
X
] satisfies the monotonicity and constancy in the Defini tion (5.1) but if
X
≥
Y
,
Ɛ
[
X
] =
Ɛ
[
Y
] if and only if
X
=
Y
P
a.s.. Suppose that
X
≥
Y
and
Ɛ
[
X
] =
Ɛ
[
Y
]. We prove it contrapositively. Suppose
X
=
Y
P
a.s. does not hold. Let
A
= {
w
∈ Ω 
X
≠
Y
} ∈ Ƒ. Then
E_{Q}
[1
_{A}X
] >
E_{Q}
[1
_{A}Y
] for each
Q
∈
Q_{c}
and there exists a
such that
E_{Q}
[1
_{A}X
] >
r
>
E_{Q}
[1
_{A}Y
]. By taking supremum on the left hand side first, we have ess sup
_{Q∈Qc}
E_{Q}
[1
_{A}Y
], >
r
>
E_{Q}
[1
_{A}Y
] and so ess sup
_{Q∈Qc}
E_{Q}
[1
_{A}X
], >
r
≥ ess sup
E_{Q}
[1
_{A}Y
], it’s a contradiction.
We need the stability property of a set
Q_{c}
to show that
Ɛ
[
X
] is a {
Ƒ
_{t}
}
_{t∈[0, T]}
 consistent expectation. In the following definitions, the stopping times
σ
and
τ
can be replaced by
t
∈ [0,
T
] without any loss.
Definition 5.7.
Let
Q
_{1}
and
Q
_{2}
be two equivalent probability measures and
σ
be a stopping time. The probability measure
[A]:=E_{Q}_{1}[Q2[AƑσ, A ∈ Ƒ_{T}
is called the pasting of
Q
_{1}
and
Q
_{2}
in
σ
.
Note that by the monotone convergence theorem for conditional expectation
is a probability measure and
Definition 5.8.
A set 𝒬 of equivalent probability measures on (Ω, Ƒ) is called stable if, for any
Q
_{1}
,
Q
_{2}
∈ 𝒬 and the stopping time
σ
, also their pasting in
σ
is contained in 𝒬.
Proposition 5.9
(
[11]
).
The set 𝒬_{c} of equivalent martingale measures is stable.
Theorem 5.10
(
[11]
).
Let 𝒬 be a set of equivalent probability measures. If 𝒬 is stable, then the following holds for X ∈ L^{2}(Ƒ_{T} )
From the Theorem (5.10), we can easily see that
Ɛ
[
X
] is a {
}
_{t ∈ [0,T]}
consistent expectation condition (5.1), Ɛ[1
_{A}X
] = Ɛ[
X

]] ∀
A
∈
.
Let us show that {
}
_{t ∈ [0,T]}
consistent expectation
Ɛ
is dominated by
Ɛ^{μ}
(μ>0). Since
Ɛ
[
X
+
Y
] −
Ɛ
[
X
] ≤ ess sup
_{Q∈𝒬c}
E_{Q}
[
Y
] and there exists 𝑔expectation
Ɛ^{μ}
with 𝑔(
t, y, z
) =
μz
satisfying
Ɛ^{μ}
[
X
] = ess sup
_{Q∈𝒬c}
E_{Q}
[
Y
] by Theorem (4.5),
Ɛ
is dominated by
Ɛ^{μ}
. Note that
Ɛ^{μ}
dominated nonlinear expectation
Ɛ
implies that
Ɛ
is lower semicontinuous
[7]
.
Finally we show that {
}
_{t ∈ [0,T]}
consistent expectation
Ɛ
satisfies the translability condition. Let
X
∈
L
^{2}
(
) and
β
∈
L
^{2}
(
). Then by the de¯nition of
Ɛ
we have
Therefore, the nonlinear expectation
Ɛ
defined as (5.5) satisfies the all the conditions of Theorem (5.4). Thus the results so far can be summarized in the following Theorem (5.11).
Theorem 5.11.
Let the nonlinear expectation Ɛ be defined as (5.5). Then there exists a unique 𝑔 which is independent of y, satisfies the assumptions (3.1) and 𝑔(t; z) ≤ μz such that
Note that the generator 𝑔 in Theorem (5.11) should be the form of 𝑔(
t
;
y
;
z
) =
μ_{t}z
which is linear in
z
and so
Ɛ
_{𝑔}
=
Ɛ^{μ}
to be consistent to the results of Theorem (4.5).
In fact, for 𝑔(
t, y, z
) =
μ_{t}z
, let us consider the BSDE
The above differential equation (5.6) is reduced to
By Girsanov’s Theorem, (
_{t}
)
_{0≤t≤T}
is a
Q
Brownian motion under
Q
defined as
Therefore we have the relations
Ɛ
_{𝑔}
[
X
] =
E_{Q}
[
X
],
Ɛ_{𝑔}
[
X

Ƒ_{t}
] =
E_{Q}
[
X

Ƒ_{t}
]
which means that 𝑔expectation is a classical mathematical expectation.
Proposition 5.12
(
[23]
)
.
Let the risk measure
be defined as
where 𝑔 satisfies the conditions (3.1). Moreover, if 𝑔 is sublinear in (y; z), i.e. positively homogeneous in (y; z) and subadditive in (y; z), then
is a dynamic coherent and timeconsistent risk measure.
Note that if 𝑔 satisfies both positive homogeneity and subadditivity, 𝑔 is indepen dent of
y
. The proposition (5.12) and Theorem (4.2) tells us that for Theorem (5.11) to hold the linearity of 𝑔 is necessary.
Acknowledgements
This work was supported by the research grant of Sungshin Women’s University in 2013.
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