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 𝑔-expectations, 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 𝑔-expectations in the framework of the continuous-time asset pricing. 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, 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. In this paper, we show that Choquet expectation is equal to 𝑔-expectation 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 𝑔-expectation, Choquet expectation and dynamic risk measure are introduced in section 3. The relationships between Choquet expectation and 𝑔-expectation are given as in the literature in section 4. It is shown that Choquet expectation is equal to 𝑔-expectation under some conditions via { Ƒ_t } _{t∈[0, T]} -consistent expectation Ɛ in section 5. 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 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. 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 real-valued, 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 𝑔-expectation of X under Ƒ_t is respectively defined by Ɛ_𝑔[X] := y₀, Ɛ_𝑔[X|Ƒ_t] := y_t, where y_t is the solution of the BSDE (3.2). Since 𝑔-expectation and conditional 𝑔-expectation 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 real-valued 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. 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 To quantify riskiness of financial positions, coherent (or convex) risk measures, Choquet expectation and 𝑔-expectation 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 ∈ L²(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 𝑔-expectation. E[ξ + η] = E[ξ] + E[η] ∀ξ, η ∈ L²(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 𝑔-expectation even if ξ and η are comonotonic. These facts means that 𝑔-expectation 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_AX] = Ɛ_𝑔[I_Aη] ∀A ∈ Ƒ_t. The η is called the conditional 𝑔-expectation 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 𝑔-expectation satisfies Ɛ_𝑔[ξ|Ƒ_t] = ess sup_Q∈QE_Q[ξ|Ƒ_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 𝑔-expectation 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 ). 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 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 ∈ L²(Ƒ_T ) where Ɛ^u is 𝑔-expectation 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 𝑔-expectation 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[X|Ƒ_t] = Ɛ_𝑔[X|Ƒ_t] ∀X ∈ L²(Ƒ_T ). Theorem 5.5 ( [11] ). For the Choquet integral with respect to a capacity c, the following are equivalent. 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 ∈ L²(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 P-a.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 _AX ] > E_Q [1 _AY ] for each Q ∈ Q_c and there exists a such that E_Q [1 _AX ] > r > E_Q [1 _AY ]. By taking supremum on the left hand side first, we have ess sup _Q∈Qc E_Q [1 _AY ], > r > E_Q [1 _AY ] and so ess sup _Q∈Qc E_Q [1 _AX ], > r ≥ ess sup E_Q [1 _AY ], 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_Q1[Q2[A|Ƒσ, 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 ∈ L²(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 _AX ] = 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 𝑔-expectation 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 semi-continuous [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)| ≤ μ|z| such that Note that the generator 𝑔 in Theorem (5.11) should be the form of 𝑔( t ; y ; z ) = μ_tz 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_tz , 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 𝑔-expectation 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 time-consistent 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.