Dynamic Monetary Risk Measures for Bounded Discrete-Time Processes

We study dynamic monetary risk measures that depend on bounded discrete-time processes describing the evolution of ﬂnancial values. The time horizon can be ﬂnite or inﬂnite. We call a dynamic risk measure time-consistent if it assigns to a process of ﬂnancial values the same risk irrespective of whether it is calculated directly or in two steps backwards in time. We show that this condition translates into a decomposition property for the corresponding acceptance sets, and we demonstrate how time-consistent dynamic monetary risk measures can be constructed by pasting together one-period risk measures. For conditional coherent and convex monetary risk measures, we provide dual representations of Legendre{Fenchel type based on linear functionals induced by adapted increasing processes of integrable variation. Then we give dual characterizations of time-consistency for dynamic coherent and convex monetary risk measures. To this end, we introduce a concatenation operation for adapted increasing processes of integrable variation, which generalizes the pasting of probability measures. In the coherent case, time-consistency corresponds to stability under concatenation in the dual. For dynamic convex monetary risk measures, the dual characterization of time-consistency generalizes to a condition on the family of convex conjugates of the conditional risk measures at diﬁerent times. The theoretical results are applied by discussing the time-consistency of various speciﬂc examples of dynamic monetary risk measures that depend on bounded discrete-time processes. (2004), et (2005) in


Introduction
Motivated by certain shortcomings of traditional risk measures such as Value-at-Risk, Artzner et al. (1997Artzner et al. ( , 1999 gave an axiomatic analysis of capital requirements and introduced the notion of a coherent risk measure. These risk measures were further developed in Delbaen (2000Delbaen ( , 2002. In Föllmer and Schied (2002a, 2002b and Frittelli and Rosazza Gianin (2002) the more general concepts of convex and monetary risk measures were established. In all these works the setting is static, that is, the risky objects are realvalued random variables describing future financial values and the risk of such financial values is only measured at the beginning of the time-period under consideration. It has been shown that in this framework, monetary risk measures can be characterized by their acceptance sets, and dual representations of Legendre-Fenchel type have been derived for coherent as well as convex monetary risk measures. For relations of coherent and convex monetary risk measures to pricing and hedging in incomplete markets, we refer to Jaschke and Küchler (2001), Carr et al. (2001), Frittelli and Rosazza Gianin (2004), and Staum (2004).
In a multi-period or continuous-time model, the risky objects can be taken to be cash-flow streams or processes that model the evolution of financial values, and risk measurements can be updated as new information is becoming available over time.
The study of dynamic consistency or time-consistency for preferences goes back at least to Koopmans (1960). For further contributions, see for instance, Epstein and Zin (1989), Duffie and Epstein (1992), Wang (2003), Epstein and Schneider (2003).
Examples and characterizations of time-consistent coherent risk measures are given in Artzner et al. (2004), Delbaen (2003), Riedel (2004), Roorda et al. (2005). Weber (2005) studies time-consistent distribution-based convex monetary risk measures that depend on final values. Rosazza Gianin (2003) and Barrieu and El Karoui (2004) give relations between dynamic risk measures and backward stochastic differential equations. Cheridito et al. (2004Cheridito et al. ( , 2005 contain representation results for static coherent and convex monetary risk measures that depend on financial values evolving continuously in time. Frittelli and Scandolo (2005) study static risk measures for cash-flow streams in a discrete-time framework.
In this paper we consider dynamic coherent, convex monetary and monetary risk measures for discrete-time processes modelling the evolution of financial values. We simply call these processes value processes. Typical examples are: -the market value of a firm's equity -the accounting value of a firm's equity -the market value of a portfolio of financial securities -the surplus of an insurance company. We first introduce coherent, convex monetary and monetary risk measures conditional on the information available at a stopping time and study the relation between such risk measures and their acceptance sets. Then, we pair the space of bounded adapted processes with the space of adapted processes of integrable variation and provide dual representations of Legendre-Fenchel type for conditional coherent and convex monetary risk measures; see Theorems 3.16 and 3.18. In Definition 3.19, we extend the notion of relevance to our setup. It plays an important role in the consistent updating of risk measures. In Proposition 3.21, Theorem 3.23 and Corollary 3.24, we relate relevance of conditional coherent and convex monetary risk measures to a strict positivity condition in the dual.
A dynamic risk measure is a family of conditional risk measures at different times. We call it time-consistent if it fulfills a dynamic programming type condition; see Definition 4.2. In Theorem 4.6, we show that for dynamic monetary risk measures, the time-consistency condition is equivalent to a simple decomposition property of the corresponding acceptance sets. The ensuing Corollary 4.8 shows that a relevant static monetary risk measure has at most one dynamic extension that is time-consistent. Also, it is shown how arbitrary one-period monetary risk measures can be pasted together to form a time-consistent dynamic risk measures. For dynamic coherent and convex monetary risk measures, we give dual characterizations of time-consistency. To this end, we introduce a concatenation operation for adapted increasing processes of integrable variation. This generalizes the pasting of probability measures as it appears, for instance, in Wang (2003), Epstein and Schneider (2003), Artzner et al. (2004), Delbaen (2003), Riedel (2004), Roorda et al. (2005). In the coherent case, time-consistency corresponds to stability under concatenation in the dual; see Theorems 4.13, 4.15 and their Corollaries 4.14, 4.16. In the convex monetary case, the dual characterization generalizes to a condition on the convex conjugates of the conditional risk measures at different times; see Theorems 4.19,4.22 and their Corollaries 4.20,4.23. The paper concludes with a discussion of the time-consistency of various examples of dynamic monetary risk measures for processes.
We also refer to the articles Bion Nadal (2004), Detlefsen and Scandolo (2005), and Ruszczyński and Shapiro (2005), which were written independently of this paper and study conditional dual representations and time-consistency of convex monetary risk measures.

The setup and notation
We denote N = {0, 1, 2, . . . } and let (Ω, F, (F t ) t∈N , P ) be a filtered probability space with F 0 = {∅, Ω}. All equalities and inequalities between random variables or stochastic processes are understood in the P -almost sure sense. For instance, if (X t ) t∈N and (Y t ) t∈N are two stochastic processes, we mean by X ≥ Y that for P -almost all ω ∈ Ω, X t (ω) ≥ Y t (ω) for all t ∈ N. Also, equalities and inclusions between sets in F are understood in the P -almost sure sense, that is, for A, B ∈ F, we write A ⊂ B if P [A \ B] = 0. By R 0 we denote the space of all adapted stochastic processes (X t ) t∈N on (Ω, F, (F t ) t∈N , P ), where we identify processes that are equal P -almost surely. The two subspaces R ∞ and A 1 of R 0 are given by where a −1 := 0 , ∆a t := a t − a t−1 , for t ∈ N , and ||a|| A 1 := E t∈N |∆a t | .
The bilinear form ., . on R ∞ × A 1 is given by σ(R ∞ , A 1 ) denotes the coarsest topology on R ∞ such that for all a ∈ A 1 , X → X, a is a continuous linear functional on R ∞ . σ(A 1 , R ∞ ) denotes the coarsest topology on A 1 such that for all X ∈ R ∞ , a → X, a is a continuous linear functional on A 1 . We call an (F t )-stopping time τ finite if τ < ∞ and bounded if τ ≤ N for some N ∈ N. For two (F t )-stopping times τ and θ such that τ is finite and 0 ≤ τ ≤ θ ≤ ∞, we define the projection π τ,θ : where ess inf denotes the essential infimum of a family of random variables (see for instance, Proposition VI.1.1 of Neveu, 1975). ||X|| τ,θ is the R ∞ -norm of the projection π τ,θ (X) conditional on F τ . Clearly, ||X|| τ,θ ≤ ||X|| R ∞ . The risky objects considered in this paper are stochastic processes in R ∞ . But since we want to consider risk measurement at different times and discuss time-consistency questions, we also need the subspaces R ∞ τ,θ := π τ,θ R ∞ . A process X ∈ R ∞ τ,θ is meant to describe the evolution of a financial value on the discrete time interval [τ, θ] ∩ N. We assume that there exists a cash account where money can be deposited at a risk-free rate and use it as numeraire, that is, all prices are expressed in multiples of one dollar put into the cash account at time 0. We emphasize that we do not assume that money can be borrowed at the same rate. A conditional monetary risk measure on R ∞ τ,θ is a mapping ρ : , assigning a value process X ∈ R ∞ τ,θ a real number that can depend on the information available at the stopping time τ and specifies the minimal amount of money that has to be held in the cash account to make X acceptable at time τ . By our choice of the numeraire, the infusion of an amount of money m at time τ transforms a value process X ∈ R ∞ τ,θ into X + m1 [τ,∞) and reduces the risk of X to ρ(X) − m.
We find it more convenient to work with negatives of risk measures. If ρ is a conditional monetary risk measure on R ∞ τ,θ , we call φ = −ρ the conditional monetary utility function corresponding to ρ.

Conditional monetary utility functions
In this section we extend the concepts of monetary, convex and coherent risk measures to our setup and prove corresponding representation results. In all of Section 3, τ and θ are two fixed (F t )-stopping times such that 0 ≤ τ < ∞ and τ ≤ θ ≤ ∞.

Basic definitions and easy properties
In the subsequent definition we extend the axioms of Artzner et al. (1999), Föllmer and Schied (2002a), Frittelli and Rosazza Gianin (2002) for static risk measures to our dynamic framework. Now, the risky objects are value processes instead of random variables, and risk assessment at a finite (F t )-stopping time τ is based on the information described by F τ .
Axiom (M) in Definition 3.1 is the extension of the monotonicity axiom in Artzner et al. (1999) to value processes. (TI), (C) and (PH) are F τ -conditional versions of corresponding axioms in Artzner et al. (1999), Föllmer and Schied (2002a), Frittelli and Rosazza Gianin (2002). The normalization axiom (N) is convenient for the purposes of this paper. Differently normalized conditional monetary utility functions on R ∞ τ,θ can be obtained by the addition of an F τ -measurable random variable.
Definition 3.1 We call a mapping φ : R ∞ τ,θ → L ∞ (F τ ) a conditional monetary utility function on R ∞ τ,θ if it has the following properties: We call a conditional monetary utility function φ on R ∞ τ,θ a conditional concave monetary utility function if it satisfies We call a conditional concave monetary utility function φ on R ∞ τ,θ a conditional coherent utility function if it satisfies For a conditional monetary utility function φ on R ∞ τ,θ and X ∈ R ∞ , we define φ(X) := φ • π τ,θ (X).
A conditional monetary risk measure on R ∞ τ,θ is a mapping ρ : R ∞ τ,θ → L ∞ (F τ ) such that −ρ is a conditional monetary utility function on R ∞ τ,θ . ρ is a conditional convex monetary risk measure if −ρ is a conditional concave monetary utility function and a conditional coherent risk measure if −ρ is a conditional coherent utility function.
Remark 3.2 It is easy to check that a mapping φ : R ∞ τ,θ → L ∞ (F τ ) is a conditional coherent utility function on R ∞ τ,θ if and only if it satisfies (M), (TI) and (PH) of Definition 3.1 together with As in the static case, the axioms (M) and (TI) imply Lipschitz-continuity. But since here, (TI) means F τ -translation invariance instead of translation invariance with respect to real numbers, we can derive the stronger F τ -Lipschitz continuity (LC) below, which implies the local property (LP). The economic interpretation of (LP) is that a conditional monetary utility function φ on R ∞ τ,θ does only depend on future scenarios that have not been ruled out by events that have occurred until time τ . Proposition 3.3 Let φ be a function from R ∞ τ,θ to L ∞ (F τ ) that satisfies (M) and (TI) of Definition 3.1. Then it also satisfies the following two properties: Proof. It follows from (M) and (TI) that for all X, Y ∈ R ∞ τ,θ , and (LC) follows by exchanging the roles of X and Y . It can be deduced from (LC) that for all X ∈ R ∞ τ,θ and A ∈ F τ , which implies (LP).
As in the proof of Proposition 3.3, it can be deduced from (M) and (TI') that Hence, Analogously, it can be shown that under (M), (LP) and (TI'), ordinary concavity implies F τ -concavity, and positive homogeneity with respect to λ ∈ R + , implies F τ -positive homogeneity.
Next, we introduce the acceptance set C φ of a conditional monetary utility function φ on R ∞ τ,θ and show how φ can be recovered from C φ . In contrast to the static case, everything is done conditionally on F τ and the local property plays an important role.
Definition 3.5 The acceptance set C φ of a conditional monetary utility function φ on R ∞ τ,θ is given by The acceptance set C φ of a condtional monetary utility function φ on R ∞ τ,θ has the following properties: If φ is a conditional concave monetary utility function, then C φ satisfies (c) F τ -Convexity: λX + (1 − λ)Y ∈ C φ for all X, Y ∈ C φ and λ ∈ L ∞ (F τ ) such that 0 ≤ λ ≤ 1.
If φ is a conditional coherent utility function, then C φ satisfies (ph) F τ -Positive Homogeneity: λX ∈ C φ for all X ∈ C φ and λ ∈ L ∞ + (F τ ), and (sa) Stability under addition: X + Y ∈ C φ for all X, Y ∈ C φ .
Proof. (n): It follows from the definition of C φ together with (N) and (TI) of Definition 3.1 that for all n ∈ N. Hence, φ(X) ≥ 0.
(lp) follows from the fact that φ satisfies the local property (LP) of Proposition 3.3. The remaining statements of the proposition are obvious.
Remark 3.8 Note that if C satisfies the local property (lp) of Proposition 3.6 and, for given is non-empty, then it is directed upwards, and hence, contains an increasing sequence (f n ) n∈N such that lim n→∞ f n = φ C (X) almost surely (see Proposition VI.1.1 of Neveu, 1975).
Proposition 3.9 Let φ be a conditional monetary utility function on R ∞ τ,θ . Then Proposition 3.10 If C is a subset of R ∞ τ,θ that satisfies (n) and (m) of Proposition 3.6, then φ C is a conditional monetary utility function on R ∞ τ,θ and C φ C is the smallest subset of R ∞ τ,θ that contains C and satisfies the conditions (cl) and (lp) of Proposition 3.6. If C satisfies (n), (m) and (c) of Proposition 3.6, then φ C is a conditional concave monetary utility function on R ∞ τ,θ . If C satisfies (n), (m), (c) and (ph) or (n), (m), (a) and (ph) of Proposition 3.6, then φ C is a conditional coherent utility function on R ∞ τ,θ .
Proof. (N) of Definition 3.1 follows from (n) of Proposition 3.6, and (M) of Definition 3.1 from (m) of Proposition 3.6. (TI) of Definition 3.1 follows directly from the definition of φ C . By Proposition 3.6, C φ C satisfies the conditions (cl) and (lp), and it obviously contains C. To show that C φ C is the smallest subset of R ∞ τ,θ that contains C and satisfies the properties (cl) and (lp) of Proposition 3.6, we introduce the set It is the smallest subset of R ∞ τ,θ containing C and satisfying (lp). Obviously,C inherits from C the monotonicity property (m) of Proposition 3.6, and by Remark 3.8, there exists for every X ∈ C φ C , an increasing sequence (f n ) n∈N in L ∞ (F τ ) such that X − f n 1 [τ,∞) ∈C and f n a.s.
→ φ C (X) ≥ 0. Set g n := f n ∧ 0. Then, X − g n 1 [τ,∞) ∈C, and g n → 0 almost surely, which shows that C φ C is the smallest subset of R ∞ τ,θ that satisfies the condition (cl) of Proposition 3.6 and containsC. It follows that C φ C is the smallest subset of R ∞ τ,θ containing C and satisfying the conditions (cl) and (lp) of Proposition 3.6.
3.2 Dual representations of conditional concave monetary and coherent utility functions on R ∞ τ,θ In this subsection we generalize duality results of Artzner et al. (1999), Delbaen (2002), Föllmer and Schied (2002a), and Frittelli and Rosazza Gianin (2002). Similar results for coherent risk measures have been obtained by Riedel (2004) and Roorda et al. (2005). We work with conditional positive linear functionals on R ∞ τ,θ that are induced by elements in A 1 . More precisely, we define and introduce the following subsets of A 1 : Processes in D τ,θ can be viewed as conditional probability densities on the product space Ω × N and will play the role played by ordinary probability densities in the static case. ByL(F) we denote the space of all measurable functions from (Ω, F) to [−∞, ∞], where we identify two functions when they are equal P -almost surely, and we set Definition 3.11 A penalty function γ on D τ,θ is a mapping from D τ,θ toL − (F τ ) with the following property: ess sup We say that a penalty function γ on D τ,θ has the local property if for all a, b ∈ D τ,θ and A ∈ F τ .
It is easy to see that for any penalty function γ on D τ,θ , the conditional Legendre-Fenchel type transform φ(X) = ess inf is a conditional concave monetary utility function on R ∞ τ,θ . Condition (3.2) corresponds to the normalization φ(0) = 0. In the following we are going to show that every conditional concave monetary utility function φ on R ∞ τ,θ satisfying the upper semicontinuity condition of Definition 3.15 below, has a representation of the form (3.3) for γ = φ # , where φ # is defined as follows: Definition 3.12 For a conditional concave monetary utility function φ on R ∞ τ,θ and a ∈ A 1 , we define φ # (a) := ess inf
(3.5) It follows from (3.5) that φ # satisfies the local property: for all a, b ∈ A 1 and A ∈ F τ .
In addition to the properties of Remarks 3.13, φ # fulfills the following two conditional versions of σ(A 1 , R ∞ )-upper semicontinuity: Proposition 3.14 Let φ be a conditional concave monetary utility function φ on R ∞ τ,θ . Then 1. For all A ∈ F τ and m ∈ R, Then, a 0 ∈ D τ,θ , and for all X ∈ C φ and µ ∈ M , Hence, 1 A X, a 0 ≥ m for all X ∈ C φ . Since C φ has the local property (lp), the set X, a 0 τ,θ | X ∈ C φ is directed downwards, and therefore it follows from Beppo Levi's monotone convergence theorem that Then, a 0 ∈ D τ,θ , and for all X ∈ C φ , µ ∈ M and A ∈ F τ , Hence, In the representation results, Theorem 3.16 and Theorem 3.18 below, the following upper semicontinuity property for conditional utility functions plays an important role.
Theorem 3.16 The following are equivalent: (1) φ is a mapping defined on R ∞ τ,θ that can be represented as for a penalty function γ on D τ,θ .
(2) φ is a conditional concave monetary utility function on R ∞ τ,θ whose acceptance set (3) φ is a conditional concave monetary utility function on R ∞ τ,θ that is continuous for bounded decreasing sequences. Proof.
(1) ⇒ (3): If φ has a representation of the form (3.6), then it obviously is a conditional concave monetary utility function on R ∞ τ,θ . To show that it is continuous for bounded decreasing sequences, let (X n ) n∈N be a decreasing sequence in R ∞ τ,θ and X ∈ R ∞ τ,θ such that lim n→∞ X n t = X t almost surely, for all t ∈ N . (3.7) It follows from Beppo Levi's monotone convergence theorem that for every fixed a ∈ D τ,θ , lim n→∞ X n , a τ,θ = X, a τ,θ , and therefore, (3) ⇒ (2): follows from Lemma 3.17 below.
(2) ⇒ (1): By (3.4) and the definition of φ * , for all X ∈ R ∞ τ,θ and a ∈ D τ,θ . Hence, (3.10) Since Y and all the processes in C φ are in R ∞ τ,θ , the process a can be chosen in A 1 τ,θ . As C φ has the monotone property (m), it follows from (3.10) that a has to be in (A 1 τ,θ ) + . Now, we can write the two sides of (3.10) as where the second equality follows from Beppo Levi's monotone convergence theorem because C φ has the local property (lp), and therefore, the set Z, a τ,θ | Z ∈ C φ is directed downwards. Hence, it follows from (3.10) that P [B] > 0, where Note that for A = 1, a τ,θ = 0 , Hence, B ⊂ 1, a τ,θ > 0 . Define the process b ∈ D τ,θ as follows: By definition of the set B, But this contradicts (3.9). Hence, X − m1 [τ,∞) ∈ C φ , and therefore, φ(X) ≥ m for all m ∈ L ∞ (F τ ) satisfying (3.9). It follows that Together with (3.8), this proves that In particular, This shows that (2) implies (1) and that φ # is a penalty function on D τ,θ .
To prove the last two statements of the theorem, we assume that φ is a conditional concave monetary utility function on R ∞ τ,θ with a representation of the form (3.6). Then, and it immediately follows that γ(a) ≤ φ * (a) = φ # (a) for all a ∈ D τ,θ . On the other hand, suppose that γ is concave, has the local property, Hence, there exists a K ≥ 1, such that P [B K ] > 0, and therefore, Let X ∈ R ∞ τ,θ and note that since γ and φ # have the local property, the sets (3.13) In particular, ess sup a∈D τ,θγ (a) = 0. By assumption, the functionγ : (3.14) The first inequality in (3.14) and the form of the set C imply that y ≤ 0. If y < 0, then it follows from (3.14) that But this contradicts (3.13). If y = 0, there exists a λ > 0 such that Lemma 3.17 Let φ be an increasing concave function from R ∞ τ,θ to L ∞ (F τ ) that is continuous for bounded decreasing sequences. Then The mapφ : R ∞ → R given bỹ is increasing, concave and continuous for bounded decreasing sequences. Denote by G the sigma-algebra on Ω × N generated by all the sets B × {t}, t ∈ N, B ∈ F t , and by ν the measure on (Ω × N, G) given by Then R ∞ = L ∞ (Ω × N, G, ν) and A 1 can be identified with L 1 (Ω × N, G, ν). Hence, it can be deduced from the Krein-Šmulian theorem that Cφ := X ∈ R ∞ |φ(X) ≥ 0 is a σ(R ∞ , A 1 )-closed subset of R ∞ (see the proof of Theorem 3.2 in Delbaen (2002) or Remark 4.3 in Cheridito et al. (2004)). Since (X µ ) µ∈M ⊂ Cφ, it follows that E [1 A φ(X)] ≥ 0, which contradicts (3.15). Hence, φ(X) ≥ 0.
Theorem 3.18 The following are equivalent: (1) φ is a mapping defined on R ∞ τ,θ that can be represented as for a non-empty subset Q of D τ,θ .
(2) φ is a conditional coherent utility function on R ∞ τ,θ whose acceptance set C φ is a σ(R ∞ , A 1 )-closed subset of R ∞ .
(3) φ is a conditional coherent utility function on R ∞ τ,θ that is continuous for bounded decreasing sequences.
is equal to the smallest σ(A 1 , R ∞ )-closed, F τ -convex subset of D τ,θ that contains Q, and the representation (3.16) also holds with Q 0 φ instead of Q.
Proof. If (1) holds, then it follows from Theorem 3.16 that φ is a conditional concave monetary utility function on R ∞ τ,θ that is continuous for bounded decreasing sequences, and it is clear that φ is coherent. This shows that (1) implies (3). The implication (3) ⇒ (2) follows directly from Theorem 3.16. If (2) holds, then Theorem 3.16 implies that φ # is a penalty function on D τ,θ , and for all X ∈ R ∞ τ,θ .
( 3.17) Since φ # has the local property, the set φ # (a) | a ∈ D τ,θ is directed upwards and there exists a sequence (a k ) k∈N in D τ,θ such that almost surely, It can easily be deduced from the coherency of φ that for all a ∈ D τ,θ , Hence, the sets A k := φ # (a k ) = 0 are increasing in k, and k∈N A k = Ω. Therefore, and φ # (a * ) = 0 by the local property of φ # . Note that for all a ∈ D τ,θ , Hence, it follows from (3.17) that It follows from Theorem 3.16 that φ # is the largest among all penalty functions on D τ,θ that induce φ. This implies Q ⊂ Q 0 φ . By Remark 3.13.2 and Proposition 3.14.
Then, it follows from the separating hyperplane theorem that there exists an X ∈ R ∞ τ,θ such that (the first equality holds because Q τ is F τ -convex, and therefore, the set { X, a τ,θ | a ∈ Q τ } is directed downwards). But, by (3.18),

Relevance
In this subsection we generalize the relevance axiom of Artzner et al. (1999)  Definition 3.19 Let φ be a conditional monetary utility function on for all ε > 0, t ∈ N and A ∈ F t∧θ , and we define Remarks 3.20 1. If φ is a θ-relevant conditional monetary utility function on R ∞ τ,θ and ξ is an (F t )stopping time such that τ ≤ ξ ≤ θ, then, obviously, the restriction of φ to R ∞ τ,ξ is ξrelevant. 2. Assume that θ is finite. Then it can easily be checked that a conditional monetary for all ε > 0 and A ∈ F θ . Also, in this case, Proposition 3.21 Let Q rel be a non-empty subset of D rel τ,θ . Then Proof. That φ is a conditional coherent utility function on R ∞ τ,θ follows from Theorem 3.18. To show that it is θ-relevant, let ε > 0, t ∈ N, A ∈ F t∧θ and choose a ∈ Q rel . Then and it remains to show that (3.20) Denote B = E 1 A j≥t∧θ ∆a j | F τ = 0 and note that This implies B ∩ A = ∅, and therefore, (3.20).
To prove the converse of Proposition 3.21 we introduce for a conditional concave monetary utility function φ on R ∞ τ,θ and a constant K ≥ 0, the set Note that it follows from Remark 3.13.2 and Proposition 3.14.2 that Q K φ is F τ -convex and σ(A 1 , R ∞ )-closed.
Lemma 3.22 Let φ be a conditional concave monetary utility function on R ∞ τ,θ that is continuous for bounded decreasing sequences and θ-relevant. Then Proof. Fix K > 0 and t ∈ N. For a ∈ D τ,θ , we denote e t (a) := j≥t∧θ ∆a j , and we define α t := sup Since Q K φ is convex and σ(A 1 , R ∞ )-closed, and, obviously, P e t (a t ) > 0 = α t .
In the next step we show that α t = 1. Assume to the contrary that α t < 1 and denote A t := e t (a t ) = 0 . Since φ is θ-relevant, and therefore also,Â By Theorem 3.16, Hence, there must exist an a ∈ D τ,θ with P [A t ∩ {e t (a) > 0}] > 0 and φ # (a) ≥ −K on A t . We then have that and P e t (c t ) > 0 > P e t (a t ) > 0 = α t . This contradicts (3.21). Therefore, we must have α t = 1 for all t ∈ N. Finally, set and note that a * ∈ Q K φ ∩ D rel τ,θ . Theorem 3.23 Let φ be a conditional concave monetary utility function on R ∞ τ,θ that is continuous for bounded decreasing sequences and θ-relevant. Then Proof. By Theorem 3.16, which immediately shows that To show the converse inequality, we choose b ∈ D τ,θ . It follows from Lemma 3.22 that there exists a process c ∈ Q 1 φ ∩ D rel τ,θ . Then, for all n ≥ 1, This shows that and therefore, φ(X) ≥ ess inf which completes the proof.
Corollary 3.24 Let φ be a conditional coherent utility function on R ∞ τ,θ that is continuous for bounded decreasing sequences and θ-relevant. Then Proof. This corollary can either be deduced from Theorem 3.18 and Lemma 3.22 like Theorem 3.23 from Theorem 3.16 and Lemma 3.22, or from Theorem 3.23 with the arguments used in the proof of the implication (2) ⇒ (1) of Theorem 3.18.

Dynamic monetary utility functions
In this section we introduce a time-consistency condition for dynamic monetary utility functions. We show that it is equivalent to a decomposition property of the corresponding acceptance sets. For dynamic coherent and concave monetary utility functions we give dual characterizations of time-consistency.
In the whole section we fix S ∈ N and T ∈ N ∪ {∞} such that S ≤ T . For τ and θ two (F t )-stopping times such that τ is finite and S ≤ τ ≤ θ ≤ T , we define the mapping φ τ,θ : and the set C τ,θ ⊂ R ∞ τ,θ by It can easily be checked that φ τ,θ defined by (4.1) is a conditional monetary utility function on R ∞ τ,θ and that the set C τ,θ given in (4.2) is the acceptance set of φ τ,θ . Moreover, if all φ t,T are concave monetary, then so is φ τ,θ ; if all φ t,T are coherent, then φ τ,θ is coherent too; and if all φ t,T are continuous for bounded decreasing sequences, then so is φ τ,θ .
for each t ∈ [S, T ] ∩ N, every finite (F t )-stopping time θ such that t ≤ θ ≤ T and all processes X ∈ R ∞ t,T . Then it can easily be seen from Definition 4.1 that for every pair of finite (F t )-stopping times τ and θ such that S ≤ τ ≤ θ ≤ T and all processes X ∈ R ∞ τ,T .
Proposition 4.4 Let τ and θ be finite (F t )-stopping times such that 0 ≤ τ ≤ θ ≤ T . Let φ τ,T be a conditional monetary utility function on R ∞ τ,T and φ θ,T a conditional monetary utility function on R ∞ θ,T . Then the following two conditions are equivalent: (2) If X and Y are two processes in R ∞ τ,T such that Proof.
(1) ⇒ (2): If X and Y are two processes in R ∞ τ,T that satisfy (4.4), then (2) ⇒ (1): Choose X ∈ R ∞ τ,θ and define Hence, it follows from (2) that Condition (2) of Proposition 4.4 requires that if a process X coincides with another process Y between times τ and θ −1 and at θ, the capital requirement for X is bigger than for Y in every possible state of the world, then also at time τ , the capital requirement for X should be bigger than for Y . A violation of this condition clearly leads to capital requirements that are inconsistent over time.
The following proposition gives two conditions under which one-time-step time-consistency implies time-consistency.
Proposition 4.5 Let (φ t,T ) t∈[S,T ]∩N be a dynamic monetary utility function that satisfies and at least one of the following two conditions: (i) T ∈ N (ii) all φ t,T are continuous for bounded decreasing sequences.
Proof. Let us first assume that (4.6) and (i) are satisfied. Then, for t ∈ [S, T ] ∩ N, an (F t )-stopping time θ such that t ≤ θ ≤ T and a process X ∈ R ∞ t,T , we denote Y = X1 [t,θ) + φ θ,T (X)1 [θ,∞) and show φ t,T (X) = φ t,T (Y ) (4.7) by induction. For t = T , (4.7) is obvious. If t ≤ T − 1, we assume that for all Z ∈ R ∞ t+1,T and every (F t )-stopping time ξ such that t + 1 ≤ ξ ≤ T . Then it follows from the normalization (N) and local property (LP) of φ that This and assumption (4.6), together with (LP) and (N) imply If (4.6) and (ii) hold but T = ∞, we choose t ∈ [S, ∞) ∩ N, a process X ∈ R ∞ t,∞ and a finite (F t )-stopping time θ ≥ t. For all N ∈ [t, ∞) ∩ N we introduce the process By the first part of the proof, (4.8) Clearly, the sequence (X N ) is decreasing and X N t → X t almost surely for all t ∈ N. Therefore, φ t,∞ (X N ) → φ t,∞ (X) almost surely. (4.9) As mentioned after Definition 4.1, φ θ,∞ is also continuous for bounded decreasing sequences. It follows that φ θ,∞ (X N ) → φ θ,∞ (X) almost surely, and hence, which, together with (4.8) and (4.9), shows that The next result characterizes time-consistency in terms of acceptance sets. Depending on the point of view, condition (2) of Theorem 4.6 can be seen as an additivity or decomposition property of the family of acceptance sets corresponding to a dynamic utility function. In Section 7 of Delbaen (2003), this property is studied for dynamic coherent utility functions that depend on random variables.
(2) ⇒ (1): ) has to be in C τ,θ . This shows that On the other hand, if X ∈ R ∞ τ,T and f ∈ L ∞ (F τ ) such that Proposition 4.7 Let (φ t,T ) t∈[S,T ]∩N be a time-consistent dynamic monetary utility function with corresponding family of acceptance sets (C t,T ) t∈[S,T ]∩N , and let τ and θ be two finite (F t )-stopping times such that S ≤ τ ≤ θ ≤ T . Then 1. 1 A X ∈ C τ,T for all X ∈ C θ,T and A ∈ F θ .
2. If φ τ,θ is θ-relevant, and X is a process in R ∞ θ,T such that 1 A X ∈ C τ,T for all A ∈ F θ , then X ∈ C θ,T .
3. If ξ is an (F t )-stopping time such that θ ≤ ξ ≤ T and φ τ,ξ is ξ-relevant, then φ θ,ξ is ξ-relevant too. In particular, if φ S,T is T -relevant, then φ τ,T is T -relevant for every finite (F t )-stopping time such that S ≤ τ ≤ T .

Proof.
1. If X ∈ C θ,T and A ∈ F θ , then also 1 A X ∈ C θ,T . Since 0 ∈ C τ,θ , it follows from Theorem 4.6 that 1 A X = 0 + 1 A X ∈ C τ,T . 2. Assume 1 A X ∈ C τ,T for all A ∈ F θ but X / ∈ C θ,T . Then there exists an ε > 0 such that P [A] > 0, where A = {φ θ,T (X) ≤ −ε}. We get from (LP) and (N) that φ θ,T (1 A X) = 1 A φ θ,T (X) ≤ −ε1 A . By Theorem 4.6, there exist Y ∈ C τ,θ and Z ∈ C θ,T such that and therefore, η ≤ −ε1 A . But then, since φ τ,θ is θ-relevant, Y cannot be in C τ,θ , which is a contradiction. 3. Let ε > 0, t ∈ N and A ∈ F t∧ξ . Set On the other hand, by (N) and (LP), Proof. Let (φ t,T ) t∈[S,T ]∩N be a time-consistent dynamic monetary utility function with φ S,T = φ and (C t,T ) t∈[S,T ]∩N the corresponding family of acceptance sets. By Proposition 4.7.3, φ t,T is T -relevant for all t ∈ [S, T ] ∩ N. Therefore, it follows from 1. and 2. of Proposition 4.7 that for all t ∈ [S, T ] ∩ N, a process X ∈ R ∞ t,T is in C t,T if and only if 1 A X ∈ C S,T for all A ∈ F t . This shows that C t,T is uniquely determined by the acceptance set C S,T of φ. Hence, φ t,T is uniquely determined by φ.  (4.10) and the sets C t,T , t ∈ [0, T ] ∩ N by

Consistent extension of the time horizon
Then ( Proof. It can easily be checked that for all t ∈ [0, S] ∩ N, the mapping φ t,T defined in (4.10) is a conditional monetary utility function on R ∞ t,T with acceptance set C t,T given by (4.11). Also, it is obvious that φ t,T inherits concavity, positive homogeneity, and continuity for bounded decreasing sequences. To prove that (φ t,T ) t∈[0,T ]∩N is time-consistent, we fix t ∈ [0, S), a process X ∈ R ∞ t,T and a finite (F t )-stopping time θ such that t ≤ θ ≤ T . Then, it follows from definition (4.10) and the time-consistency of (φ For T ∈ N, time-consistent dynamic monetary utility functions can be defined by backwards induction as follows: For all t = 0, . . . , T , there exists only one conditional monetary utility function φ t,t on R ∞ t,t . It is given by and its acceptance set is Now, for every t = 0, . . . T − 1, let φ t,t+1 be an arbitrary conditional monetary utility function on R ∞ t,t+1 with acceptance set C t,t+1 . It can easily be checked that for all t = 0, . . . T −1, the dynamic monetary utility function (φ s,t+1 ) t+1 s=t is time-consistent. Therefore, it follows from Proposition 4.9 that a time-consistent dynamic monetary utility function (φ t,T ) t∈[0,T ]∩N can be defined recursively by The corresponding acceptance sets are given by C t,T := C t,t+1 + C t+1,t+2 + · · · + C T −1,T , t = 0, 1, . . . , T − 1 .

Concatenation of elements in
For the dual characterization of time-consistency of dynamic coherent and concave monetary utility functions, the following concatenation operation in A 1 + will play an important role: Definition 4.10 Let a, b ∈ A 1 + , θ a finite (F t )-stopping time and A ∈ F θ . Then we define the concatenation a ⊕ θ A b as follows: We call a subset Q of A 1 + c2-stable if a ⊕ θ A b ∈ Q for all a, b ∈ Q , every finite (F t )-stopping time θ and all A ∈ F θ .

Remark 4.11
Let Q be a c1-stable subset of A 1 + . Then a ⊕ θ A b ∈ Q for all a, b ∈ Q, each bounded (F t )-stopping time θ, and A ∈ F θ , Indeed, if Q is c1-stable, set for each (F t )-stopping time θ and A ∈ F θ , A n := A ∩ {θ = n}, n ∈ N. Then all of the following processes are in Q: A 0 b , a n := a n−1 ⊕ n An b , n ≥ 1 .
If θ is bounded, then a n = a ⊕ θ A b for all n such that n ≥ θ. If θ is finite, then a n → a ⊕ θ A b in ||.|| 1 A , as n → ∞.

Time-consistent dynamic coherent utility functions
In this subsection we show how time-consistency of dynamic coherent utility functions is related to stability under concatenation in A 1 + . This generalizes results of Artzner et al. (2004), Riedel (2004) and Roorda et al. (2005). Examples will be discussed in Subsections 5.1, 5.2, 5.4 and 5.5 below.
Remark 4.12 Let (φ t,T ) t∈[0,T ]∩N be a time-consistent dynamic coherent utility function such that for all t ∈ [0, T ] ∩ N and X ∈ R ∞ t,T , φ t,T (X) = ess inf a∈Q t,T X, a t,T , for a non-empty subset Q t,T of D t,T . Then, it can easily be checked that for all finite stopping times τ ≤ T , where Q τ,T is given by In the following theorem and corollary, we provide necessary dual conditions for timeconsistency of dynamic coherent utility functions (φ t,T ) t∈[0,T ]∩N such that all φ t,T are continuous for bounded decreasing sequences. where for all finite (F t )-stopping times τ ≤ T , the set Q 0 τ,T is given by Then, for every pair of finite stopping times τ and θ with 0 ≤ τ ≤ θ ≤ T , the following hold: for all a, b ∈ Q 0 τ,T and A ∈ F θ .
Proof of Corollary 4.14. It follows from Theorem 3.18 and Corollary 3.24 that By Theorem 4.13.3, Q 0 0,T is c2-stable, which immediately implies that also Q 0,rel 0,T is c2stable. To prove the rest of the corollary, let (C t,T ) t∈[0,T ]∩N be the family of acceptance sets corresponding to (φ t,T ) t∈[0,T ]∩N and fix a finite (F t )-stopping time τ ≤ T . Parts 1 and 2 of Proposition 4.7 imply that for every X ∈ R ∞ τ,T , X ∈ C τ,T ⇔ 1 A X ∈ C 0,T for all A ∈ F τ , and therefore, X ∈ C τ,T ⇔ 1 A X, a 0,T ≥ 0 for all A ∈ F τ and a ∈ Q 0,rel 0,T ⇔ X, a τ,T ≥ 0 for all a ∈ Q 0,rel 0,T .
This shows that φ τ,T and the conditional coherent utility function ess inf have the same acceptance set. Hence, they must be equal. It is clear that ess inf On the other hand, since X − φ τ,T (X)1 [τ,∞) ∈ C τ,T , it follows that 1 A X − φ τ,T (X)1 [τ,∞) , a 0,T ≥ 0 , for all A ∈ F τ and a ∈ Q 0 τ,T , and therefore, which shows that ess inf The subsequent theorem and its corollary give sufficient dual conditions for timeconsistency of dynamic coherent utility functions. (i) For every a ∈ Q t,T there exists b ∈ Q 0 t+1,T such that a 1, a t+1,T 1 [t+1,∞) = b on the set 1, a t+1,T > 0 for all a ∈ Q t,T , b ∈ Q t+1,T and A ∈ F t+1 . Then Corollary 4.16 Let Q rel be a non-empty subset of D rel 0,T such that for all a, b ∈ Q rel , every defines a time-consistent dynamic coherent utility function such that φ τ,T (X) = ess inf a∈Q rel X, a τ,T 1, a τ,T (4.12) for every finite (F t )-stopping time τ ≤ T and X ∈ R ∞ τ,T . In particular, φ τ,T (X) is Trelevant for every finite (F t )-stopping time τ ≤ T .
Proof of Theorem 4.15. By Theorem 3.18, all φ t,T are continuous for bounded decreasing sequences. Hence, by Proposition 4.5, it is enough to show that Hence, X, a t+1,T = 1, a t+1,T X, b t+1,T ≥ 1, a t+1,T φ t+1,T (X), and therefore, To show the converse inequality, we introduce the set an F t+1 -measurable partition of Ω and note that it induces φ t+1,T . Also, it follows from condition (ii) that for all a ∈ Q t,T and b ∈Q t+1,T .
Choose X ∈ R ∞ t+1,T . The set X, b t+1,T | b ∈Q t+1,T is directed downwards. Therefore, there exists a sequence (b n ) n≥1 inQ t+1,T such that X, b n t+1,T φ t+1,T (X) almost surely, and hence, for every a ∈ Q t,T , T almost surely, which shows that Proof of Corollary 4.16. Time-consistency follows from Theorem 4.15, the representation (4.12) from Remark 4.12, and T -relevance from Proposition 3.21.
As a consequence of Corollaries 4.14 and 4.16 we get the following stability result for subsets of D rel 0,T .
Corollary 4.17 Let Q rel be a non-empty subset of D rel 0,T and denote by Q rel the σ(A 1 , R ∞ )closed, convex hull of Q rel . If then the sets Q rel and Q rel ∩ D rel 0,T are c2-stable. In particular, if Q rel is c1-stable, then Q rel and Q rel ∩ D rel 0,T are c2-stable.
Proof. Define the dynamic coherent utility function By Corollary 4.16, (φ t,T ) t∈[0,T ]∩N is time-consistent. Hence, it follows from Corollary 4.14 that Q rel and Q rel ∩ D rel 0,T are c2-stable.

Time-consistent dynamic concave monetary utility functions
In this subsection we give necessary and sufficient conditions for time-consistency of dynamic concave monetary utility functions in terms of penalty functions. We use the following standard conventions for the definition of conditional expectations of random variables that are not necessarily integrable: Let f ∈L(F) and τ a finite (F t )-stopping time . If there exists a g ∈ L 1 (F) such that If there exists a g ∈ L 1 (F) such that f ≤ g, we define If X is an adapted process on (Ω, F, (F t ) t∈N , P ) taking values in the interval [m, ∞] for some m ∈ R, we define for all a ∈ A 1 + , X, a τ,θ := lim n→∞ X ∧ n, a τ,θ .
Remark 4.18 Let (φ t,T ) t∈[0,T ]∩N be a dynamic concave monetary utility function such that for each t ∈ [0, T ] ∩ N, φ t,T is given by for a penalty function γ t,T on D t,T that satisfies the local property. Then it can easily be checked that for all finite (F t )-stopping times τ ≤ T , where γ τ,T is the penalty function on D τ,T given by The subsequent theorem and corollary give necessary dual conditions for time-consistency of dynamic concave monetary utility functions (φ t,T ) t∈[0,T ]∩N such that all φ t,T are continuous for bounded decreasing sequences. Then (4.14) for every pair of finite (F t )-stopping times τ, θ such that 0 ≤ τ ≤ θ ≤ T and all a ∈ D τ,T .
Corollary 4.20 Let (φ t,T ) t∈[0,T ]∩N be a time-consistent dynamic concave monetary utility function such that φ 0,T is T -relevant and continuous for bounded decreasing sequences. Then for every finite (F t )-stopping time τ ≤ T , and for every pair of finite (F t )-stopping times τ, θ such that 0 ≤ τ ≤ θ ≤ T and all a ∈ D τ,θ .
Proof of Theorem 4.19. Let τ and θ be two finite (F t )-stopping times such that 0 ≤ τ ≤ θ ≤ T , and (C t,T ) t∈[0,T ]∩N the acceptance sets corresponding to (φ t,T ) t∈[0,T ]∩N . It follows from Remark 4.3 and Theorem 4.6 that for all a ∈ D τ,T , (4.15) and for all a ∈ D τ,T and b ∈ D θ,T , By Remark 4.18, Since φ # θ,T has the local property, the set φ # θ,T (b) | b ∈ D θ,T is directed upwards, and therefore, ess sup which together with (4.15), proves (4.14).
Proof of Corollary 4.20. By Theorem 3.16, C 0,T is a σ(R ∞ , A 1 )-closed subset of R ∞ . Let τ ≤ T be a finite (F t )stopping time and (X µ ) µ∈M a net in C τ,T such that X µ → X in σ(R ∞ , A 1 ) for some X ∈ R ∞ . Then, X ∈ R ∞ τ,T , and for each A ∈ F τ , 1 A X µ → 1 A X in σ(R ∞ , A 1 ). By Proposition 4.7.1, (1 A X µ ) µ∈M is a net in C 0,T . Hence, 1 A X ∈ C 0,T , which by Proposition 4.7.2, implies that X ∈ C τ,T . This shows that C τ,T is σ(R ∞ , A 1 )-closed. Hence, it follows from Theorem 3.16 that (4.16) By Proposition 4.7.3, φ τ,T is T -relevant, which by Theorem 3.23, implies that (4.17) By Theorem 4.19, it follows from (4.16) that for every pair of finite (F t )-stopping times τ, θ such that 0 ≤ τ ≤ θ ≤ T and all a ∈ D τ,θ .
In the proof of Theorem 4.19 we showed that for all a ∈ D τ,T and b ∈ D θ,T , and it follows from (4.16) and (4.17) that ess sup and therefore, ess sup In the next theorem and corollary we give sufficient dual conditions for time-consistency of dynamic concave monetary utility functions. For their formulation we need the following notation: Definition 4.21 Let θ be a finite (F t )-stopping time such that θ ≤ T .
For every a ∈ D 0,T , we define the process →θ a ∈ D θ,T as follows: for a penalty function γ t,T on D t,T with the local property. If for each t ∈ [0, T ) ∩ N and a ∈ D t,T , γ t,T (a) = ess sup Proof of Theorem 4.22. Fix t ∈ [0, T ) ∩ N and X ∈ R ∞ t,T . Note that for all a ∈ D t,T and b ∈ D t+1,T . Since γ t+1,T has the local property, the set It can easily be checked that for all a ∈ D t,T and b ∈ D t+1,T , (4.20) Therefore, it follows from (4.18) that for all a ∈ D t,T and n ≥ 1, This shows that for all a ∈ D t,T , It follows from (4.19) and (4.20) that for all a ∈ D t,T and b ∈ D t+1,T , Hence, for fixed a ∈ D t,T , the inequality This shows that

Special cases and examples
In much of this section the pasting of probability measures plays an important role. It can be viewed as a special case of the concatenation operation introduced in Subsection 4.3 and has appeared under different names in various contexts; see for instance, Wang (2003), Epstein and Schneider (2003), Artzner et al. (2004), Delbaen (2003), Riedel (2004), Roorda et al. (2005).
We describe probability measures on (Ω, F) which are absolutely continuous with respect to P by their Radon-Nikodym derivatives dQ/dP ∈ h ∈ L 1 (F) | h ≥ 0 , E [h] = 1 . We recall that a probability measure Q absolutely continuous with respect to P is equivalent to P if and only if f = dQ/dP > 0, in which case, for finite (F t )-stopping time θ, For all T ∈ N ∪ {∞}, we denotẽ Definition 5.1 For T ∈ N ∪ {∞}, f, g ∈D T , a finite (F t )-stopping time θ ≤ T and A ∈ F θ , we define the pasting f ⊗ θ A g by We call a subset P ofD T m1-stable if it contains f ⊗ s A g for all f, g ∈ P, every s ∈ [0, T ]∩N and A ∈ F s , and m2-stable if it contains f ⊗ θ A g for all f, g ∈ P, every finite (F t )-stopping time θ ≤ T and A ∈ F θ .
Remark 5.2 It can be shown as in Remark 4.11 that for T ∈ N, m1-stability is equivalent to m2-stability.

Dynamic coherent utility functions that depend on final values
Let T ∈ N and P a non-empty subset ofD T . Then is a non-empty subset of D 0,T , and the concatenation of two elements a = f 1 [T,∞) and b = g1 [T,∞) in Q(P) at an (F t )-stopping time θ ≤ T for a set A ∈ F θ , is equal to This shows that Q(P) is c1-stable if and only if P is m1-stable. (For T ∈ N, c1-stability is equivalent to c2-stability and m1-stability equivalent to m2-stability).
If P rel is a non-empty subset ofD rel T , then Q(P rel ) is a non-empty subset of D rel 0,T , and φ t,T (X) := ess inf defines a dynamic coherent utility function such that φ t,T is T -relevant for every t = 0, . . . , T . If P rel is m1-stable, it follows from Corollary 4.16 that (φ t,T ) T t=0 is time-consistent. On the other hand, if (φ t,T ) T t=0 is time consistent, then by Corollary 4.14, the σ(A 1 , R ∞ )closed convex hull of Q(P rel ) is c1-stable, which implies that the σ(L 1 , L ∞ )-closed, convex hull of P rel is m1-stable. This class of time-consistent dynamic coherent utility functions appears in Artzner et al. (2004), Riedel (2004), Roorda et al. (2005) and in a continuous-time setup, in Delbaen (2003).

Dynamic coherent utility functions defined by worst stopping
Let T ∈ N ∪ {∞} and P rel a non-empty m1-stable subset ofD rel and for all X ∈ R ∞ t,T , Then (φ t,T ) t∈[0,T ]∩N is a time-consistent dynamic coherent utility function such that every φ t,T is T -relevant. To see this, note that φ 0,T is a T -relevant coherent utility function on R ∞ 0,T that can be represented as φ 0,T (X) = inf where Q(P rel ) is the non-empty subset of D 0,T given by Note that, unless T ∈ N and ξ = T , an element of Q(P rel ) of the form E [f | F ξ ] 1 [ξ,∞) does not belong to D rel 0,T . But it follows from Theorem 3.18 and Corollary 3.24 that φ 0,T can also be represented as Let θ ≤ T be a finite (F t )-stopping time, A ∈ F θ and a, b two processes in Q(P rel ) of the form a = f a 1 [ξa,∞) and where ξ a ≤ T and ξ b ≤ T are finite (F t )-stopping times, f a = E f a | F ξa and f f =f a ⊗ θ Bfb and ξ = 1 B c ξ a + 1 B ξ b . It follows from the m1-stability of P rel that Q(P rel ) is c1-stable. By Theorem 3.18, Q 0 0,T is the σ(A 1 , R ∞ )-closed, convex hull of Q(P rel ). Hence, it follows from Corollary 4.17 that Q 0 0,T and Q 0,rel 0,T are c2-stable. Therefore, by Corollary 4.16, φ t,T (X) := ess inf a∈Q 0,rel 0,T X, a t,T 1, a t,T , t ∈ [0, T ] ∩ N , X ∈ R ∞ t,T , defines a time-consistent dynamic coherent utility function such that everyφ t,T is Trelevant. It can easily be checked thatφ t,T = φ t,T for all t ∈ [0, T ] ∩ N. For finite time horizon T , this class of time-consistent dynamic coherent utility functions is also discussed in Artzner et al. (2004) and in a continuous-time setup in Delbaen (2003). Up to signs they are of the same form as the super-hedging prices of American contingent claims discussed in Karatzas and Kou (1998), see also Sections 6.5, 7.3, 9.3 and 9.4 in Föllmer and Schied (2004).
By Proposition 3.3, ψ t also satisfies (LP) ψ t (1 A Y + 1 A c Z) = 1 A ψ t (X) + 1 A c ψ t (Y ) for all Y, Z ∈ L ∞ (F T ) and A ∈ F t .
Denote by Θ t,T the set of all (F t )-stopping times ξ such that t ≤ ξ ≤ T , and define a dynamic monetary utility function by φ t,T (X) := ess inf ξ∈Θ t,T ψ t (X ξ ) , t ∈ [0, T ] ∩ N , X ∈ R ∞ t,T .
Note that except linearity and σ-additivity, the operators ψ t have all the properties of conditional expectations. For T < ∞, we can proceed as in Section VI.1 of Neveu (1975) and define for all X ∈ R ∞ 0,T the process (S t (X)) T t=0 recursively by S T (X) := X T S t (X) := X t ∧ ψ t (S t+1 (X)) , for t ≤ T − 1 . It can easily be checked by backwards induction that S t (X) = ψ t (X ξ t ) = φ t,T (X) for all t = 0, . . . , T .

Dynamic coherent utility functions that depend on the infimum over time
Let T ∈ N ∪ {∞} and P rel a non-empty subset ofD rel T . For all t ∈ [0, T ] ∩ N, define and φ t,T (X) := ψ t inf s∈[t,T ]∩N X s , X ∈ R ∞ t,T .

Dynamic coherent utility functions that depend on an average over time
Let T ∈ N ∪ {∞} and P rel a non-empty subset ofD rel T . For all t ∈ [0, T ] ∩ N, define ψ t (Y ) := ess inf Then (φ t,T ) t∈[0,T ]∩N is a dynamic coherent utility function such that every φ t,T is Trelevant. If P rel is m1-stable, then (φ t,T ) t∈[0,T ]∩N is time-consistent. To show the latter, we denote for f ∈ P rel by J(f ) the process a ∈ D rel 0,T given by for all t = 0, . . . , T − 1 and X ∈ R ∞ t,T , which by Proposition 4.5, implies that (φ t,T ) T t=0 is time-consistent.
The functions ψ t are conditional robust versions of the mapping which assigns a random variable Y ∈ L ∞ (F T ) its certainty equivalent under expected exponential utility. For the relation of entropic utility functions to pricing in incomplete markets we refer to Frittelli (2000), Rouge and El Karoui (2000), and Delbaen et al. (2002). Entropic risk measures can be found in Schied (2002a, 2004) and Weber (2005). Conditional entropic risk measures and their dynamic properties are also studied in Frittelli and Rosazza Gianin (2004), Barrieu and El Karoui (2004), Mania and Schweizer (2005), and Detlefsen and Scanolo (2005).