El e c t ro nic
Jou r na l of
Pr
o ba b i l i t y
Vol. 11 (2006), Paper no. 3, pages 57-106.
Journal URL
http://www.math.washington.edu/∼ejpecp/
Dynamic Monetary Risk Measures for Bounded Discrete-Time Processes
Patrick Cheridito∗
ORFE Princeton University Princeton, NJ 08544
USA dito@princeton.edu
Freddy Delbaen†
Departement f¨ur Mathematik ETH Z¨urich
8092 Z¨urich Switzerland delbaen@math.ethz.ch
Michael Kupper†
Departement f¨ur Mathematik ETH Z¨urich
8092 Z¨urich Switzerland kupper@math.ethz.ch
Abstract
We study dynamic monetary risk measures that depend on bounded discrete-time processes describing the evolution of financial values. The time horizon can be finite or infinite. We call a dynamic risk measure time-consistent if it assigns to a process of financial 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 de- composition property for the corresponding acceptance sets, and we demonstrate how time-consistent dynamic monetary risk measures can be constructed by pasting to- gether 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
∗Supported by NSF grant DMS-0505932
†Supported by Credit Suisse
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 different times. The theoretical results are applied by discussing the time-consistency of various specific examples of dynamic monetary risk measures that depend on bounded discrete-time processes.
Key words: Conditional monetary risk measures, Conditional monetary utility func- tions, Conditional dual representations, Dynamic monetary risk measures, Dynamic monetary utility functions, Time-consistency, Decomposition property of acceptance sets, Concatenation of adapted increasing processes of integrable variation.
AMS 2000 Subject Classification: 91B30, 91B16.
Submitted to EJP on June 23, 2005. Final version accepted on January 3, 2006.
1 Introduction
Motivated by certain shortcomings of traditional risk measures such as Value-at-Risk, Artzner et al. (1997, 1999) gave an axiomatic analysis of capital requirements and intro- duced the notion of a coherent risk measure. These risk measures were further developed in Delbaen (2000, 2002). In F¨ollmer and Schied (2002a, 2002b, 2004) 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 real- valued 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 K¨uchler (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 mea- surements 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. (2004, 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 mea- sures 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 Def- inition 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 corre- sponding acceptance sets. The ensuing Corollary 4.8 shows that a relevant static mone- tary 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), Ro- orda 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 Ruszczy´nski and Shapiro (2005), which were written independently of this paper and study conditional dual representations and time-consistency of convex monetary risk measures.
2 The setup and notation
We denote N = {0,1,2, . . .} and let (Ω,F,(Ft)t∈N, P) be a filtered probability space withF0 ={∅,Ω}. All equalities and inequalities between random variables or stochastic processes are understood in theP-almost sure sense. For instance, if (Xt)t∈N and (Yt)t∈N are two stochastic processes, we mean byX ≥ Y that for P-almost all ω ∈Ω, Xt(ω) ≥ Yt(ω) for all t ∈N. Also, equalities and inclusions between sets in F are understood in theP-almost sure sense, that is, forA, B∈ F, we writeA⊂B ifP[A\B] = 0. ByR0 we denote the space of all adapted stochastic processes (Xt)t∈N on (Ω,F,(Ft)t∈N, P), where we identify processes that are equal P-almost surely. The two subspaces R∞ and A1 of R0 are given by
R∞:=©
X ∈ R0 | ||X||R∞ <∞ª , where
||X||R∞ := inf
½
m∈R| sup
t∈N|Xt| ≤m
¾
and
A1 :=©
a∈ R0| ||a||A1 <∞ª ,
where
a−1:= 0, ∆at:=at−at−1, fort∈N, and ||a||A1 := E
"
X
t∈N
|∆at|
# .
The bilinear formh., .ion R∞× A1 is given by hX, ai:= E
"
X
t∈N
Xt∆at
# .
σ(R∞,A1) denotes the coarsest topology onR∞ such that for all a∈ A1,X 7→ hX, ai is a continuous linear functional on R∞. σ(A1,R∞) denotes the coarsest topology on A1 such that for allX∈ R∞,a7→ hX, ai is a continuous linear functional on A1.
We call an (Ft)-stopping timeτ finite ifτ <∞and bounded ifτ ≤N for someN ∈N. For two (Ft)-stopping times τ and θ such that τ is finite and 0≤τ ≤θ ≤ ∞, we define the projectionπτ,θ:R0 → R0 by
πτ,θ(X)t:= 1{τ≤t}Xt∧θ, t∈N. ForX∈ R∞, we set
||X||τ,θ:= ess inf
½
m∈L∞(Fτ)|sup
t∈N|πτ,θ(X)t| ≤m
¾ ,
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 πτ,θ(X) conditional on Fτ. Clearly, ||X||τ,θ ≤ ||X||R∞. The risky objects considered in this paper are stochastic processes inR∞. But since we want to consider risk measurement at different times and discuss time-consistency questions, we also need the subspaces
R∞τ,θ :=πτ,θR∞.
A processX∈ 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 onR∞τ,θ is a mapping
ρ:R∞τ,θ →L∞(Fτ),
assigning a value process X ∈ R∞τ,θ a real number that can depend on the information available at the stopping timeτ and specifies the minimal amount of money that has to be held in the cash account to makeX acceptable at timeτ. By our choice of the numeraire, the infusion of an amount of moneymat time τ transforms a value process X∈ R∞τ,θ into X+m1[τ,∞) and reduces the risk of X toρ(X)−m.
We find it more convenient to work with negatives of risk measures. Ifρis a conditional monetary risk measure onR∞τ,θ, we callφ=−ρ the conditional monetary utility function corresponding toρ.
3 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 andθ are two fixed (Ft)-stopping times such that 0≤τ <∞and τ ≤θ≤ ∞.
3.1 Basic definitions and easy properties
In the subsequent definition we extend the axioms of Artzner et al. (1999), F¨ollmer 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 (Ft)-stopping time τ is based on the information described by Fτ.
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 Fτ-conditional versions of cor- responding axioms in Artzner et al. (1999), F¨ollmer 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 onR∞τ,θ can be obtained by the addition of an Fτ-measurable random variable.
Definition 3.1 We call a mapping φ : R∞τ,θ → L∞(Fτ) a conditional monetary utility function onR∞τ,θ if it has the following properties:
(N) Normalization: φ(0) = 0
(M) Monotonicity: φ(X)≤φ(Y) for all X, Y ∈ R∞τ,θ such that X≤Y
(TI) Fτ-Translation Invariance: φ(X+m1[τ,∞)) = φ(X) +m for all X ∈ R∞τ,θ and m∈L∞(Fτ)
We call a conditional monetary utility functionφonR∞τ,θ a conditional concave monetary utility function if it satisfies
(C) Fτ-Concavity: φ(λX+ (1−λ)Y) ≥λφ(X) + (1−λ)φ(Y) for all X, Y ∈ R∞τ,θ and λ∈L∞(Fτ) such that0≤λ≤1
We call a conditional concave monetary utility functionφ on R∞τ,θ a conditional coherent utility function if it satisfies
(PH)Fτ-Positive Homogeneity: φ(λX) =λφ(X)for allX∈ R∞τ,θandλ∈L∞+(Fτ) :=
{f ∈L∞(Fτ)|f ≥0}.
For a conditional monetary utility function φ on R∞τ,θ and X ∈ R∞, we define φ(X) :=
φ◦πτ,θ(X).
A conditional monetary risk measure onR∞τ,θ is a mapping ρ:R∞τ,θ → L∞(Fτ) such that
−ρ is a conditional monetary utility function onR∞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∞(Fτ) is a conditional coherent utility function onR∞τ,θif and only if it satisfies (M), (TI) and (PH) of Definition 3.1 together with
(SA) Superadditivity: φ(X+Y)≥φ(X) +φ(Y) for all X, Y ∈ R∞τ,θ.
As in the static case, the axioms (M) and (TI) imply Lipschitz-continuity. But since here, (TI) means Fτ-translation invariance instead of translation invariance with respect to real numbers, we can derive the stronger Fτ-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 timeτ.
Proposition 3.3 Letφbe a function from R∞τ,θ toL∞(Fτ) that satisfies(M)and(TI) of Definition 3.1. Then it also satisfies the following two properties:
(LC)Fτ-Lipschitz Continuity: |φ(X)−φ(Y)| ≤ ||X−Y||τ,θ, for allX, Y ∈ R∞τ,θ. (LP) Local Property: φ(1AX+ 1AcY) = 1Aφ(X) + 1Acφ(Y) for all X, Y ∈ R∞τ,θ and A∈ Fτ.
Proof. It follows from (M) and (TI) that for allX, Y ∈ R∞τ,θ,
φ(X)≤φ(Y +||X−Y||τ,θ) =φ(Y) +||X−Y||τ,θ,
Hence, φ(X)−φ(Y) ≤ ||X−Y||τ,θ, and (LC) follows by exchanging the roles of X and Y. It can be deduced from (LC) that for allX∈ R∞τ,θ and A∈ Fτ,
|1Aφ(X)−1Aφ(1AX)|= 1A|φ(X)−φ(1AX)| ≤1A||X−1AX||τ,θ= 0,
which implies (LP). ¤
Remark 3.4 It can be shown with an approximation argument that every function φ: R∞τ,θ →L∞(Fτ) satisfying (M), (LP) and the real translation invariance:
(TI’)φ(X+m1[τ,∞)) =φ(X) +m for allX ∈ R∞τ,θ and m∈R,
also fulfills theFτ-translation invariance (TI). Indeed, it follows from (TI’) and (LP) that (TI’) also holds formof the form
m=
K
X
k=1
mk1Ak, (3.1)
where mk ∈ R and Ak ∈ Fτ for k = 1, . . . , K. For general m ∈ L∞(Fτ), there exists a sequence (mn)n≥1 of elements of the form (3.1) such that||m−mn||L∞ → ∞, as n→ ∞.
As in the proof of Proposition 3.3, it can be deduced from (M) and (TI’) that
|φ(X)−φ(Y)| ≤ ||X−Y||R∞ for allX, Y ∈ R∞τ,θ.
Hence,
φ(X+m1[τ,∞)) = lim
n→∞φ(X+mn1[τ,∞)) = lim
n→∞φ(X) +mn=φ(X) +m .
Analogously, it can be shown that under (M), (LP) and (TI’), ordinary concavity im- pliesFτ-concavity, and positive homogeneity with respect toλ∈R+, impliesFτ-positive homogeneity.
Next, we introduce the acceptance set Cφ of a conditional monetary utility function φ on R∞τ,θ and show how φ can be recovered from Cφ. In contrast to the static case, everything is done conditionally onFτ and the local property plays an important role.
Definition 3.5 The acceptance set Cφ of a conditional monetary utility function φ on R∞τ,θ is given by
Cφ:=©
X∈ R∞τ,θ|φ(X)≥0ª .
Proposition 3.6 The acceptance set Cφ of a condtional monetary utility function φ on R∞τ,θ has the following properties:
(n) Normalization: ess inf©
f ∈L∞(Fτ)|f1[τ,∞)∈ Cφ
ª= 0.
(m) Monotonicity: X∈ Cφ,Y ∈ R∞τ,θ, X ≤Y ⇒ Y ∈ Cφ
(cl) Fτ-Closedness: (Xn)n∈N⊂ Cφ, X∈ R∞τ,θ, ||Xn−X||τ,θ a.s.
→ 0 ⇒ X ∈ Cφ. (lp) Local Property1AX+ 1AcY ∈ Cφ for allX, Y ∈ Cφ and A∈ Fτ.
If φis a conditional concave monetary utility function, then Cφ satisfies
(c) Fτ-Convexity: λX + (1−λ)Y ∈ Cφ for all X, Y ∈ Cφ and λ ∈ L∞(Fτ) such that 0≤λ≤1.
If φis a conditional coherent utility function, then Cφ satisfies
(ph)Fτ-Positive Homogeneity: λX ∈ Cφ for allX ∈ Cφ and λ∈L∞+(Fτ), and (sa) Stability under addition: X+Y ∈ Cφ for all X, Y ∈ Cφ.
Proof. (n): It follows from the definition ofCφ together with (N) and (TI) of Definition 3.1 that
ess inf©
f ∈L∞(Fτ)|f1[τ,∞)∈ Cφ
ª= ess inf©
f ∈L∞(Fτ)|φ(f1[τ,∞))≥0ª
= ess inf{f ∈L∞(Fτ)|φ(0) +f ≥0}= ess inf{f ∈L∞(Fτ)|f ≥0}= 0. (m) follows directly from (M) of Definition 3.1.
(cl): Let (Xn)n∈N be a sequence in Cφ and X ∈ R∞τ,θ such that ||Xn−X||τ,θ a.s.
→ 0. By (LC) of Proposition 3.3,
φ(X)≥φ(Xn)− ||Xn−X||τ,θ, for alln∈N. Hence,φ(X)≥0.
(lp) follows from the fact that φsatisfies the local property (LP) of Proposition 3.3.
The remaining statements of the proposition are obvious. ¤
Definition 3.7 For an arbitrary subsetC of R∞τ,θ, we define for all X ∈ R∞τ,θ, φC(X) := ess sup©
f ∈L∞(Fτ)|X−f1[τ,∞)∈ Cª , with the convention
ess sup∅:=−∞.
Remark 3.8 Note that if C satisfies the local property (lp) of Proposition 3.6 and, for givenX ∈ R∞τ,θ, the set
©f ∈L∞(Fτ)|X−f1[τ,∞)∈ Cª
is non-empty, then it is directed upwards, and hence, contains an increasing sequence (fn)n∈N such that limn→∞fn = φC(X) almost surely (see Proposition VI.1.1 of Neveu, 1975).
Proposition 3.9 Letφbe a conditional monetary utility function onR∞τ,θ. ThenφCφ=φ.
Proof. For all X∈ R∞τ,θ,
φCφ(X) = ess sup©
f ∈L∞(Fτ)|X−f1[τ,∞)∈ Cφ
ª
= ess sup©
f ∈L∞(Fτ)|φ(X−f1[τ,∞))≥0ª
= ess sup{f ∈L∞(Fτ)|φ(X)≥f}=φ(X).
¤ Proposition 3.10 If C is a subset of R∞τ,θ that satisfies (n) and (m) of Proposition 3.6, thenφC is a conditional monetary utility function on R∞τ,θ and CφC is the smallest subset of R∞τ,θ that contains C and satisfies the conditions(cl) and (lp) of Proposition 3.6.
IfC satisfies(n),(m)and(c)of Proposition 3.6, thenφC 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 φC 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 φC. By Proposition 3.6, CφC satisfies the conditions (cl) and (lp), and it obviously containsC. To show thatCφC is the smallest subset ofR∞τ,θ that contains C and satisfies the properties (cl) and (lp) of Proposition 3.6, we introduce the set
C˜:=
( K X
k=1
1AkXk |K≥1,(Ak)Kk=1 anFτ-partition of Ω, Xk∈ C for all k )
.
It is the smallest subset of R∞τ,θ containing C and satisfying (lp). Obviously, ˜C inherits fromCthe monotonicity property (m) of Proposition 3.6, and by Remark 3.8, there exists for everyX ∈ CφC, an increasing sequence (fn)n∈N inL∞(Fτ) such thatX−fn1[τ,∞)∈C˜
and fn a.s.
→ φC(X) ≥ 0. Set gn := fn∧0. Then, X−gn1[τ,∞) ∈ C, and˜ gn → 0 almost surely, which shows that CφC is the smallest subset of R∞τ,θ that satisfies the condition (cl) of Proposition 3.6 and contains ˜C. It follows that CφC is the smallest subset of R∞τ,θ containingC 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), F¨ollmer 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 ele- ments in A1. More precisely, we define
hX, aiτ,θ := E
X
t∈[τ,θ]∩N
Xt∆at| Fτ
, X ∈ R∞, a∈ A1,
and introduce the following subsets of A1: A1+ := ©
a∈ A1 |∆at≥0 for all t∈Nª (A1τ,θ)+ := πτ,θA1+
and Dτ,θ := n
a∈(A1τ,θ)+| h1, aiτ,θ = 1o .
Processes inDτ,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.
By ¯L(F) we denote the space of all measurable functions from (Ω,F) to [−∞,∞], where we identify two functions when they are equalP-almost surely, and we set
L¯−(F) :=©
f ∈L(F)¯ |f ≤0ª .
Definition 3.11 A penalty function γ on Dτ,θ is a mapping from Dτ,θ to L¯−(Fτ) with the following property:
ess sup
a∈Dτ,θ
γ(a) = 0. (3.2)
We say that a penalty functionγ on Dτ,θ has the local property if γ(1Aa+ 1Acb) = 1Aγ(a) + 1Acγ(b), for alla, b∈ Dτ,θ and A∈ Fτ.
It is easy to see that for any penalty function γ on Dτ,θ, the conditional Legendre–
Fenchel type transform
φ(X) = ess inf
a∈Dτ,θ
nhX, aiτ,θ−γ(a)o
, X∈ R∞τ,θ, (3.3)
is a conditional concave monetary utility function onR∞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 functionφon R∞τ,θsatisfying the upper semicontinuity condition of Definition 3.15 below, has a representation of the form (3.3) for γ =φ#, whereφ# is defined as follows:
Definition 3.12 For a conditional concave monetary utility function φ on R∞τ,θ and a∈ A1, we define
φ#(a) := ess inf
X∈Cφ
hX, aiτ,θ and
φ∗(a) := ess inf
X∈R∞τ,θ
nhX, aiτ,θ−φ(X)o .
Remarks 3.13
Letφbe a conditional concave monetary utility function on R∞τ,θ. 1. Obviously, for alla∈ A1,φ#(a) and φ∗(a) belong to ¯L−(Fτ), and
φ#(a)≥φ∗(a) for all a∈ A1. Moreover,
φ#(a) =φ∗(a) for all a∈ Dτ,θ (3.4) because forX ∈ R∞τ,θ and a∈ Dτ,θ,
hX, aiτ,θ−φ(X) =
X−φ(X)1[τ,∞), a®
τ,θ , and X−φ(X)1[τ,∞)∈ Cφ. 2. It can easily be checked that
φ#(λa+ (1−λ)b)≥λφ#(a) + (1−λ)φ#(b), for alla, b∈ A1 and λ∈L∞(Fτ) such that 0≤λ≤1, and
φ#(λa) =λφ#(a) for all a∈ A1 and λ∈L∞+(Fτ). (3.5) It follows from (3.5) thatφ# satisfies the local property:
φ#(1Aa+ 1Acb) = 1Aφ#(a) + 1Acφ#(b) for alla, b∈ A1 and A∈ Fτ.
In addition to the properties of Remarks 3.13,φ#fulfills the following two conditional versions ofσ(A1,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∈ Fτ and m∈R, n
a∈ Dτ,θ|Eh
1Aφ#(a)i
≥mo
is aσ(A1,R∞)-closed subset of A1. 2. For every f ∈L¯−(Fτ),
na∈ Dτ,θ|φ#(a)≥fo is aσ(A1,R∞)-closed subset of A1.
Proof.
1. Let (aµ)µ∈M be a net in ©
a∈ Dτ,θ|E£
1Aφ#(a)¤
≥mª
and a0 ∈ A1 such thataµ→a0 inσ(A1,R∞). Then,a0 ∈ Dτ,θ, and for all X∈ Cφand µ∈M,
h1AX, aµi= Eh
1AhX, aµiτ,θi
≥Eh
1Aφ#(aµ)i
≥m . Hence,
1AX, a0®
≥ m for all X ∈ Cφ. Since Cφ has the local property (lp), the set n
X, a0®
τ,θ|X ∈ Cφ
o is directed downwards, and therefore it follows from Beppo Levi’s monotone convergence theorem that
Eh
1Aφ#(a0)i
= E
·
1A ess inf
X∈Cφ
X, a0®
τ,θ
¸
= inf
X∈Cφ
Eh 1A
X, a0®
τ,θ
i
= inf
X∈Cφ
1AX, a0®
≥m . 2. Let (aµ)µ∈M be a net in ©
a∈ Dτ,θ |φ#(a)≥fª
and a0 ∈ A1 such that aµ → a0 in σ(A1,R∞). Then,a0∈ Dτ,θ, and for all X∈ Cφ,µ∈M andA∈ Fτ,
h1AX, aµi= Eh
1AhX, aµiτ,θi
≥Eh
1Aφ#(aµ)i
≥E [1Af]. Hence,
Eh 1A
X, a0®
τ,θ
i=
1AX, a0®
≥E [1Af], which shows that
X, a0®
τ,θ ≥f , for all X∈ Cφ
and therefore,φ#(a0)≥f. ¤
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.
Definition 3.15 We call a function φ:R∞τ,θ →L∞(Fτ) continuous for bounded decreas- ing sequences if
n→∞lim φ(Xn) =φ(X) almost surely
for every decreasing sequence(Xn)n∈N in R∞τ,θ andX ∈ R∞τ,θ such that Xtna.s.
→ Xt for all t∈N. Theorem 3.16 The following are equivalent:
(1) φis a mapping defined on R∞τ,θ that can be represented as φ(X) = ess inf
a∈Dτ,θ
nhX, aiτ,θ−γ(a)o
, X∈ R∞τ,θ, (3.6) for a penalty functionγ on Dτ,θ.
(2) φ is a conditional concave monetary utility function on R∞τ,θ whose acceptance set Cφ is a σ(R∞,A1)-closed subset of R∞.
(3) φ is a conditional concave monetary utility function on R∞τ,θ that is continuous for bounded decreasing sequences.
Moreover, if(1)–(3)are satisfied, thenφ#is a penalty function onDτ,θ, the representation (3.6)also holds with φ# instead ofγ,γ(a)≤φ#(a) for all a∈ Dτ,θ, and γ =φ# provided thatγ is concave, has the local property of Definition 3.11and{a∈ Dτ,θ |E [1Aγ(a)]≥m}
is aσ(A1,R∞)-closed subset of A1 for all A∈ Fτ and m∈R. 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 (Xn)n∈N be a decreasing sequence in R∞τ,θ and X ∈ R∞τ,θ such that
n→∞lim Xtn=Xt almost surely, for all t∈N. (3.7) It follows from Beppo Levi’s monotone convergence theorem that for every fixeda∈ Dτ,θ,
n→∞lim hXn, aiτ,θ =hX, aiτ,θ , and therefore,
n→∞lim φ(Xn) = inf
n∈Nφ(Xn) = inf
n∈Ness inf
a∈Dτ,θ
n
hXn, aiτ,θ−γ(a)o
= ess inf
a∈Dτ,θ
n∈infN
nhXn, aiτ,θ−γ(a)o
= ess inf
a∈Dτ,θ
nhX, aiτ,θ−γ(a)o
=φ(X).
(3)⇒ (2): follows from Lemma 3.17 below.
(2)⇒ (1): By (3.4) and the definition of φ∗,
φ#(a) =φ∗(a)≤ hX, aiτ,θ−φ(X) for allX ∈ R∞τ,θ and a∈ Dτ,θ. Hence,
φ(X)≤ess inf
a∈Dτ,θ
n
hX, aiτ,θ−φ#(a)o
for all X∈ R∞τ,θ. (3.8) To show the converse inequality, fixX∈ R∞τ,θ, let m∈L∞(Fτ) with
m≤ess inf
a∈Dτ,θ
nhX, aiτ,θ−φ#(a)o
, (3.9)
and assume thatY =X−m1[τ,∞) ∈ C/ φ. SinceCφis a convex,σ(R∞,A1)-closed subset of R∞, it follows from the separating hyperplane theorem that there exists an a∈ A1 such that
hY, ai< inf
Z∈Cφ
hZ, ai . (3.10)
SinceY and all the processes inCφare inR∞τ,θ, the processacan be chosen inA1τ,θ. AsCφ
has the monotone property (m), it follows from (3.10) thatahas to be in (A1τ,θ)+. Now, we can write the two sides of (3.10) as
hY, ai= Eh
hY, aiτ,θi
and inf
Z∈Cφ
hZ, ai= E
· ess inf
Z∈Cφ
hZ, aiτ,θ
¸ ,
where the second equality follows from Beppo Levi’s monotone convergence theorem be- causeCφ has the local property (lp), and therefore, the setn
hZ, aiτ,θ|Z ∈ Cφ
o
is directed downwards. Hence, it follows from (3.10) that P[B]>0, where
B :=
½
hY, aiτ,θ <ess inf
Z∈Cφ
hZ, aiτ,θ
¾ . Note that for A=n
h1, aiτ,θ= 0o , 1A
¯
¯
¯hZ, aiτ,θ¯
¯
¯≤1Ah|Z|, aiτ,θ≤1A||Z||τ,θh1, aiτ,θ= 0 for all Z ∈ R∞τ,θ. Hence,B ⊂n
h1, aiτ,θ>0o
. Define the process b∈ Dτ,θ as follows:
b:= 1B
a
h1, aiτ,θ + 1Bc1[τ,∞). By definition of the setB,
hX, biτ,θ−m=hY, biτ,θ<ess inf
Z∈Cφ
hZ, biτ,θ=φ#(b) on B .
But this contradicts (3.9). Hence, X −m1[τ,∞) ∈ Cφ, and therefore, φ(X) ≥ m for all m∈L∞(Fτ) satisfying (3.9). It follows that
φ(X)≥ess inf
a∈Dτ,θ
nhX, aiτ,θ−φ#(a)o . Together with (3.8), this proves that
φ(X) = ess inf
a∈Dτ,θ
nhX, aiτ,θ−φ#(a)o
for all X∈ R∞τ,θ. In particular,
0 =φ(0) = ess inf
a∈Dτ,θ
n−φ#(a)o .
This shows that (2) implies (1) and thatφ# is a penalty function onDτ,θ.
To prove the last two statements of the theorem, we assume that φ is a conditional concave monetary utility function onR∞τ,θ with a representation of the form (3.6). Then,
γ(a)≤ hX, aiτ,θ−φ(X) for all X∈ R∞τ,θ and a∈ Dτ,θ,
and it immediately follows thatγ(a)≤φ∗(a) =φ#(a) for alla∈ Dτ,θ. On the other hand, suppose that γ is concave, has the local property, {a∈ Dτ,θ|E [1Aγ(a)]≥m} is for all A∈ Fτ and m∈ Ra σ(A1,R∞)-closed subset of A1, and there exists ana0 ∈ Dτ,θ such that
P[γ(a0)< φ#(a0)]>0. (3.11) Note that
n
γ(a0)< φ#(a0)o
= [
k≥1
Bk, whereBk :=n
(−k)∨γ(a0)< φ#(a0)o .
Hence, there exists aK ≥1, such that P[BK]>0, and therefore,
−∞ ≤˜γ(a0)<φ˜#(a0)≤0, (3.12) where
˜
γ(a) := E [1BKγ(a)] and φ˜#(a) := Eh
1BKφ#(a)i
, a∈ Dτ,θ. LetX∈ R∞τ,θ and note that since γ and φ# have the local property, the sets
nhX, aiτ,θ−γ(a)|a∈ Dτ,θ
o and n
hX, aiτ,θ−φ#(a)|a∈ Dτ,θ
o
are directed downwards. Hence,
a∈Dinfτ,θ
{h1BKX, ai −˜γ(a)}= E
·
1BK ess inf
a∈Dτ,θ
n
hX, aiτ,θ−γ(a)o¸
= E [1BKφ(X)]
= E
·
1BK ess inf
a∈Dτ,θ
nhX, aiτ,θ−φ#(a)o¸
= inf
a∈Dτ,θ
nh1BKX, ai −φ˜#(a)o .
(3.13)
In particular, ess supa∈Dτ,θγ˜(a) = 0. By assumption, the function ˜γ : Dτ,θ → [−∞,0]
is concave and {a∈ Dτ,θ|γ˜(a)≥m} is for all m ∈ R, a σ(A1,R∞)-closed subset of A1. Therefore, the set
C :={(a, z)∈ Dτ,θ×R|˜γ(a)≥z}
is a non-empty, convex,σ(A1×R,R∞×R)-closed subset ofA1×Rwhich, by (3.12), does not contain (a0,φ˜#(a0)). Since γ has the local property, (a, z)∈ Dτ,θ×R is in C if and only if (1BKa+ 1BKc 1[τ,∞), z) is inC. Therefore, it follows from the separating hyperplane theorem that there exists (Y, y)∈ R∞τ,θ×Rsuch that
1BKY, a0®
+yφ˜#(a0)< inf
(a,z)∈C{h1BKY, ai+yz} ≤ inf
a∈Dτ,θ,˜γ(a)>−∞{h1BKY, ai+y˜γ(a)}. (3.14) The first inequality in (3.14) and the form of the setC imply that y ≤0. If y <0, then it follows from (3.14) that
¿
−1BK
1 yY, a0
À
−φ˜#(a0)< inf
a∈Dτ,θ
½¿
−1BK
1 yY, a
À
−˜γ(a)
¾ .
But this contradicts (3.13). Ify= 0, there exists aλ >0 such that
1BKλY, a0®
−φ˜#(a0)< inf
a∈Dτ,θ,˜γ(a)>−∞h1BKλY, ai ≤ inf
a∈Dτ,θ,γ(a)>−∞˜ {h1BKλY, ai −γ˜(a)} , which again contradicts (3.13). Hence, γ =φ# if γ is concave, has the local property of Definition (3.11) and{a∈ Dτ,θ|E [1Aγ(a)]≥m} is a σ(A1,R∞)-closed subset of A1 for
allA∈ Fτ andm∈R. ¤
Lemma 3.17 Let φ be an increasing concave function from R∞τ,θ to L∞(Fτ) that is continuous for bounded decreasing sequences. Then Cφ := n
X∈ R∞τ,θ|φ(X)≥0o is a σ(R∞,A1)-closed subset of R∞.
Proof. Let (Xµ)µ∈M be a net in Cφ and X ∈ R∞ such that Xµ → X in σ(R∞,A1). It follows thatX∈ R∞τ,θ. Assume
φ(X)<0 onA for someA∈ Fτ with P[A]>0. (3.15) The map ˜φ:R∞→Rgiven by
φ(X) = E [1˜ Aφ◦πτ,θ(X)] , X ∈ R∞,
is increasing, concave and continuous for bounded decreasing sequences. Denote byG the sigma-algebra on Ω×N generated by all the sets B× {t}, t∈ N, B ∈ Ft, and byν the measure on (Ω×N,G) given by
ν(B× {t}) = 2−(t+1)P[B], t∈N, B ∈ Ft.
Then R∞ = L∞(Ω×N,G, ν) and A1 can be identified with L1(Ω×N,G, ν). Hence, it can be deduced from the Krein–ˇSmulian theorem that Cφ˜ :=n
X ∈ R∞|φ(X)˜ ≥0o is a σ(R∞,A1)-closed subset ofR∞(see the proof of Theorem 3.2 in Delbaen (2002) or Remark 4.3 in Cheridito et al. (2004)). Since (Xµ)µ∈M ⊂ Cφ˜, it follows that E [1Aφ(X)]≥0, which
contradicts (3.15). Hence,φ(X)≥0. ¤
Theorem 3.18 The following are equivalent:
(1) φis a mapping defined on R∞τ,θ that can be represented as φ(X) = ess inf
a∈Q hX, aiτ,θ , X ∈ R∞τ,θ, (3.16) for a non-empty subsetQ of Dτ,θ.
(2) φ is a conditional coherent utility function on R∞τ,θ whose acceptance set Cφ is a σ(R∞,A1)-closed subset of R∞.
(3) φ is a conditional coherent utility function on R∞τ,θ that is continuous for bounded decreasing sequences.
Moreover, if(1)–(3) are satisfied, then the set Q0φ:=n
a∈ Dτ,θ|φ#(a) = 0o
is equal to the smallest σ(A1,R∞)-closed, Fτ-convex subset of Dτ,θ that contains Q, and the representation(3.16) also holds with Q0φ 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 thatφis 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 onDτ,θ, and
φ(X) = ess inf
a∈Dτ,θ
nhX, aiτ,θ−φ#(a)o
for all X∈ R∞τ,θ. (3.17) Sinceφ# has the local property, the set©
φ#(a)|a∈ Dτ,θ
ªis directed upwards and there exists a sequence (ak)k∈N inDτ,θ such that almost surely,
φ#(ak)%ess sup
a∈Dτ,θ
φ#(a) = 0, ask→ ∞. It can easily be deduced from the coherency of φthat for alla∈ Dτ,θ,
n
φ#(a) = 0o
∪n
φ#(a) =−∞o
= Ω.
Hence, the setsAk:=©
φ#(ak) = 0ª
are increasing in k, and S
k∈NAk= Ω. Therefore, a∗ := 1A0a0+X
k≥1
1Ak\Ak−1ak∈ Dτ,θ,
andφ#(a∗) = 0 by the local property ofφ#. Note that for alla∈ Dτ,θ, 1{φ#(a)=0}a+ 1{φ#(a)=−∞}a∗∈ Q0φ.
Hence, it follows from (3.17) that φ(X) = ess inf
a∈Q0φ hX, aiτ,θ , for all X∈ R∞τ,θ. (3.18) It remains to show that Q0φ is equal to the σ(A1,R∞)-closed Fτ-convex hull hQiτ of Q.
It follows from Theorem 3.16 that φ# is the largest among all penalty functions on Dτ,θ
that induce φ. This implies Q ⊂ Q0φ. By Remark 3.13.2 and Proposition 3.14.2, Q0φ is Fτ-convex and σ(A1,R∞)-closed. Hence, hQiτ ⊂ Q0φ. Now, assume that there exists a b∈ Q0φ\ hQiτ. Then, it follows from the separating hyperplane theorem that there exists anX ∈ R∞τ,θ such that
hX, bi< inf
a∈hQiτhX, ai= E
· ess inf
a∈hQiτhX, aiτ,θ
¸
= E
· ess inf
a∈Q hX, aiτ,θ
¸
= E [φ(X)] (3.19) (the first equality holds because hQiτ is Fτ-convex, and therefore, the set {hX, aiτ,θ|a∈ hQiτ} is directed downwards). But, by (3.18),
hX, bi −E [φ(X)] = Eh
hX, biτ,θ−φ(X)i
≥0
for allb∈ Q0φ, which contradicts (3.19). Hence,Q0φ\ hQiτ is empty, that is,Q0φ⊂ hQiτ.¤
3.3 Relevance
In this subsection we generalize the relevance axiom of Artzner et al. (1999) to our framework and show representation results for relevant conditional concave monetary and coherent utility functions. In Artzner et al. (1999) a monetary risk measure is called relevant if it is positive for future financial positions that are non-positive and negative with positive probability. In the following definition we give an Fτ-conditional version of this concept. It has consequences for the dual representation of conditional concave monetary and coherent utility functions and plays an important role for the uniqueness of time-consistent dynamic extensions of static monetary risk measures; see Proposition 4.8 below.
Definition 3.19 Let φ be a conditional monetary utility function on R∞τ,θ. We call φ θ-relevant if
A⊂©
φ(−ε1A1[t∧θ,∞))<0ª
for allε >0, t∈N and A∈ Ft∧θ, and we define Dτ,θrel :=
a∈ Dτ,θ |P
X
j≥t∧θ
∆aj >0
= 1 for all t∈N
.
Remarks 3.20
1. If φ is a θ-relevant conditional monetary utility function on R∞τ,θ and ξ is an (Ft)- stopping time such that τ ≤ ξ ≤ θ, then, obviously, the restriction of φ to R∞τ,ξ is ξ- relevant.
2. Assume that θ is finite. Then it can easily be checked that a conditional monetary utility functionφon R∞τ,θ isθ-relevant if and only if
A⊂©
φ(−ε1A1[θ,∞))<0ª for allε >0 andA∈ Fθ. Also, in this case,
Dτ,θrel ={a∈ Dτ,θ|P [∆aθ >0] = 1} . Proposition 3.21 Let Qrel be a non-empty subset of Dτ,θrel. Then
φ(X) = ess inf
a∈Qrel hX, aiτ,θ , X∈ R∞τ,θ is aθ-relevant conditional coherent utility function onR∞τ,θ.
Proof. That φ is a conditional coherent utility function on R∞τ,θ follows from Theorem 3.18. To show that it isθ-relevant, let ε >0,t∈N,A∈ Ft∧θ and choose a∈ Qrel. Then
φ(−ε1A1[t∧θ,∞))≤ −ε
1A1[t∧θ,∞), a®
τ,θ =−εE
1A X
j≥t∧θ
∆aj | Fτ
,
and it remains to show that E
1A
X
j≥t∧θ
∆aj | Fτ
>0 on A . (3.20)
DenoteB=n Eh
1AP
j≥t∧θ∆aj | Fτ
i
= 0o
and note that
0 = E
1BE
1A
X
j≥t∧θ
∆aj | Fτ
= E
1B1A
X
j≥t∧θ
∆aj
.
This impliesB∩A=∅, and therefore, (3.20). ¤
To prove the converse of Proposition 3.21 we introduce for a conditional concave mon- etary utility functionφon R∞τ,θ and a constant K≥0, the set
QKφ :=n
a∈ Dτ,θ|φ#(a)≥ −Ko .
Note that it follows from Remark 3.13.2 and Proposition 3.14.2 thatQKφ isFτ-convex and σ(A1,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 andθ-relevant. Then
QKφ ∩ Dτ,θrel is non-empty for all K >0. Proof. FixK >0 andt∈N. Fora∈ Dτ,θ, we denote
et(a) := X
j≥t∧θ
∆aj, and we define
αt:= sup
a∈QKφ
P[et(a)>0]. (3.21)
Let (at,n)n∈N be a sequence in QKφ with
n→∞lim P£
et(at,n)>0¤
=αt. SinceQKφ is convex and σ(A1,R∞)-closed,
at:=X
n≥1
2−nat,n is still in QKφ , and, obviously,
P£
et(at)>0¤
=αt.
In the next step we show thatαt = 1. Assume to the contrary that αt < 1 and denote At:=©
et(at) = 0ª
. Since φis θ-relevant, At⊂©
φ(−K1At1[t∧θ,∞))<0ª , and therefore also,
Aˆt:= \
B∈Fτ, At⊂B
B ⊂ ©
φ(−K1At1[t∧θ,∞))<0ª .
By Theorem 3.16,
φ(−K1At1[t∧θ,∞)) = ess inf
a∈Dτ,θ
n
−K1At1[t∧θ,∞), a®
τ,θ−φ#(a)o .
Hence, there must exist an a∈ Dτ,θ with P [At∩ {et(a)>0}]> 0 and φ#(a) ≥ −K on Aˆt. We then have that
bt:= 1Aˆ
ta+ 1Aˆcat∈ QKφ , ct:= 1 2bt+1
2at∈ QKφ , and P£
et(ct)>0¤
> P£
et(at)>0¤
= αt. This contradicts (3.21). Therefore, we must haveαt= 1 for allt∈N. Finally, set
a∗ =X
t≥1
2−tat,
and note thata∗ ∈ QKφ ∩ Dτ,θrel. ¤
Theorem 3.23 Let φ be a conditional concave monetary utility function on R∞τ,θ that is continuous for bounded decreasing sequences andθ-relevant. Then
φ(X) = ess inf
a∈Drelτ,θ
nhX, aiτ,θ−φ#(a)o
, for all X∈ R∞τ,θ. Proof. By Theorem 3.16,
φ(X) = ess inf
a∈Dτ,θ
nhX, aiτ,θ−φ#(a)o
, for all X∈ R∞τ,θ, which immediately shows that
φ(X)≤ess inf
a∈Drelτ,θ
nhX, aiτ,θ−φ#(a)o
, for all X∈ R∞τ,θ.
To show the converse inequality, we choose b ∈ Dτ,θ. It follows from Lemma 3.22 that there exists a processc∈ Q1φ∩ Drelτ,θ. Then, for all n≥1,
bn:= (1− 1 n)b+ 1
nc∈ Drelτ,θ,
n→∞lim hX, bniτ,θ= lim
n→∞
½ (1− 1
n)hX, biτ,θ+ 1
nhX, ciτ,θ
¾
=hX, biτ,θ almost surely, and
φ#(bn) = ess inf
X∈Cφ
hX, bniτ,θ ≥(1− 1
n) ess inf
X∈Cφ
hX, biτ,θ+ 1 ness inf
X∈Cφ
hX, ciτ,θ
= (1− 1
n)φ#(b) + 1
nφ#(c) → φ#(b) almost surely. This shows that
hX, biτ,θ−φ#(b)≥ess inf
a∈Dτ,θrel
nhX, aiτ,θ−φ#(a)o , and therefore,
φ(X)≥ess inf
a∈Drelτ,θ
nhX, aiτ,θ−φ#(a)o ,
which completes the proof. ¤
