CONDITIONAL RISK AND ACCEPTABILITY MAPPINGS AS BANACH-LATTICE VALUED MAPPINGS

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1 CONDITIONAL RISK AND ACCEPTABILITY MAPPINGS AS BANACH-LATTICE VALUED MAPPINGS RAIMUND KOVACEVIC Abstract. Conditional risk and acceptability mappings quantify the desirability of random variables e.g. nancial returns by accounting for available information. In this paper the focus lies on acceptability mappings, considered as concave translation-equivariant monotone mappings L p F L p Ω, F, P with 1 p p, where the σ algebras F F describe the available information. Based on the order completeness of L p Ω, F, P-spaces, we analyze superdierentials and concave conjugates of conditional acceptability mappings. The related results are used to show properties of two important classes of multi-period acceptability functionals: SECfunctionals and additive acceptability compositions. In particular, we derive a chain rule for superdierentials and use it in order to characterize the conjugates of additive acceptability compositions and SEC-functionals. Key words: conditional risk measure, conditional acceptability mapping, monetary utility function, multi-period acceptability functional, convex analysis, chain rule 1. Introduction Measuring the risk or acceptability of nancial positions and their related income streams is an important issue. Since the days of Markowitz [29], this has led to an increasing literature on risk measures and desirable properties of risk measures. An axiomatic theory for the very important class of coherent risk measures was given by Artzner et al. [4, 2] in 1999 and the development of this theory set standards for the subsequent discussion. The role of convexity instead of subadditivity was emphasized in [16, 19]. Many important results can be found in [17, 18, 47, 35], and we mention also the original works [22, 26, 34, 41, 25, 9]. In this article the main results will be formulated in terms of concave acceptability functionals and mappings see [35, 42], which are also known as monetary utility functions e.g. [23]. Up to sign, such functionals are identical with convex risk measures [16], [19]. In principle it is easily possible to state the whole message of this paper in terms of convex risk functions. Acceptability is preferred here, because in addition to risk management we are also interested in multi-stage decision making, and acceptability related to the notion of utility can be applied in a very natural way in this context. While risk and acceptability functionals assign a real number to a random variable, multiperiod risk and acceptability functionals assign a value to a whole stochastic process. Furthermore, dynamic risk measures describe the stochastic development of a valuation related to an end payment over time. Both notions are relevant for valuation as well as for optimal decision making under uncertainty: We refer to [39, 7, 12, 36, 8, 20, 5]. Conditional mappings are the basic building blocks of dynamic risk measurement and can be used for the construction of certain multi-period functionals: Risk or acceptability functionals evaluate in terms of a real number, how favorable a random variable or distribution is. The valuation is done at the beginning of a single period, assuming that there is no nontrivial This work was supported by the Austrian Research Fund FWF. 1

2 CONDITIONAL ACCEPTABILITY MAPPINGS 2 information available. On the other hand, conditional risk and acceptability mappings give a valuation of a random variable with respect to a single period, but the valuation is done with respect to some nontrivial σ algebra, representing information that will be available in future. Such mappings were introduced and analyzed by Artzner et al. [3, 5], Detlefsen and Scandolo [11], Ruszczy«ski and Shapiro [43] and Pug and Römisch [35]. In this paper we consider conditional acceptability mappings as mappings between spaces L p Ω, F, P and L p Ω, F, P with F F and develop a simple calculus that nally will allow to analyze certain multi-period acceptability mappings in a direct way. Existing literature focuses mainly on the case L Ω, F, P see e.g. the detailed studies [17, 10] on functionals, and [11, 5] on conditional mappings. On the other hand, extreme events play an increasing role in practical risk management. Hence, the theory of valuation should apply to more than just essentially bounded random variables. Therefore, the extension to p < aims at bringing the theoretical discussion of risk and acceptability measures more closely to the empirically oriented literature on application of stable distributions e.g. [38, 37] and extreme value distributions e.g. [13, 30] to risk management. Only recently, some papers deal with unconditional risk functionals on L p Ω, F, P spaces [21, 31, 14, 15, 46, 24], or even more general spaces [19, 6, 9]. So far, for conditional mappings literature is even more sparse. [35] aims at mappings L p Ω, F, P L p Ω, F, P. Based on the theory of nonsmooth analysis on partially ordered vector spaces e.g. [33, 27] in particular in Banach lattices, this article develops the theory of conditional acceptability mappings L p Ω, F, P L p Ω, F, P with p p. A simple calculus for conditional acceptability mappings, compositions, and some other types of multi-period functionals is developed. 2. Conditional Acceptability Mappings and Multi-period Acceptability Functionals In the following, we work on probability spaces L p Ω, F, P and the related extended spaces L p Ω, F, P = L p Ω, F, P }. The value denotes a representant of the class of random variables, that have almost surely value. In both cases we assume 1 p. The state space Ω and the measure P are considered as xed, hence we will write L p F = L p Ω, F, P and L p F = L p Ω, F, P. In order to compare random variables dened on this spaces we use the partial order based on the cones K + = X L p F : X 0 a.s.}. Having stated this, we will understand all inequalities or equations between random variables in the sense of holds almost surely. Recall that a set S L p F is minorized by an element X 0 L p F, if X 0 X whenever X S, and X 0 is called a minorant. If S is minorized and there exists a X 0 L p F such that X 0 Y for any minorant Y then X 0 is the inmum of S see [45], p 3, and we write X 0 = inf S. The inmum of a set S L p F is dened in the same way, if we replace L p F by L p F in the last paragraph. If S L p F is minorized by a nite element X L p F, then the inmum is also nite, a property known as order completeness: It is well known that L p F-spaces are Riesz spaces for 0 p. Moreover, if 1 p they are Banach lattices in addition e.g. [1], Theorem 13.5, because they possess lattice norms and are Cauchy-complete. Finally, because p-norms are order continuous norms for 1 p <, it can be shown [1], Theorem 13.7 that L p F- spaces are order complete Banach lattices for 1 p. We will not go into these details here, but refer to the Appendix, where a short alternative proof is presented. Consider now random variables X, Y L p F. Those can be interpreted as nancial cash ows, payable at the end of a period. The σ algebra F can be interpreted as information available at the end of the considered time period. In addition, consider a σ algebra F F representing the information available at its beginning. Conditional acceptability mappings then

3 CONDITIONAL ACCEPTABILITY MAPPINGS 3 give a valuation of the end cash ow with respect to information available at the beginning of the period. Slightly extending 2.50 in [35], we dene Conditional acceptability mappings as upper semicontinuous mappings A F : L p F L p F with F F and 1 p p such that for all X, Y L p F the following properties hold: CA1 Monotonicity. X Y AX F AY F CA2 Predictable Translation Equivariance. AX + X 1 F = AX F + X 1 for every X 1 L p F. CA3 Concavity. A λ X + 1 λ Y F λ A X F + 1 λ A Y F for λ [0, 1] and dom A = X L p F : AX F }is a convex set. Remark 1. Note that implicitly, we restrict our analysis to proper mappings, because our definition of extended spaces does not include the value +, and recall [28], [33, denition 5.3] that a mapping f : X Ȳ is said to be lower semicontinuous l.s.c. at x dom f, if for any neighborhood V of zero in Y and for any b Y satisfying b C fx, there exists a neighborhood U of x in X such that f U b + V + C. A mapping f is called upper semicontinuous u.s.c., if f is l.s.c. Remark 2. As pointed out before, acceptability functionals are closely related to the notion of convex risk mappings: If A is an acceptability mapping, then the mapping given by ρy F := AY F is a convex risk mapping. If the functional A F is positively homogeneous for all F i.e., AλY F = λay F holds for all λ 0 and Y, then the functional ρ is a coherent risk mapping. Sometimes we will write shortly A 1 := A F. According to the denition, AY F is either in L p F or it is. The rst reason for usage of the extended space including is purely technical in the context of convex analysis see e.g. [27, 33], e.g. the presence of the value is related to the usage of an indicator 0, if X S function I S X =. Hence the value can be used to identify arguments else that lie outside of the domain of the mapping. With regards to content, the extension allows to identify absolutely unfavorable random variables with value -. In principle this means that the paper aims at mappings with nite value, but it is possible to restrict them in the style of: 2.1 AY F EY F = if Y 0 otherwise, which maps L 1 F to L 1 F. It should be kept in mind that direct application of an unconditional functional to all conditional distributions separately is not always possible for restricted functionals: The componentwise extension of 2.2 AY = EY if Y 0 otherwise to a fully conditional mapping is given by 2.3 AY F EY F on the set where Y F 0 =, otherwise, where Y F denotes the conditional distributions. However, this may lead to 0 < P AY F = } < 1, which is forbidden.

4 CONDITIONAL ACCEPTABILITY MAPPINGS 4 In the following we will consider linear continuous mappings between spaces L p F and L p F as our basic building block. At rst glance it seems to be natural to use conditional expectations E X Z F. Such mappings with Z L q F and 1 p + 1 q = 1 are clearly linear and continuous, but map into L 1 F. The following Lemma shows that a restricted dual space is more reasonable in our context: Lemma 2.1. For 1 p p < the conditional expectations X E X Z F with arguments X L p F and Z L s F L q F are linear, continuous and map into L p F, if s p p p p. For mappings L F L p F such conditional expectations have these properties, if Z L p F. Proof. Conditional expectations X E X Z F are linear. Moreover, Z are dual variables such that X E X Z F maps into L p F, if Z L q F Y : X Y L p F, X L p F }. For s = p p p p, which is equivalent to 1 p + 1 s = 1 p, the generalized Hölder inequality 2.4 X Z p X p Z s implies that X Z is p integrable if Z is s-integrable, because X is p-integrable by assumption. To ensure continuity as a mapping into L p F, assume now that X n converges to X in the p-norm, which means that X n X p 0, as n. Apply Jensen's inequality and get 2.5 E X n X Z F p X n X Z p. Using again the generalized Hölder inequality 2.4, we conclude E X n X Z F p X n X p Z s. Z is s-integrable and X n converges to X, hence E X n Z F converges to E X Z F. Example. For mappings L p F L 1 F the random variable Z is an element of the natural dual space. For mappings L p F L p F, we have p =. Based on Lemma 2.1 and the general theory of convex analysis on partially ordered vector spaces see [33], [27], we introduce superdierentials and conjugates. We will restrict ourselves to acceptability mappings A F : L p F L p F such that the related sets A X 0 F = Z L s : A X F A X 0 F + E X X 0 Z F, X doma}, where 1 p + 1 s = 1 p, are nonempty at any X 0 intdoma. This means that the mapping has supergradients in the more general sense of [33], [27] of the form E X Z F. The set A X 0 F in fact a subset of the full superdierential is called superdierential and its elements, respectively the related Z, are called supergradients in the following. It is well known that conditional acceptability mappings are continuous, if they are locally bounded at some element X 0 L p F [32], Theorem 4, and that A X F is nonempty, if the mapping is continuous at some point X 0 doma [33], Lemma 3.2. The concave conjugate see [33, denition 5.5] of a mapping A F : L p F L p F with nonempty sets A X F is given by a mapping with 1 p + 1 s = 1 p, and A + Z F = A + : L s F L p F inf E X Z F A X F }. X L pf

5 CONDITIONAL ACCEPTABILITY MAPPINGS 5 The concave biconjugate is a mapping, A ++ : L p F L p F dened by A ++ X F = E X Z F A + Z F }. inf Z L sf Note that in the light of lemma 2.1 the dual variable Z can be restricted in order to modify a mapping with given conjugate representation, such that it maps into a desired space L p F. The restriction to mappings with supergradients of the form E [XZ F ] for some Z might be seen as somehow arbitrary, but allows for a straightforward generalization of classical results about the most important class of unconditional acceptability functionals: concave upper semicontinuous functionals, which by the Fenchel-Moreau-Rockafellar Theorem [40], Theorem 5 can be represented as 2.6 A X = inf Z L qf E X Z A + Z }, where 1 p + 1 q = 1 and A+ Z = inf X LpF E X Z A X} is the concave or Fenchel- Moreau conjugate of the functional A. See [35] for many examples. Conditional versions of such functionals can be dened by restricting Z to the space L s as discussed above, replacing any occurrence of an expectation by the appropriate conditional expectation, and using the almost sure inmum instead of the ordinary inmum. Clearly such modied mappings have supergradients in L s and the theory developed in this paper can be applied. The following proposition is based on the existence of supergradients as discussed above and summarizes the main relations between conjugates and supergradients in the context of acceptability mappings. Proposition 2.2. Let A F be a concave mapping. Then the inequality 2.7 A X F E X Z F A + Z F, holds for all X L p F, Z L s F with 1 p + 1 s = 1 p. Moreover, if A X F then the equation 2.8 A X F = A ++ X F is valid, and A F is u.s.c. at X. In this case the inmum inf E X Z F A + Z F : Z doma + } is attained and Z 2.9 A X F = E X Z F A + Z F Z A X F. Proof. Inequality 2.7 follows directly from the denition of the conjugate, and the second assertion follows from Theorem 5.8 in [33]. Furthermore, A X F = E X Z F A + Z F holds i A X F + E Y Z F A Y F E X Z F for any Y L p F. a First assume A X F = E X Z F A + Z F. Using the denition of the conjugate mapping we have A X F + E Y Z F A Y F A X F + inf Z E Y Z F A Y F } Using the above assumption we get for any Y. = A X F + A + Z F. A X F + E Y Z F A Y F E X Z F

6 CONDITIONAL ACCEPTABILITY MAPPINGS 6 b For the converse assume A X F + E Y Z F A Y F E X Z F. Remember that this inequality should hold for all Y - which means that E Y Z F A Y F has a lower bound. Then the inmum exists and it follows that or A X F + inf Z E Y Z F A Y F } E X Z F A X F + A + Z F E X Z F Together, we have A X F + A + Z F = E X Z F. Moreover, A X F + E Y Z F A Y F E X Z F holds i A Y F A X F + E Y X Z F, which means that Z A X F. Together with inequality 2.7, this yields equation 2.9. Conditional acceptability mappings can be characterized even sharper by the following proposition compare [35, Theorems 2.51, 2.52]: Proposition 2.3. Within the considered class of mappings a concave conditional mapping A F : L p F L p F is an acceptability mapping, if and only if the representation A X F = inf Z L s E X Z F A + Z F : Z 0; E Z F = 1; Z Z }, with Z L s F, holds for each point in int doma. Remark. The set Z represents additional constraints on Z. Proof. By Proposition 2.2 and using the presumed existence of supergradients, we have at each point in int doma A X F = inf E X Z F A + Z F } = E X Z X F A + Z X F Z for some Z X and Z X is a supergradient. In the following we analyze the conditions Z 0 and E Z F = 1. If one of these constraints is analyzed, Z a will denote the set of the other constraints, including Z Z. If we assume E Z F = 1 rst, and X 1 is F -measurable, it is possible to infer A X + X 1 F = inf E X + X1 Z F A + Z F : E Z F = 1 } Z Z a = inf E X Z F + X 1 E Z F A + Z F : E Z F = 1 } Z Z a = inf Z Z a E X Z F A + Z F } + X 1 = A X F + X 1. On the other hand let E Z F ω 1 on a set S F with positive probability. Then we can conclude A X F + X 1 = inf Z Z a E X Z F + X 1 A + Z F } inf E X Z F + X 1 E Z F A + Z F } Z Z a = A X + X 1 F on S. This would contradict the assumption of predictable translation equivariance.

7 CONDITIONAL ACCEPTABILITY MAPPINGS 7 Assume now that Z 0 holds a.s. and choose some Y 0. Then, for any random variable X we have X + Y X. Because Y and Z both are nonnegative, it follows that E Y Z F 0. Furthermore, let Z be a supergradient of A at X + Y. Using Proposition 2.2, we infer A X + Y F = E X + Y Z F A + Z F E X Z F A + Z F inf E X Z F A + Z F } Z Z a = A X F. Conversely, assume now that for a supergradient Z of A at X we have Z < 0 on a set S F with positive probability, and choose Y 0 such that E Y Z F < 0, on S, which is always possible. Then we have A X F = E X Z F A + Z F > E X Z F + E Y Z F A + Z F inf E X Z F + E Y Z F A + Z F } Z Z a = A X + Y F on the set S, which contradicts monotonicity. 3. Application to Multi-period Functionals In the following we consider a nite time horizon T <. Time t = 0,..., T is discrete, F = F 0,..., F T is a ltration with F T = F, and F 0 =, Ω}, denotes the trivial σ-algebra. Within this setup we want to valuate adapted stochastic processes X t t 1,...,T }, representing random nancial payos at t = 1,..., T. Multi-period functionals evaluate the acceptability of such a process in terms of real numbers. Such functionals can be described as mappings A ; F from product spaces T L pt F t, which are Banach lattices, into R. The related norm is given by X = E X t pt 1 p t, 1 pt, into the extended real line R. An important class of valuation functionals are multi-period acceptability functionals [35]. We summarize the main facts: A multi-period functional A X; F is called multi-period valuation functional, if it satises MA1 Concavity: The mapping X A X; F is concave in X. MA2 Monotonicity: X t Y t for all t implies A X; F A Y ; F A multi-period valuation functional is a multi-period acceptability mapping, if in addition MA3 Translation Equivariance: A X 1,..., X t + c,..., X T ; F = A X 1,..., X t,..., X T ; F + c holds for all periods t, if c is a real number. For concave multi-period functionals A X; F it is possible to dene their concave conjugate A + Z; F = inf X, Z A X; F : X T L p F t }. If A X; F is in addition proper and upper semicontinuous, then the Fenchel-Moreau-Rockafellar Theorem [40], Theorem 5 ensures that the functional equals its biconjugate: A X; F = A ++ X; F. The supergradient representation of multi-period acceptability functionals is well known see Theorem 3.21 in [35]: Let A ; F be an upper semicontinuous multi-period functional satisfying

8 CONDITIONAL ACCEPTABILITY MAPPINGS 8 MA1, MA2 and MA3. Then the representation T } 3.1 A X; F = inf E X t Z t A + Z; F : Z t 0; E Z t = 1 Z holds for every X T L p F t. Conversely, if A ; F can be represented by a dual representation 3.1 and the conjugate A + is proper, then A is proper, upper semicontinuous and satises MA1,MA2,MA3. In the following, we will apply the facts of the previous section to special types of multi-period functionals that can be constructed from conditional mappings, namely additive compositions of acceptability functionals, and SEC-functionals Additive Acceptability Compositions. Additive compositions are formed by recursive application of conditional acceptability mappings to the values X t. Up to now, conditional probability mappings have been dened as mappings L p F L p F. If we want to compose such mappings, we have to modify this denition slightly, taking care of the possibility of innite valued arguments: We will understand conditional acceptability probability mappings as mappings L p F L p F, and assume that A F =. In addition we use the short notations X t = X t, X t+1,..., X T and F t = F t,..., F T. With these agreements it is possible to dene additive acceptability compositions in the following way see [43, 44]: Let A t F t, t = 1,..., T be a sequence of conditional monotone mappings. An additive acceptability composition B F can be constructed recursively for any X T L j=1 pj F j, : B T 1 X T F T 1 = A T 1 X T F T 1 for T 1, and B t 1 X t F t 1 = A t 1 X t + B t X t+1 F t F t 1 for t < T 1. The related additive acceptability composition B F is given by BX F = B 0 X F. Remark. Notice that meaningful values will occur only if all random variables Y t belong to the respective domains and all mappings A t map into the correct spaces. The latter requirement can be ensured by restricting the supergradient space according to Lemma 2.1. The mappings B t will be called nested conditional acceptability mappings and can be understood as dynamic acceptability mapping. If the ltration is clear, we might write B t X t+1. In the following we will use the framework of section 2, in order to extend results of [35]. Based on convexity and monotonicity it is possible to state a chain rule for acceptability compositions. Proposition 3.1. Let A t } t 1,...,T 1} be a collection of conditional acceptability functionals with T 2. Given supergradients A T 1 X T and Z k A k 1 X k + B k X k+1,..., X T F k for k t 0 + 1,..., T }, a supergradient for the nested conditional acceptability mappings B t0 F t 0 at the base points X t0+1,..., X T is given by a T t-tupel M = M t0+1,..., M T with M t0+1 Z t0+1 and M k+1 = M k Z k+1

9 CONDITIONAL ACCEPTABILITY MAPPINGS 9 for t 0 < k T 1. In the case of t 0 = 0, a supergradient for the additive acceptability composition BX F is given by the recursion M 1 Z 1 and for 0 < k T 1. M k+1 = M k Z k+1 Proof. We use backward induction to prove the result: For t = T 1 the B T 1 X T ; [A T 1 ] = A T 1 X T holds, and is a supergradient. Let the Proposition be true for all t t 0. Then, a supergradient for the nested mapping B t0 Y t0+1,..., Y T ; [A t0,...,a T 1 ] is given by M t0+1,..., M T. That means: B t0 X t0+1 + Y t0+1,..., X T + Y T B t0 X t0+1,..., X T + E t0 Yt0+1 Z t Et0 YT 1... Z t0+1. Using monotonicity, we can conclude: B t0 1 X t0 + Y t0, X t0+1 + Y t0+1,..., X T + Y T = = A t0 1 X t0 + Y t0 + B t0 X t0+1 + Y t0+1,..., X T + Y T A t0 1 X t0 + B t0 X t0+1,..., X T + Y t0 + E t0 Y k k=t 0+1 As Z t0 is a supergradient of A t0 1 at X t0 + B t0 X t0+1,..., X T it follows that B t0 1 X t0 + Y t0, X t0+1 + Y t0+1,..., X T + Y T A t0 1 X t0 + B t X t0+1,..., X T + [ + E t0 1 Z t0 Y t0 + E t0 Yt0+1 Z t Et0 Y T T t=t 0+1 k j=t 0+1 Z t ] = Z j = B t0 1 X t0, X t0+1,..., X T + E t0 1 Yt0 Z t0 + Et0 1 Yt0+1 Z t0+1 Z t E t0 1 YT 1... Z t0+1 Z t0 The values of A t F t are p t -integrable, and the Z t are supergradients. This means that the conditional expectations E t 1 Xt Z t = E Xt Z t F t 1 are pt 1 -integrable by Proposition 2.1. This means that all the conditional expectations involved in each nesting-step are p t - integrable. It should be noted that supergradients of additive acceptability compositions form nonnegative martingales: Corollary 3.2. Let A t } t 1,...,T } be a collection of u.s.c. integrability adapted conditional acceptability mappings. Then the process M t } t t0+1,...,t } of the supergradients dened in Proposition 3.1 is a positive martingale. Proof. We know that M k+1 = M k Z k+1. For all k we have Z k 0 by monotonicity, hence M k 0 follows. From Proposition 2.3 we know that E Zt F t 1 = 1, because of predictable translation equivariance. Moreover, M t } is adapted to F t. Therefore E M t F t 1 = E M t 1 Z t F t 1 = Wt 1 E Zt F t 1 = Mt 1.

10 CONDITIONAL ACCEPTABILITY MAPPINGS 10 So far, we have characterized the supergradients of additive acceptability compositions. Using this martingale representation and Proposition 2.3 it is also possible to calculate the dual representation of compositions by using the concave conjugates of the constituent conditional mappings. Proposition 3.3. Let B 0 be an additive acceptability composition constructed by a sequence A t } t=0,...,t of conditional acceptability mappings with concave conjugates A t X F t = inf Z t+1 L st Et X t+1 Z t A + t Z t+1 : Z t+1 0; E t Z t+1 = 1, Z t Z t }. In addition let F = F i i=0,...,t be a ltration. Under the assumptions of Theorem 3.1 the acceptability composition BX F can be represented by 3.2 B 0 X 1, X 2,..., X T = T with inf Z 1,..., T E 0 X t M t A + 0 Z 1 1 E 0 A + t Z t+1 M t : Z Z } Z = Z t : M 1 = Z 1 ; M t+1 = M t Z t; ; Z t 0; E t 1 Z t = 1, Z t Z t }. Proof. Theorem 2.3 can be used to replace the conditional mappings A t by their concave conjugates in a recursive manner. Here is one step: First note that by Proposition A t 1 Y t = inf Z t Et 1 Y t Z t A + t 1 Z t : Z t 0; E t 1 Z t = 1 }. In the following, all the inma must be understood with respect to the constraints Z t 0, E Z t F = 1, Z t 1 0 and E Z t 1 F = 1 for all t 1. Using 2.5, we get A T 2 X T 1 + A T 1 X T = inf ET 2 [X T 1 + A T 1 X T ] 1 A + T 2 1 } 1 [ = inf E T 2 [X T 1 1 ] + inf ET 2 [X T 1 ] E T 2 A + T 1 ] } } 1 A + T = inf ET 2 [X T 1 1 ] + E T 2 [X T 1 ] A + T 2 Z [ T 1 E T 2 A + T 1 Z ]} T 1 1, Iterating and dening the M 1 = Z 1 and M t+1 = M t Z t+1 we get the statement. Here we have used that [ ] E infey T F T 1 A T 1 } 1 F T 2 = inf E [EY T F T 1 1 A T 1 1 F T 2 ]} In order to see this, notice that [ ] E infey T F T 1 A T 1 } 1 F T 2 = inf E [Y T 1 F T 2 ] E [A T 1 1 F T 2 ]} inf E [EY T F T 1 A T 1 1 F T 2 ]}.

11 CONDITIONAL ACCEPTABILITY MAPPINGS 11 Since the inmum is attained, there is a Z YT such that [ ] E infey T F T 1 A T 1 1 } F T 2 = E [EY T Z YT F T 1 A T 1 Y ZT 1 F T 2 ] inf E [EY T 1 F T 1 A T 1 1 F T 2 ]}. The M t are martingales and constitute the supergradient of the composition see Proposition 3.1 and Corollary 3.2. So, B 0 X 1, X 2,..., X T = T inf Z 1,..., E X t M t A + 0 Z 1 T 1 E A + t Z t+1 M t : Z Z basically gives the conjugate representation of the composition, and the mapping A + 0 Z 1 + T E A + t Z t+1 M t is the related conjugate mapping B + 0. It is possible to restate Proposition 3.3 in a way, such that only the supergradients M t are used: Corollary 3.4. Under the assumptions of Proposition 3.1, and with ψm t, M t+1 = sup Z t+1 E A + t Z t+1 M t : Mt+1 = M t Z t+1, Z t+1 Z t }, an acceptability composition can be represented by B 0 X 1, X 2,..., X T = T inf M 1,M 2,...,M T E X t M t A + 0 M 1 T 1 ψm t, M t+1 : M t M with M = M t M t 0; E M t F t 1 = M t 1 }. We can outline the properties of such mappings in the following way compare [35], Theorem 3.33: Proposition 3.5. An acceptability composition is an u.s.c. concave MA1, monotonic MA2 multi-period probability functional that is also translation equivariant MA3. Proof. As a composition of monotone concave mappings the multi-period functional must be concave itself. It is also monotone because M t 0 for all t. From M 0 = 1 and E M t F t 1 = M t 1 we can infer the equation E M t = 1, which, by 3.1, is the criterion for weak translation equivariance. Upper semicontinuity follows from the comparison with 3.1. } }, 4. Separable Expected Conditional Functionals Another important approach for dening multi-period functionals takes sums of expectations of conditional acceptability mappings. Such functionals are called separable expected conditional SEC, [35], p 145: A multi-period acceptability functional is called separable expected

12 CONDITIONAL ACCEPTABILITY MAPPINGS 12 conditional SEC if it is of the form A X; F = E A t 1 X t F t 1, where the A t 1 F t 1 are conditional u.s.c. acceptability mappings. An important SEC-functional is the multi-period average value at risk [36] α X; F = E α X t F t 1. A multi-period functional A is SEC if and only if its dual is SEC Proposition 3.27 in [35]. This means that the concave conjugate of any SEC-functional can be represented in the form A + Z; F = E A + t 1 Z t F t 1. Building on Proposition 3.1, we can to characterize the supergradients of SEC-functionals: Proposition 4.1. Let A X; F = E A t 1 X t F t 1 be a SECfunctional, Z = Z 1,..., a vector of supergradients of the constituent conditional acceptability mappings A t. Then Z is also a supergradient of the SEC-functional. Proof. Each addend E A t X t F t 1 can be seen as an acceptability composition, with no action between the beginning and period t. The conjugate of the expectation equals zero at one and is unbounded elsewhere. Therefore, the supergradient of the expectation must be one almost sure. Supergradients for the conditional acceptability mappings A t are given by Z t. Applying Proposition 3.1, we see that for each t the product Z t 1 = Z t must be a supergradient of the associated acceptability composition E A t 1 X t F t 1, which means that or E A t 1 X t + Y t F t 1 E A t 1 X t F t 1 + E Y t Z t. Summing over all t we get E A t 1 X t + Y t F t 1 E A t 1 X t F t 1 + A X + Y ; F A X; F + E Y t Z t. This shows that Z is a supergradient for the whole SEC-functional. Appendix E Y t Z t, Proposition. Let S = X ϑ : ϑ Θ} be an arbitrary set of nonnegative functions from L p F with p 1. Then this set has an inmum X L p F. Proof. Let A be the directed set of nite subsets of Θ and dene a decreasing net X α } α A with the following elements: 4.1 X α = inf X ϑ1,..., X ϑk }, for α = ϑ 1,..., ϑ k }. Furthermore, dene s α = X α dp for each X α and observe that s α must be bounded below, because X α 0 is bounded below. This means that s = inf s α } must exist.

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