CONDITIONAL ACCEPTABILITY MAPPINGS AS BANACH-LATTICE VALUED MAPPINGS

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1 CONDITIONAL ACCEPTABILITY MAPPINGS AS BANACH-LATTICE VALUED MAPPINGS RAIMUND KOVACEVIC Department of Statistics and Decision Support Systems, University of Vienna, Vienna, Austria Abstract. Conditional acceptability mappings quantify the desirability of random variables e.g. modeling nancial returns by accounting for available non-trivial information. They are dened as mappings from spaces L p Ω, F, µ to spaces L p Ω, F 1, µ, where the σ algebras F 1 F describe the available information. Moreover, such mappings have to be concave, translationequivariant and monotonically increasing. Based on the order completeness of L p Ω, F, µ-spaces, we will dene and analyze superdierentials and concave conjugates for conditional acceptability mappings. In contrast to most of the existing literature, special emphasis is placed on applicability to arbitrary Banach lattices L p Ω, F, µ and L p Ω, F 1, µ with 1 p p. In particular, it will be possible to consider mappings with p < in a straightforward manner. Furthermore, the above results are used to show properties of such multiperiod acceptability functionals that are based on conditional acceptability mappings, in particular, SEC-functionals and additive acceptability compositions. Finally, a chain rule for superdierentials is derived and used to characterize the conjugates of additive acceptability compositions and SECfunctional. 1. Introduction A probability functional is an extended real valued function dened on a set of random variables, which are dened on some probability space Ω, F, P. If a functional is interpreted in the sense of higher values being preferable to lower values we call it a monotonic or acceptability-type [7] functional. Such functionals can be used to judge the desirability of random nancial payments. Acceptability functionals are monotonic functionals with additional properties: Denition 1.1. A probability functional A : L p Ω, F, P R = R is called an acceptability functional, if the following properties hold for all random variables X, Y L p Ω, F, P: A1 Monotonicity. X Y AX AY. A2 Translation Equivariance. AX + c = AX + c for all constants c. A3 Concavity. A λ X + 1 λ Y λ A X + 1 λ A Y for all λ [0, 1]. The domain of a probability functional A is dened as the set dom A = X L p Ω, F, P : AY >. Funded by FWF. 1

2 CONDITIONAL ACCEPTABILITY MAPPINGS 2 We allow p [1, ], but point out that p = might be a very optimistic assumption, particularly in some areas of empirical nance. From the Fenchel-Moreau-Rockafellar Theorem [9], Theorem 5, it follows that concave upper semicontinuous functionals can be represented as 1.1 A X = inf L qω,f,µ E X A +, where 1 p + 1 q = 1 and A+ = inf X LpΩ,F,µ E X A X is the concave or Fenchel-Moreau conjugate of the functional A. This representation is called supergradient representation and is very useful for analyzing properties of concave functionals. A simple but useful example is the average value at risk also called tail value at risk, conditional value at risk or expected shortfall. For a random variable X with distribution function G, it is dened as 1.2 α X = 1 G 1 x dx. α 0 This functional is a positive homogeneous acceptability functional and it is well known that its conjugate representation is given by ˆ α 1.3 α X = min E X : E = 1, 0 1 α. 2. Conditional Acceptability Mappings A natural generalization of acceptability functionals are conditional acceptability mappings. Such mappings valuate the acceptability of a random variable relative to additional nontrivial information. A typical example would be the valuation of a random variable at a future point in time: More information will be available, currently it is unknown which state of the system will actually be reached. Literature agrees on how to dene such mappings. However, there are dierent views about how to analyze them, and particularly their conjugate mappings. An important part of literature concentrates, based on the notion of coherent risk measures, on mappings from L -spaces into L -spaces with the additional requirement of positive homogeneity e.g. [2]. Detlefsen in [3] drops the assumption of positive homogeneity but retains the usage of bounded random variables. Pug and Römisch in [7] drop both assumptions but use an indirect denition of conjugate mappings. In the present work, which is a short version of [4], we want to give a general theory of conditional acceptability mappings as possibly nonhomogeneous mappings between L p -spaces for p 1. We regard such mappings as Banach lattice valued mappings, based on the adequate partial order a.s. and use this concept to dene and analyze supergradients and concave conjugate mappings in a straightforward way. Compared with [3] our approach will be more general, while compared with [7] it will be more straightforward and explicit, in particular regarding the used L p -spaces Basic Denitions. Again we consider random variables X L p Ω, F 1, P that can be interpreted as uncertain nancial payments at the end of a time period. For technical reasons we augment any used L p -space with the value. Denition 2.1. For any space L p Ω, F, P, the related extended space is given by L p Ω, F, P = L p Ω, F, P.

3 CONDITIONAL ACCEPTABILITY MAPPINGS 3 Remark. The set represents the functions with a.s. constant value. Denition 2.2. Conditional probability mappings are mappings A F 1 : L p Ω, F, P L p Ω, F 1, P with F 1 F and 1 p p. In addition we assume A F 1 =. In the rst instance, we are interested in monotonic mappings, particularly in the conditional counterparts of acceptability functionals. Denition 2.3. A conditional monotonic mapping is a conditional probability mapping such that CA1 Monotonicity. X Y a.s. AX F 1 AY F 1 a.s. holds for all random variables X, Y L p Ω, F, P. A conditional acceptability mapping is a conditional monotonic mapping that fullls the following additional assumptions for all X, Y L p Ω, F, P. CA2 Predictable Translation Equivariance. AX +X 1 F 1 = AX F 1 +X 1 holds a.s. for every X 1 L p Ω, F 1, P. CA3 Concavity. A λ X + 1 λ Y F 1 λ A X F λ A Y F 1 holds a.s. for λ [0, 1]. Sometimes we will shortly write A 1 := A F 1. In this setup, the σ algebra F can be interpreted as the information available at the end of a future time period and F 1 as the information available at its beginning. If F 1 is the trivial σ algebra, Ω, conditional mappings are reduced to unconditional probability functionals. Denition 2.4. The domain of a conditional acceptability mapping A is given by dom A = X L p Ω, F, P : AX F 1. This means that AY F 1 is either in L p Ω, F 1, P or it is a.s. Implicitly, we restrict our analysis to proper mappings, because our denition of extended spaces does not include the value +. Complementary to the conditions CA1-CA3 and dependent on the application, further properties can be required: A conditional mapping A is called positively homogeneous if A λ X F 1 = λ A X F 1 for any positive λ and X L p Ω, F, P. It is called centered at zero if A 0 F 1 = Conjugate Representations of Acceptability Mappings. Generalizing the denition of conditional expectation, Pug and Römisch use a trick from probability theory and derive a representation theorem for conditional acceptability mappings Theorem in [7]: For any u.s.c. conditional acceptability mapping A Y F 1 and for every Y L p Ω, F, µ and B F 1, the conjugate functional of the restricted expectation E A Y F 1 1 B can be written as E A Y F 1 1 B = inf E Y A B : 0, E F 1 = 1 B, L qω,f,µ where A B = inf E Y E A Y F 1 1 B. Y L pω,f,µ This meets the concerns for measurability and allows to use the full duality theory for real valued functionals. However, such a representation is an implicit one, which sometimes makes the reasoning more complicated.

4 CONDITIONAL ACCEPTABILITY MAPPINGS 4 If we want to dene concave conjugates for conditional acceptability mappings directly in the style of equation 1.1, we have to consider the fact that the values of such mappings are random variables. Three basic questions have to be answered: 1 What kind of partial order should be used to dene supergradients and conjugates? 2 Is the inmum with respect to this order suciently compatible with the L p -spaces and their norms? In particular: Is the resulting lattice order complete? 3 What type of continuous linear mappings should be used to generalize the expectation E X in equation 1.1? The rst question has already been answered implicitly in the preceding subsection: Every inequality is understood in the almost sure sense. The inmum of a set of random variables with respect to this ordering can be dened in straightforward manner: Denition 2.5. Let V = L p Ω, F, P be a random space and a.s. the almost sure order. A random variable X 0 V is the inmum of a nonempty subset A V, if a X 0 is a lower bound of the set A, i.e. Y X 0 a.s. for any Y A b Y for all Y A implies X 0 a.s. If V = L p Ω, F, P, then X 0 must hold in addition. We denote the inmum of a set A L p Ω, F, P by inf A. It will be clear from the context, whether the notation inf means the inmum of a set of real numbers or the inmum of a set of random variables. As pointed out before, the almost sure order is only a partial order. Therefore, the question arises, whether the inmum in this case is a reasonable concept. At least, order completeness has to be required. Fortunately this request is fullled for L p Ω, F, P-spaces. Recall that Riesz spaces vector lattices are ordered vector spaces V, that are also lattices, which means that any pair of elements v V has an inmum supremum in V. Furthermore, a subset A V is called order bounded from below if there is a vector v V that is dominated by each element of A. Denition 2.6. A Riesz space V is order complete Dedekind complete if every nonempty subset A V that is order bounded from below has an inmum. If the space V is dened as an extended space V = V of a Riesz space V with v for any v V, we call the extended space V order complete, if V is order complete. It is a well known fact that L p Ω, F, P-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 Ω, F, P-spaces are order complete Banach Lattices for 1 p. We will not go into these details here, but refer to the Appendix, where an alternative proof for the order completeness of L p Ω, F, P-spaces with 1 p is presented. Regarding the third question above, it obviously would be natural to use some kind of conditional expectation E X F 1. Such mappings with X L p Ω, F, P

5 CONDITIONAL ACCEPTABILITY MAPPINGS 5 and L q Ω, F, P with 1 p + 1 q = 1 clearly are linear and continuous as mappings into L 1 Ω, F 1, P. However, if we want to cover mappings into L p Ω, F, P with 1 < p p we have to restrict the set of feasible. Proposition 2.7. For 1 p <, 1 p p the conditional expectations X E X F 1 with arguments X L p Ω, F, P and dual variable L s Ω, F, P L q Ω, F, P are linear, continuous and map into L p Ω, F, P, if s p p p p. For mappings L p Ω, F, P L p Ω, F, P with p = the conditional expectations above have these properties, if L p Ω, F, P. Proof. It is clear that the conditional expectation X E X F 1 is linear. Moreover, are dual variables such that X E X F 1 maps into L p Ω, F, P, if L q Ω, F, P Y : X Y L p Ω, F 1, P, X L p Ω, F 1, P. This means that X must be p -integrable. For s = p p p p, which is equivalent to 1 p + 1 s = 1 p, the generalized Hölder inequality 2.1 X p X p s can be used. As X is p-integrable by assumption, this means that X will be p integrable if is s-integrable. To ensure continuity as a mapping into L p Ω, F, P, assume now that X n converges to X in the p-norm, which means that X n X p 0, as n. Because is a convex function we can apply Jensen's inequality and get 2.2 E X n X F p X n X p From this using again the generalized Hölder inequality 2.1 we can conclude E X n X F p X n X p s. If is s-integrable and X n converges to X in the p-norm, it follows that E X n F converges to E X F. Example. For mappings L p Ω, F, P L 1 Ω, F 1, P the variables can be from the natural dual space L q Ω, F, P because q = s holds in this cases. Until now the main focus in literature was on mappings L p Ω, F, P L p Ω, F 1, P, in particular p =. From Proposition 2.7 we see that the dual variables must be from L Ω, F, P in this case. This is a strong restriction and coincides with the natural dual only for the case p = 1. With this preparations, we dene superdierentials and conjugates. We use the principles of convex analysis for partially ordered vector spaces, essentially following the approach of Papageorgiou in [6]. Denition 2.8. The topological superdierential of a proper concave conditional mapping A F 1 : L p Ω, F, P L p Ω, F 1, P at X 0 L p Ω, F, P is given by the set A X 0 F 1 = = L s Ω, F, P : A X F 1 A X 0 F 1 + E X X 0 F 1, X doma, where 1 p + 1 s = 1 p. The elements of superdierentials are called supergradients.

6 CONDITIONAL ACCEPTABILITY MAPPINGS 6 The superdierential will be nonempty for a broad range of concave mappings, at least in the interior of their domains. Proposition 2.9. If a concave mapping A F 1 : L p Ω, F, P L p Ω, F 1, P is continuous at some point X 0 doma then A X F 1 for all X int doma. Proof. L p -spaces are metric spaces hence they are normal spaces. Using this fact as a premise we can apply Lemma 3.2 from [6]. Proposition A conditional acceptability mapping A : L p Ω, F, P L p Ω, F 1, P is continuous if it is locally bounded at some element X 0 L p Ω, F, P. Proof. See [5], Theorem 4. Next, we dene concave conjugates for conditional mappings. Based on our considerations about the inmum, this can be done in a straightforward manner. Denition The concave conjugate of a mapping is given by a mapping with 1 p + 1 s = 1 p : A + F 1 = A F 1 : L p Ω, F, P L p Ω, F 1, P A + : L s Ω, F, P L p Ω, F 1, P inf E X F 1 A X F 1. X L pω,f,p The concave biconjugate is a mapping, A ++ : L p Ω, F, P L p Ω, F 1, P dened by A ++ X F 1 = inf L sω,f,p E X F1 A + F 1. Like in the case of unconditional functionals, it is possible to characterize mappings that are equal to their own biconjugate. The following proposition is based on the existence of supergradients. Denition A conditional mapping A F 1 with closed domain is called upper semicontinuous u.s.c. at X 0, if and only if for every ε > 0 a.s. there exists a neighborhood U of X 0 such that A X F 1 A X 0 F 1 + ε a.s. for all X U, or if A X 0 F 1 = +. A mapping A F 1 is called lower semicontinuous l.s.c. at a point X 0 if A F 1 is u.s.c. at the point X 0. Remark. If a mapping A F 1 is nite, l.s.c. and u.s.c. at a point X 0 then it is continuous at this point [6], Theorem Proposition Let A F 1 be a proper, concave mapping. Then for all X L p Ω, F, P, L s Ω, F, P with 1 p + 1 s = 1 p the inequality 2.3 A X F 1 E X F 1 A + F 1, holds a.s..

7 CONDITIONAL ACCEPTABILITY MAPPINGS 7 Moreover, if A X 0 F 1 then the equation 2.4 A X 0 F 1 = A ++ X 0 F 1 holds a.s., and A F 1 is u.s.c. at X 0. In this case the inmum inf E X F 1 A + F 1 : doma + is attained and 2.5 A X F 1 = E X F 1 A + F 1 a.s. A X F 1. If in addition A is continuous at some point inside doma, equation 2.4 holds for each point in int doma. Proof. The inequality follows directly from the denition of the conjugate: A + F 1 := inf E X F 1 A X F 1 it follows that X A + F 1 E X F 1 A X F 1 a.s. or A X F 1 E X F 1 A + F 1. The second assertion follows from Theorem 5.8 in [6]. The a.s. equation A X F 1 = E X F 1 A + F 1 holds if and only if A X F 1 + E Y F 1 A Y F 1 E X F 1 holds a.s. for any Y L p Ω, F, P. a First assume A X F 1 = E X F 1 A + F 1 a.s.. Using the denition of the conjugate mapping we have A X F 1 + E Y F 1 A Y F 1 A X F 1 + inf E Y F 1 A Y F 1 Using the above assumption we get for any Y. = A X F 1 + A + F 1. A X F 1 + E Y F 1 A Y F 1 E X F 1 b For the other direction assume A X F 1 + E Y F 1 A Y F 1 E X F 1 a.s.. Remember that this inequality should hold for all Y - which means that E Y F 1 A Y F 1 has a lower bound. Then the inmum exists and it follows that or A X F 1 + inf E Y F 1 A Y F 1 E X F 1 A X F 1 + A + F 1 E X F 1 Together with inequality 2.3 the equation A X F 1 + A + F 1 = E X F 1 can be concluded. Moreover, holds, if and only if A X F 1 + E Y F 1 A Y F 1 E X F 1 A Y F 1 A X F 1 + E Y X F 1, which means A X F 1, is true. Together with inequality 2.3, the last inequality yields equation 2.5. The statement about continuity follows from Proposition 2.9. As

8 CONDITIONAL ACCEPTABILITY MAPPINGS 8 For conditional acceptability functionals we can give an even sharper characterization: Proposition Assume that the conditions of Proposition 2.9 or Proposition 2.10 are fullled. Then a concave conditional mapping A F 1 : L p Ω, F, P L p Ω, F 1, P is an acceptability mapping if and only if a dual representation A X F 1 = inf L sω,f,p E X F1 A + F 1 : 0; E F 1 = 1 a.s., holds for each point in int doma. The set represents additional constraints on, restricting e.g. the conjugate A + to be nite. Proof. By Proposition 2.13 and because A is concave and continuous at some point in doma for each point in int doma we have : A X F 1 = inf E X F1 A + F 1 = E X F 1 A + F 1 where is a supergradient. If E F 1 = 1 a.s. and F 1 -measurable X 1, it is possible to infer A X + X 1 F 1 = E X + X 1 F 1 A + F 1 = E X F 1 + E X 1 F 1 A + F 1 = E X F 1 + X 1 E F 1 A + F 1 = E X F 1 + X 1 A + F 1 = A X F 1 + X 1. On the other hand let E F 1 1 on a set S with positive probability. Then we can conclude A X F 1 + X 1 = E X F 1 + X 1 A + F 1 E X F 1 + X 1 E F 1 A + F 1 = A X + X 1 F 1 on this set. This would contradict the assumption of predictable translation equivariance. Assume now that 0 holds a.s. and let Y 0 a.s.. Then, for any random variable X we have X + Y X. Because Y and both are nonnegative it follows that E Y F 1 0. Using Proposition 2.13, we infer A X + Y F 1 = E X + Y F 1 A + F 1 = E X F 1 + E Y F 1 A + F 1 E X F 1 A + F 1 inf E X F1 A + F 1 = A X F 1.

9 CONDITIONAL ACCEPTABILITY MAPPINGS 9 For the other direction assume now that < 0 on a set S with positive probability, or E Y F 1 < 0 on this set. Then we have A X F 1 = E X F 1 A + F 1 > E X F 1 + E Y F 1 A + F 1 inf E X F1 + E Y F 1 A + F 1 = A X + Y F 1 on the set S. But this would mean that the mapping can not be monotonic. Using the structural anity between conjugate representations for unconditional functionals and conditional mappings, Proposition 2.14 can be used to dene acceptability mappings. If any occurrence of an expectation in the conjugate representation of an acceptability functional is replaced by the appropriate conditional expectation, the result will be a conditional acceptability mapping. Based on equation 1.3, we dene the conditional Average Value at Risk as an example: Denition The conditional average value at risk AV@R is dened as a mapping α F 1 : L p Ω, F, P L p Ω, F 1, P by a generalized LP: α X F 1 = inf E X F 1 : 0 1 α a.s., E F 1 = 1 a.s. The dual variables in this denition are bounded almost surely and therefore, they are in L Ω, F, P. This means that the conditional α is able to map a space L p Ω, F 1, P into itself. Corollary The α is a continuous mapping and its superdierential is a nonempty set for any X L p Ω, F, P. Proof. Because of Proposition 2.7 the α is continuous. Then from Proposition 2.9 nonemptyness of the superdierential follows. 3. Application to Multi-period Functionals For multi-period valuation we consider the following general setup: Time t = 0,..., T is discrete. We restrict ourselves to a nite time horizon T <. F = F 0,..., F T is a ltration with F T = F. Consider now a stochastic process X t t 1,...,T, representing random nancial payos at points in time 1,..., T. Multi-period functionals evaluate the acceptability of such a process in terms of a real number, related to the actual period of decision. Such multi-period functionals can be described as mappings A ; F from product spaces T L pt Ω, F t, P, which are Banach lattices together with T the norm X = E X t pt 1 p t, 1 pt, into the extended real line R. Denition 3.1. We will call a multi-period functional A X; F a multi-period acceptability functional, if it is proper and satises Ω,, F 1 1,..., F 1 T, MA0 Information Monotonicity: If F 1 = F 2 = Ω,, F 2 1,..., F 2 T are ltrations such that F 1 t F 2 t for

10 CONDITIONAL ACCEPTABILITY MAPPINGS 10 MA1 all t, then A X; F 1 A X; F 2 for any X dom A T L p Ω, F t, P. Translation Equivariance: For all periods t the equation A X 1,..., X t + c,..., X T ; F = A X 1,..., X t,..., X T ; F + c holds, if c is a real number. MA2 Concavity: The mapping X A X; F is concave. MA3 Monotonicity: X t Y t a.s.for all t implies A X; F A Y ; F Remark. This Denition goes back to [7], where a stronger version of MA1 was used. For concave multi-period functionals A X; F it is possible to dene the concave conjugate A + ; F = inf X, A X; F : X T L p Ω, F t, P. If A X; F is in addition proper and upper semicontinuous, then the Fenchel- Moreau-Rockafellar Theorem [9], 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: Proposition 3.2. Let A ; F be an upper semicontinuous multi-period functional satisfying MA1, MA2 and MA3. Then the representation T 3.1 A X; F = inf E X t t A + ; F : t 0; E t = 1 holds for every X T L p Ω, F t, P. 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 and MA3. Proof. See Theorem 3.21 in [7]. Obviously, there are many ways for constructing multi-period functionals. For example, Pug and Römisch [7] analyze separable functionals, risk functionals of the value-of-information type, compositions of conditional acceptability mappings and polyhedral multi-period acceptability functionals. In the following, we will apply the above results to two special types of multiperiod functionals that can be constructed from conditional mappings, namely additive compositions of acceptability mappings and SEC-functionals Additive Acceptability Compositions. Again we consider stochastic processes X t t 1,...,T, adapted to a ltration F = F 0,..., F T and representing nancial cash ows. It is assumed that F 0 = Ω, is the trivial σ-algebra, hence a mapping A 0 F 0 can be used to represent an unconditional functional. In order to shorten notation we will frequently use the abbreviation A t = A t F t in the following. Conditional expectations will be abbreviated by E F t = E t.

11 CONDITIONAL ACCEPTABILITY MAPPINGS 11 Additive compositions will recursively apply conditional acceptability mappings to the values X t. Up to now, conditional probability mappings have been dened as mappings L p Ω, F, P L p Ω, F 1, P. If we want to compose such mappings we have to slightly modify this denition, taking care of the possibility of innite valued arguments: We will understand conditional probability mappings as mappings L p Ω, F, P L p Ω, F 1, P. In addition we have to make sure that the conditional mappings map into the correct spaces. Denition 3.3. Let p be a sequence of real numbers p = p 0 p 1... p T, with 1 p t. We will call a sequence of mappings A t t 0,...,T 1 p integrability adapted if A t 1 maps from L pt Ω, F t, P into L pt 1 Ω, F t 1, P for all t 1,..., T 1. With these agreements it is possible to dene additive monotonic compositions in the following way. Denition 3.4. Let X t t 1,...,T be a stochastic process with X t L pt Ω, F t, P and let A t F t, t = 1,..., T be an integrability adapted sequence of conditional monotonic mappings. An additive monotonic composition is a multi-period probability functional B F which for any X T j=t L pj Ω, F j, P is constructed in the following way: for T 1 and B T 1 X T F T 1 = A T 1 X T F T 1 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. Then the multiperiod functional B F is given by BX F = B 0 X F. The intermediate mappings B t F T 1 are called nested conditional acceptability mappings. If all of the mappings A t F t are conditional acceptability mappings we call the composition an additive acceptability composition. Remark. For additive acceptability compositions we have B t X t+1 ; F = A t A t+1 A T 1 T k=t+1 X k. Based on the convexity and monotonicity of the constituent conditional mappings it is possible to state a chain rule for additive acceptability mappings. Proposition 3.5. Let A t t 1,...,T 1 be a collection of integrability adapted conditional acceptability functionals with T 2. Given supergradients T A T 1 X T and k A k 1 X k + B X k+1,..., X T

12 CONDITIONAL ACCEPTABILITY MAPPINGS 12 for k t 0 + 1,..., T, a supergradient for the nested conditional acceptability functional B t0 ; [A t0,,..., A T 1 ]; F at the base points X t0+1,..., X T is given by a T t-tupel M = M t0+1,..., M T with and M t0+1 t0+1 M k+1 = M k k+1 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 and M 1 1 for 0 < k T 1. M k+1 = M k k+1 Proof. We use backward induction to prove the result: For t = T 1 the equation CA T 1 X T ; [A T 1 ] = A T 1 X T holds and T is a supergradient. Let the Proposition be true for all t t 0. Then, a supergradient for the nested mapping CA t0 Y t0+1,..., Y T ; [A t0,...,a T 1 ] is given by M t0+1,..., M T. That means: CA t0 X t0+1 + Y t0+1,..., X T + Y T CA t0 X t0+1,..., X T +E t0 Yt0+1 t Et0 YT T T 1... t0+1. Using monotonicity we have for t 0 1: CA t0 1 X t0 + Y t0, X t0+1 + Y t0+1,..., X T + Y T = = A t0 1 X t0 + Y t0 + CA t0 X t0+1 + Y t0+1,..., X T + Y T T k A t0 1 X t0 + CA t0 X t0+1,..., X T + Y t0 + E t0 Y k k=t 0+1 j=t 0+1 j As t0 is a supergradient of A t0 1 at X t0 + CA t0 X t0+1,..., X T it follows that CA t0 1 X t0 + Y t0, X t0+1 + Y t0+1,..., X T + Y T A t0 1 X t0 + CA t0 X t0+1,..., X T + [ + E t0 1 t0 Y t0 + E t0 Yt0+1 t Et0 Y T T t=t 0+1 t ] = = CA t0 1 X t0, X t0+1,..., X T +E t9 1 Yt0 t0 +Et0 1 Yt0+1 t0+1 t E t0 1 YT T T 1... t0+1 t0 The mappings A t F t are integrability adapted and the t are supergradients, hence the conditional expectations E X t t F t 1 are also pt 1 -integrable by Proposition 2.7. This means that all the conditional expectations involved in each nesting-step are p t -integrable. Because we made no restriction on T it is shown that M = M t0,..., M T as dened above is a supergradient for CA t for any t T.

13 CONDITIONAL ACCEPTABILITY MAPPINGS 13 It should be noted that supergradients of additive acceptability compositions form a martingale: Corollary 3.6. 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 3.5 is a martingale. Proof. From Proposition 2.14 we know that E t F t 1 = 1 holds because of predictable translation equivariance. Moreover, as the process M t is adapted to F t, the random variable M t 1 is F t 1 -measurable. Therefore holds. E M t F t 1 = E M t 1 t F t 1 = Wt 1 E t F t 1 = Mt 1 So far we have characterized the supergradients of additive acceptability compositions. Using this martingale representation and Proposition 2.14 it is also possible to calculate the dual representation of compositions by using the concave conjugates of the constituent conditional mappings. Proposition 3.7. Let CA 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 = Et X t+1 t A + t t+1 : t+1 0; E t t+1 = 1, t t. inf t+1 L st Ω,F,P Under the assumptions of Theorem 3.5 the acceptability composition CA 0 can be represented by 3.2 CA 0 X 1, X 2,..., X T = T with inf 1,..., T T E X t M t A E A + t t+1 M t : = t : M 1 = 1 ; M t+1 = M t t; ; t 0; E t F t 1 = 1, t t. Proof. Theorem 2.14 can be used to replace the conditional mappings A t by their concave conjugates in a recursive manner. In this way the assertion can be proved by backward induction. Here is one step: First note that 3.3 A t 1 Y t = inf t Et 1 Y t t A + t 1 t : t 0; E t 1 t = 1 holds by Theorem Using 2.2 we get for a two period additive acceptability compositions: A T 2 X T 1 + A T 1 X T = inf E T 2 [X T 1 + A T 1 X T ] T 1 A + T 2 T 1 T 1 [ = inf E ] T 2 X T 1 + inf E T 1 X T T A + T 1 T 1 T T 1 A + T 2 T 1 T = inf T 1 E T 2 X T 1 T 1 + E T 2 inf E T 1 X T T A + T 1 T T 1 A + T 2 T 1 T

14 CONDITIONAL ACCEPTABILITY MAPPINGS 14 Let now T be a supergradient of A T 1 X T. Then this supergradient must also be minimizer of E T 1 X T T F T 1 A + T 1 T by Theorem Because of [ monotonicity all the t are nonnegative and hence T is also a minimizer of ET 1 X T T A + T 1 T ] T 1. Therefore we can interchange inmum and conditional expectation. This is possible because the inmum in the conjugate representation mapping must be attained if the superdierential is not empty - which is the case, since - by assumption - we know the supergradients. A T 2 X T 1 + A T 1 X T = ] = inf E T 2 X T 1 T 1 + inf E T 2 [E T 1 X T T A + T 1 T 1 T T ] = inf E T 2 X T 1 T 1 + E T 2 [E T 1 X T T A + T 1, T 1 T T T 1 A + T 2 T 1 T 1 A + T 2 T 1 = inf E T 2 X T 1 T 1 + E T 2 X T T T 1 E T 2 A + T 1, T 1 T T 1 A + T 2 T 1 T All the inma must be understood with respect to the constraints T 0 E T F 1 = 1 and T 1 0 E T 1 F 1 = 1. Iterating this step and dening the variable M t as M 1 = 1 and M t+1 = M t t+1 we get the statement of the theorem. In the light of Proposition 3.5 and Corollary 3.6 the M t are martingales and constitute a supergradient of the composition. So the equation CA 0 X 1, X 2,..., X T = T inf 1,..., T E X t M t A T 1 E A + t t+1 M t : basically gives the conjugate representation of the composition and the mapping A T E A + t t+1 M t is the related conjugate mapping CA + 0. It is possible to restate Proposition 3.7 in a way, such that only the supergradients M t are used: Corollary 3.8. Under the assumptions of Proposition 3.5, and with ψm t, M t+1 = sup E A + t t+1 M t : Mt+1 = M t t+1, t+1 t, the supergradient representation of an acceptability composition is given t+1 by CA 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. Based on the results on supergradients and concave conjugates of additive acceptability compositions, we can outline the properties of such mappings in the following way:,

15 CONDITIONAL ACCEPTABILITY MAPPINGS 15 Proposition 3.9. An acceptability composition is a proper, u.s.c. concave MA2, monotonic MA3 multi-period probability functional that is also translation equivariant MA1. Proof. Properness and upper semicontinuity follows from 3.2. As a composition of monotonic concave mappings the multiperiod functional must be concave itself. It is also monotonic because of M t 0 a.s. 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 Proposition 3.2, is the criterion for weak translation equivariance. Proof. An alternative proof for MA1', MA2 and MA3, which is independent of the dual representation 3.2 is given in [7], Theorem Example Based on Proposition 3.8 the nested average value at risk - constructed by composing conditional α s for the later periods with an unconditional α for the rst period - has the following representation: α X; F = T E X t M t : 0 M t 1 α M t 1, E M t F t 1 = M t 1, M 0 = 1 inf M 1,M 2,...,M T 4. Separable Expected Conditional Functionals Another approach for dening multi-period functionals consists in applying singleperiod functionals to the random variables X t of the process and summing up the results: T A X; F = A [t] X t Such multi-period functionals are called separable functionals. If the constituent single-period functionals are concave, the resulting separable functional will be also concave MA2. In addition, separable functionals are weakly translationequivariant MA1' and monotonic MA3 if this is also true for all their constituents. Unfortunately, it is not automatically guaranteed that separable functionals are information monotonic for arbitrary single period acceptability functionals. That means that generally such multi-period functionals are not acceptability functionals without additional measures to assure this property. For example, sums of expectations or sums of as well as sums of other typical single-period functionals are not information monotonic. Nevertheless, there is an approach for dening separable functionals that account for information in the right way: it is characterized by taking sums of expectations of conditional acceptability mappings. Such functionals are called separable expected conditional SEC, [7], p 145. Denition 4.1. SEC-functional A multi-period acceptability functional is called separable expected conditional SEC if it is of the form T A X; F = E A [t] X t F t 1,

16 CONDITIONAL ACCEPTABILITY MAPPINGS 16 where the A [t] F t 1 are conditional u.s.c. acceptability mappings. The most important and best known SEC-functional is the multi-period average T value at risk [8] E α X t F t 1. SEC-functionals are separable functionals and hence they fulll MA1', MA2 and MA3. It is well known that SEC-functionals based on conditional versions of distortion functionals AX = 0,1] αx dmα are information monotonic [7], Proposition 3.9. A multi-period functional A is separable if and only if its dual is separable, and it is SEC if and only if its dual is SEC Proposition 3.27 in [7]. That means that the concave conjugate of any SEC-functional can be represented in the form T A ; F = E A [t] t F t 1. Building on the chain-rule given in Proposition 3.5, we can characterize the supergradients of SEC-functionals: Proposition 4.2. Let A X; F = T E A [t] X t F t 1 be a SEC-functional and = 1,..., T a vector of supergradients of the constituent conditional acceptability mappings A [t]. Then is also a supergradient of the SEC-functional. Proof. Each summand E A [t] X t F t 1 can be interpreted 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 t. Applying Proposition 3.5, we see that for each t the product t 1 = t must be a supergradient of the associated acceptability composition E A [t] X t F t 1, which means that E A [t] X t + Y t F t 1 E A [t] X t F t 1 + E Y t t. or Summing over all t we get T T E A [t] X t + Y t F t 1 E A [t] X t F t 1 + A X + Y ; F A X; F + T E Y t t. T E Y t t, This shows that is a supergradient for the whole SEC-functional. Appendix Proposition. Let S = X ϑ : ϑ Θ be an arbitrary set of nonnegative functions from L p Ω, F, P with p 1. Then this set has an almost sure inmum X L p Ω, F, P.

17 CONDITIONAL ACCEPTABILITY MAPPINGS 17 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 is bounded below. This means that s = inf s α must exist. Select now a sequence α n n N with α n α n+1 such that s αn s. By 4.1, this sequence of sets induces a decreasing sequence X αn n N of random variables in S. Moreover, the set S was assumed to be bounded below by zero and to contain integrable random variables. Hence, because of X αn 0 X αn dp we can use monotoneˆconvergence to conclude that there exists an X L 1 Ω, F, P with X αn C and X dp = lim n Xαn dp = s. X is the inmum of X αn n N and must also be a lower bound for A, hence also for S: Assume that some X α is not dominated by X. Because of the ordered structure of X αn n N A this means that X α X a.s. and X α ω Xω must hold on some set with positive probability. But this would imply X α dp < X dp = s, which contradicts the fact that s = inf Xα dp. On the other hand X must be the largest lower bound: If any Y exists with Y X a.s. and Y ω > Xω on some set with positive probability then Y dp > X dp = s follows, which means that Y can not be a lower bound of A. We can also conclude that X L p Ω, F, P, because S contains only p-integrable functions X ϑ, which means that 0 X X ϑ and hence p-integrability of X must hold for any ϑ Θ. References [1] C. D. Aliprantis and K. C. Border. Innite Dimensional Analysis - A Hitchhiker's Guide. Springer, 3d edition, [2] P. Artzner, F. Delbaen, D. Heath, and H. Ku. Coherent multiperiod risk adjusted values and bellman's principle. Annals of Operations Research, 1521:522, [3] K. Detlefsen and G. Scandolo. Conditional and dynamic convex risk measures. Finance and Stochastics, 9:539561, [4] R. Kovacevic. Conditional Acceptability mappings: Convex Analysis in Banach Lattices. Dissertation, University of Vienna, [5] K. Nikodem. Continuity properties of convex-type set-valued mappings. Journal of Inequalities in Pure and Applied Mathematics, 43:Article [6] N. S. Papageorgiou. Nonsmooth analysis on partially ordered vector spaces: Part 1 - convex case. Pacic Journal of Mathematics, 1072:403458, [7] G. Pug and W. Römisch. Modeling, Measuring and Managing Risk. World Scientic, August [8] G. Pug and A. Ruszczy«ski. Measuring risk for income streams. Computational Optimization and Applications, 32:161178, [9] R. Rockafellar. Conjugate duality and optimization. In CBMS-NSF Regional Conference Series in Applied Mathematics, volume 16. SIAM, Philadelphia, 1974.