Representation of convex operators and their static and dynamic sandwich extensions
aa r X i v : . [ m a t h . F A ] M a y Representation of convex operators andtheir static and dynamic sandwich extensions
Jocelyne Bion-Nadal ∗ and Giulia Di Nunno †‡ May 31, 2016
Abstract
Monotone convex operators and time-consistent systems of oper-ators appear naturally in stochastic optimization and mathematicalfinance in the context of pricing and risk measurement. We study thedual representation of a monotone convex operator when its domainis defined on a subspace of L p , with p ∈ [1 , ∞ ], and we prove a sand-wich preserving extension theorem. These results are then applied tostudy systems of such operators defined only on subspaces. We pro-pose various dynamic sandwich preserving extension results dependingon the nature of time: finite discrete, countable discrete, and continu-ous. Of particular notice is the fact that the extensions obtained aretime-consistent. The literature on extension theorems for functionals features some funda-mental results. For all we just mention two: first is the Hahn-Banach the-orem and its various versions, that provides e.g. a majorant preservingextension and then the K¨onig theorem that provides a sandwich preservingone (see e.g. [19]). Both cases give results for linear functionals with val-ues in R . This paper presents sandwich preserving extension theorems for convex monotone operators defined in a subspace L in L p ( B ) with values in L p ( A ) ( A ⊆ B ), for p ∈ [1 , ∞ ].Other results of this type are studied in the case of linear operators, see [2]for p ∈ [1 , ∞ ) and [9] for p = ∞ . Indeed the need for working in an operatorsetting taking values in L p ( A ) is motivated by pricing and risk measurement ∗ UMR 7641 CNRS - Ecole Polytechnique. Ecole Polytechnique, 91128 Palaiseau Cedex,France. Email: [email protected] † Centre of Mathematics for Applications (CMA), Department of Mathematics, Univer-sity of Oslo, P.O. Box 1053 Blindern, N-0316 Oslo Norway. Email: [email protected] ‡ Norwegian School of Economics and Business Administration (NHH), Helleveien 30,N-5045 Bergen, Norway.
1n mathematical finance. To explain for any two fixed points in time, say s ≤ t , a financial asset with payoff X ∈ L t ⊆ L p ( F t ) has a price x s,t ( X )evaluated at s . This value is x s,t ( X ) ∈ L p ( F s ), where F s ⊆ F t .These price operators are linear if the market model benefits of assumptionsof smoothness, such as no transaction costs, no liquidity risk, perfect clearingof the market, no constrains in trading, etc. However, they are convex (whenconsidered from the seller’s perspective, the so-called ask-prices) when suchmarket model assumptions are not fulfilled. Convex operators of such formalso appear as value processes in the case of dynamic stochastic optimizationand often this is in fact a way to obtain such price processes.It is reasonable to have the domain of these operators defined on a subspace L of the corresponding L p space. In fact, in general, not all positions areactually available for purchase in the market. It is only in the idealisticassumption of a complete market that we find that all positions are alwaysfeasible, i.e. the subspace is actually the whole L p space. Strictly speaking,though less discussed in the literature, also the risk measurement is usuallyperformed more reasonably on a subspace L ⊆ L p . In this case L representsthose risks for which there is grounded measurement in terms of e.g. statisti-cal knowledge, time series analysis, and general good information. For risksoutside this set, one can always resolve with a conservative risk evaluationwhich corresponds to high (even too high to be competitive) hedging prices.The reasonable evaluation of risk and corresponding prices is relevant froman insurance perspective.When dealing with a dynamic approach to pricing, we consider an informa-tion flow represented by a filtration ( F t ) t ∈ [0 ,T ] ( T < ∞ ) and then a system ofprice operators is naturally appearing: ( x s,t ) s,t ∈T , where T ⊆ [0 , T ]. For thefixed times s, t : s ≤ t the price operator is x s,t : L t −→ L p ( F s ) where thedomain is the subspace L t ⊆ L p ( F t ). An important necessary property ofthese system of operators is time-consistency , which models the consistencyof prices or measures of risk over time. Namely, for s ≤ t ≤ u and X ∈ L u ,the evaluation x s,u ( X ) at time s is required to coincide with the two stepsevaluation x s,t ( x t,u ( X )).The question we address is how to extend the whole family of operators,so that the domains reach the whole L p ( F t ) in such a manner that time-consistency is preserved together with some sandwich property. The sand-wich property itself is a control from above and below reasonably given onsuch operators, as it happens, in their own context, for the Hahn-Banach andK¨onig theorems. In applications this may assume various meanings. In [3],[2], [16] there are different studies on some aspects of the fundamental theo-rem of asset pricing with controls on tail events, first in a multiperiod marketand then in a continuous time market. In [9] the majorant and minorantoperators are linked to no-good-deal dynamic bounds and the correspondingpricing measures. From the application perspective, the feasibility of these2ricing rules is directly linked to the existence of the corresponding time-consistent sandwich preserving extension of the system of price operators.So far this link has been explored only for linear pricing. The present paperprovides fundamental results to address some questions related to convexpricing, e.g. no-good-deal bounds for convex pricing and its connectionswith indifferent pricing.We stress that to obtain a time-consistent extension it is not enough tocollect all the extensions of the single operators in one family. It is only viasome careful procedure of extension that we can obtain such result.Also it is important to mention that the representation result we obtain forconvex operators defined on a subspace L of L p ( B ) taking values in L p ( A )( A ⊆ B ) is crucial for the development of the extensions. Representationtheorems for convex operators have been studied in the context of risk-measures in the recent years. The first results were obtained for the staticcase, corresponding to operators with real values ( A trivial). Here we haveto mention [18] (for p ∈ [1 , ∞ ]) and [17] (for p = ∞ ), where the domainof the operators is the whole L p ( B ) space, and the paper [4], where a verygeneral framework is proposed, which also includes the case of a subspace L of L p ( B ) (for p ∈ [1 , ∞ ]) with lattice property on L . In both cases themappings take real values. We also mention [5] and [14] for a representationin the case of operators defined on the whole L ∞ ( B ) with values in L ∞ ( A )studied in the context of conditional risk measures. Our contribution inthis area provides a representation theorem for convex operators defined on L ⊆ L p ( B ) (for p ∈ [1 , ∞ ]) without requiring the lattice property of L .In a summary our contribution provides various elements of novelty:1. A representation theorem for a convex operator defined on a subspace L ⊆ L p ( B ) with values in L p ( A ) with A ⊆ B not trivial and withoutrequiring the lattice property on L .2. A sandwich preserving extension theorem for such convex operators,where the sandwich bounds are sublinear and superlinear operatorsdefined on the whole L p ( B ).3. Various time-consistent sandwich preserving extensions theorems deal-ing with a family of convex operators x s,t defined on L t ⊆ L ( F t ) withvalues in L p ( F s ), for s ≤ t . We have to stress that • Time-consistency is a crucial property necessary in many appli-cations such as in financial price evolution. • Time-consistency is not preserved by independent extensions ofthe single operators x s,t . • Time-consistent extensions require careful procedures in order tomaintain the delicate relationships of the operators x s,t over time.3he paper is organized as follows. In Section 2 we give a precise presen-tation of the operators, the spaces, and the topology we consider. Thenthe representation theorem is proved. Section 3 is dedicated to the sand-wich extension of such convex operators. The sequel of the paper deals withtime-consistent systems of operators. In Section 4 the sandwich extensionis studied in the case of discrete time. In Section 5 we reach out to obtainthe sandwich extension for continuous time systems of operators. L p and their representation Let (Ω , B , P ) be a complete probability space. Here we consider B to be the P -completed σ -algebra generated by a countable family of sets in Ω. Alsolet A ⊆ B be a P -augmented countably generated σ -algebra . For example,any Borel σ -algebra of a metrizable separable space completed by the P -nullevents satisfies this assumption.For any p ∈ [1 , ∞ ] we consider the L p ( B ) := L p (Ω , B , P ) of real valuedrandom variables with the finite norms: k X k p := ( ( E [ | X | p ]) /p , p ∈ [1 , ∞ ) , esssup | X | , p = ∞ . We equip these spaces with a topology. In the cases p ∈ [1 , ∞ ), we con-sider the usual topology derived from the norm. In the case p = ∞ , wefix the weak* topology σ ( L ∞ , L ). We will denote by the superscript “+”,e.g. L p ( B ) + , the cones of the non-negative random variables with the cor-responding induced topology.In the sequel we deal with a linear sub-space L ⊆ L p ( B ). We always consider L equipped with the topology induced by the corresponding L p ( B ) space.Motivated by the applications we assume that:i) 1 ∈ L ,ii) for the σ -algebra A ⊆ B we have the property that 1 A X ∈ L for every A ∈ A and every X ∈ L .iii) For all X ∈ L and every sequence X n ∈ L such that sup n || X n || p < ∞ which converges P -a.s. to X , X belongs to L .Fix p ∈ [1 , ∞ ] and the sub-space L ⊆ L p ( B ) as above. We consider anoperator x : L −→ L p ( A ) (2.1)that is: This assumption will be implicitly used in the sandwich extension theorems. It is notnecessary for the upcoming representation theorem. monotone , i.e. for any X ′ , X ′′ ∈ L , x ( X ′ ) ≥ x ( X ′′ ) , X ′ ≥ X ′′ , • convex , i.e. for any X ′ , X ′′ ∈ L and λ ∈ [0 , x (cid:0) λX ′ + (1 − λ ) X ′′ (cid:1) ≤ λx ( X ′ ) + (1 − λ ) x ( X ′′ ) • Fatou property , for all sequence X n in L such that sup n || X n || L p < ∞ which converges P .a.s. to X ,lim inf n →∞ x ( X n ) ≥ x ( X ) (2.2) • weak A -homogeneous , i.e. for all X ∈ Lx (1 A X ) = 1 A x ( X ) , A ∈ A , • projection property x ( f ) = f, f ∈ L p ( A ) ∩ L. In particular we have x (0) = 0 and x (1) = 1.Note that, if p ∈ [1 , ∞ ) and the operator x is monotone and linear (as in[2]) the assumption of weak A -homogeneity is equivalent to A -homogeneity,i.e. for all X ∈ L , we have x ( ξX ) = ξx ( X )for all ξ ∈ L p ( A ) such that ξX ∈ L . If p = ∞ and the operator is linear andsemi-continuous, then the same result holds (see [9]). Our first result is a representation theorem for L p -valued convex operatorsof the type above. This can be regarded as a non-trivial extension of [21,Theorem 5]. The result by Rockafellar is written for functionals and can beretrieved setting A to be the trivial σ -algebra up to P -null events. Theorem 2.1.
Let x be an operator of the type (2.1) . Then the followingrepresentation holds: x ( X ) = esssup V ∈V n V ( X ) − x ∗ ( V ) o , X ∈ L, (2.3)5 here x ∗ ( V ) := esssup X ∈ L n V ( X ) − x ( X ) o , V ∈ V , and V is the space of the linear, non-negative, continuous, A -homogeneousoperators V : L p ( B ) −→ L p ( A ) such that E [ x ∗ ( V )] < ∞ .Moreover, the operator x also admits representation in the form: x ( X ) = esssup V ∈ V n V ( X ) − x ∗ ( V ) o , X ∈ L, (2.4) where x ∗ ( V ) := esssup X ∈ L n V ( X ) − x ( X ) o , V ∈ V , (2.5) and V is the space of the linear, non-negative, continuous, A -homogeneousoperators V : L p ( B ) −→ L p ( A ) . For future reference we borrow the terminology proper of the literature onrisk measures and we call the operator x ∗ minimal penalty .Before the proof of the theorem we present a couple of technical lemmas. Lemma 2.2. If V = 1 A V + 1 A c V , for V , V ∈ V , A ∈ A , and A c := Ω \ A ,then x ∗ ( V ) = 1 A x ∗ ( V ) + 1 A c x ∗ ( V ) . (2.6) Moreover the set (cid:8) V ( X ) − x ∗ ( V ) : V ∈ V (cid:9) is a lattice upward directed.Proof. For any
X, Y ∈ L we have1 A (cid:0) V ( X ) − x ( X ) (cid:1) + 1 A c (cid:0) V ( Y ) − x ( Y ) (cid:1) = V (1 A X + 1 A c Y ) − x (1 A X + 1 A c Y ) ≤ esssup Z ∈ L (cid:8) V ( Z ) − x ( Z ) (cid:9) . Hence,1 A esssup X ∈ L (cid:8) V ( X ) − x ( X ) (cid:9) + 1 A c esssup Y ∈ L (cid:8) V ( Y ) − x ( Y ) (cid:9) ≤ esssup Z ∈ L (cid:8) V ( Z ) − x ( Z ) (cid:9) . Namely, we have 1 A x ∗ ( V ) + 1 A c x ∗ ( V ) ≤ x ∗ ( V ). On the other hand, forany Z ∈ L , we have V ( Z ) − x ( Z ) = 1 A (cid:0) V ( Z ) − x ( Z ) (cid:1) + 1 A c (cid:0) V ( Z ) − x ( Z ) (cid:1) ≤ A esssup Z ∈ L (cid:8) V ( Z ) − x ( Z ) (cid:9) + 1 A c esssup Z ∈ L (cid:8) V ( Z ) − x ( Z ) (cid:9) . Therefore, x ∗ ( V ) ≤ A x ∗ ( V ) + 1 A c x ∗ ( V ). So (2.6) holds.To prove the lattice property, it is enough to consider for any V , V ∈ V , theset A := (cid:8) V ( X ) − x ∗ ( V ) ≥ V ( X ) − x ∗ ( V ) (cid:9) ∈ A and V = 1 A V + 1 A c V .From (2.6) we have that: V ( X ) − x ∗ ( V ) = 1 A (cid:0) V ( X ) − x ∗ ( V ) (cid:1) + 1 A c (cid:0) V ( X ) − x ∗ ( V ) (cid:1) = sup (cid:8) V ( X ) − x ∗ ( V ) , V ( X ) − x ∗ ( V ) (cid:9) . By this we end the proof. 6 emma 2.3.
For any V ∈ V , the set (cid:8) V ( X ) − x ( X ) : X ∈ L (cid:9) is a latticeupward directed.Proof. We consider X , X ∈ L and we set A := (cid:8) V ( X ) − x ( X ) ≥ V ( X ) − x ( X ) (cid:9) ∈ A . Consider X = 1 A X + 1 A c X . Then V ( X ) − x ( X ) = 1 A (cid:0) V ( X ) − x ( X ) (cid:1) + 1 A c (cid:0) V ( X ) − x ( X ) (cid:1) = sup (cid:8) V ( X ) − x ( X ) , V ( X ) − x ( X ) (cid:9) . By this we end the proof.
Lemma 2.4.
Let x be an operator of type (2.1) and consider h ( X ) := E (cid:2) x ( X ) (cid:3) , X ∈ L. Then h is a convex form lower semi-continuous on L .Proof. Let c in R , let C = { X ∈ L, h ( X ) ≤ c } .For p < ∞ , we have to prove that C is closed in L p . L p is a Banach space,it is thus enough to consider a sequence X n in C with limit X in L p . Wecan assume that sup n || X n || p < ∞ . There is a subsequence converging to XP -a.s. It follows from the assumption iiii) on L and the Fatou property that X ∈ C .In case p = ∞ , we have to prove that C is closed for the weak* topology σ ( L ∞ , L ). The proof is inspired by the proof of Theorem 4.31 in [17] The set C being convex it is enough from the Krein Smulian Theorem ([15] TheoremV 5 7) to prove that for all r C r = C ∩ {|| X || ∞ ≤ r } is closed for the weak*topology. From Lemma A 64 in [17] it is enough to prove that C r is closed in L for the norm topology. Let X n be a sequence in C r with limit X for the L norm. There is a subsequence of X n converging P a.s. to X . Necessarily || X || ∞ ≤ r . It follows from assumption iii) on L that X belongs to L . Fromthe Fatou property it follows that X belongs to C r .We are now ready to prove Theorem 2.1. Proof.
Set h ( X ) := E (cid:2) x ( X ) (cid:3) , X ∈ L. Clearly h is a non-negative, monotone, convex, and proper (i.e., h ( X ) > −∞ and finite for some X , see [21, p. 1]) functional. By Lemma 2.4 it is alsolower semi-continuous. Thus, by application of [21, Theorem 4 and Theorem5], we have the representation h ( X ) = sup v ∈ L ∗ n v ( X ) − h ∗ ( v ) o (2.7)where h ∗ ( v ) := sup X ∈ L n v ( X ) − h ( X ) o
7s the
Fenchel transform of h and L ∗ is the set of continuous linear formson L . Note that we can restrict to v ∈ L ∗ such that h ∗ ( v ) < ∞ , and inthis case v is a non-negative linear form. Recall that we always consider theusual L p -norm topology in the case p ∈ [1 , ∞ ) and the weak* topology inthe case p = ∞ . Then we distinguish the two cases.If p ∈ [1 , ∞ ), the classical Hahn-Banach theorem guarantees that v ( X ), X ∈ L , can be extended to a non negative continuous linear form v ( X ), X ∈ L p ( B ), and the extension admits the representation v ( X ) = E h f X i , X ∈ L p ( B ) , for some f ∈ L q ( B ) with q = p ( p − − and f ≥ p = ∞ , then we refer to a version of the Hahn-Banach theorem for locallyconvex topological spaces as in [10, Chapter II] and we proceed as follows.Recall that the weak* topology on L ∞ ( B ), defined by the family of semi-norms p g ( · ) := E h g · i , for every g ∈ L ( B ) : g ≥ , is locally convex. For every non-negative linear form v on L , continuous forthe weak* topology, there is a semi-norm p g such that | v ( X ) | ≤ p g ( X ) . Hence, applying the above mentioned corollary, we can extend v to a non-negative weak* continuous linear form on L ∞ ( B ). The extension admits therepresentation v ( X ) = E h f X i , X ∈ L ∞ ( B ) , for some f ∈ L ( B ) such that f ≥ p ∈ [1 , ∞ ], the convex functional h ( X ), X ∈ L , in (2.7) canbe rewritten as: h ( X ) = sup f ∈ L q ( B ): f ≥ (cid:8) E [ f X ] − h ∗ ( E [ f · ]) (cid:9) = sup f ∈W (cid:8) E [ f X ] − h ∗ ( E [ f · ]) (cid:9) (2.8)where W := (cid:8) f ∈ L q ( B ) : f ≥ , h ∗ ( E [ f · ]) < ∞ (cid:9) . (2.9)Note that W 6 = ∅ , because h is real valued.We remark immediately that E [ f |A ] = 1, for every f ∈ W . Indeed, considerany A ∈ A and X = 1 A . For any α ∈ R we have E [ f α A ] − h ∗ ( E [ f · ]) ≤ h ( α A ) = E [ x ( α A )] = αE [1 A ] . Hence, α (cid:0) E [ f A ] − P ( A ) (cid:1) ≤ h ∗ ( E [ f · ]) < ∞ . α → ±∞ , we see that E [ f A ] = P ( A ), A ∈ A . Namely E [ f |A ] = 1.For every f ∈ W , denote V ( X ) := E [ f X |A ] , X ∈ L p ( B ) . (2.10)Hereafter we show that V ∈ V . First of all note that the operator V isnaturally non-negative, linear, and A -homogeneous. It is also continuous.Indeed for the case p ∈ [1 , ∞ ) it is immediate from the conditional H¨olderinequality.For the case p = ∞ , we consider a neighborhood of E [ f X |A ] for the weak*topology: U := (cid:8) Y ∈ L ∞ ( A ) : ∀ g i ∈ L ( A ) , i = 1 , ..., h, | E (cid:2) g i E [ f X |A (cid:3) − E (cid:2) g i Y (cid:3) | < ǫ (cid:9) Since f ≥ E [ f |A ] = 1, then g i f ∈ L ( B ), i = 1 , ..., h , and the set˜ U := (cid:8) Z ∈ L ∞ ( B ) : ∀ g i ∈ L ( A ) , i = 1 , ..., h, | E (cid:2) g i f X (cid:3) − E (cid:2) g i f Z (cid:3) | < ǫ (cid:9) is a neighborhood of X in L ∞ ( B ) in the weak* topology and for all Z ∈ ˜ U , E [ f Z |A ] ∈ U . This proves the continuity of V for the weak* topology. Thus V belongs to V .Define x ∗ ( V ) := esssup X ∈ L { V ( X ) − x ( X ) } , for V in (2.10). We show that E [ x ∗ ( V )] < ∞ . From the lattice property of Lemma 2.3, from [20, Prop VI1.1.], and the monotone convergence theorem, we have: E [ x ∗ ( V )] = sup X ∈ L (cid:8) E [ V ( X )] − E [ x ( X )] (cid:9) = sup X ∈ L (cid:8) E [ f X ] − h ( X ) (cid:9) = sup X ∈ L (cid:8) v ( X ) − h ( X ) (cid:9) = h ∗ ( v ) < ∞ . Then we can conclude that V as in (2.10) is an element of V .We are now ready to prove the representation (2.3). For every V ∈ V , define x ∗ ( V ) := esssup X ∈ L n V ( X ) − x ( X ) o . Note that from x (0) = 0, we have that x ∗ ( V ) ≥
0. For every V ∈ V and X ∈ L , we have x ∗ ( V ) ≥ V ( X ) − x ( X )or, equivalently, x ( X ) ≥ V ( X ) − x ∗ ( V ) . Thus x ( X ) ≥ esssup V ∈V n V ( X ) − x ∗ ( V ) o . (2.11)9o conclude we need to show the reverse inequality. To this aim it is enoughto show that E h x ( X ) i ≤ E h esssup V ∈V n V ( X ) − x ∗ ( V ) oi , X ∈ L. (2.12)Indeed we have: E [ x ( X )] = h ( X ) = sup f ∈W (cid:8) E [ f X ] − h ∗ ( E [ f · ]) (cid:9) = sup f ∈W (cid:8) E [ E [ f X |A ]] − h ∗ ( E [ f · ]) (cid:9) ≤ sup V ∈V (cid:8) E [ V ( X )] − E [ x ∗ ( V )] (cid:9) = E [esssup V ∈V (cid:8) V ( X ) − x ∗ ( V ) (cid:9) ]where the last equality is due to the lattice property of Lemma 2.2 and [20,Proposition VI.1.1]. We have then proved the representation (2.3).To prove the representation (2.4), we note that V ⊆ V . From equation (2.3)we have x ( X ) = esssup V ∈V n V ( X ) − x ∗ ( V ) o ≤ esssup V ∈ V n V ( X ) − x ∗ ( V ) o . From the definition of x ∗ in (2.5) we have that x ∗ ( V ) ≥ V ( X ) − x ( X ), thatis x ( X ) ≥ V ( X ) − x ∗ ( V ), for every V ∈ V and X ∈ L . So, we conclude that x ( X ) ≥ esssup V ∈ V (cid:8) V ( X ) − x ∗ ( V ) (cid:9) . By this we end the proof.
Corollary 2.5.
Let x be of type (2.1) . Then the following representationholds: x ( X ) = esssup f ∈ D n E (cid:2) f X |A (cid:3) − x ∗ ( E (cid:2) f · |A (cid:3) ) o , X ∈ L, (2.13) where D := (cid:8) f ∈ L q ( B ) : f ≥ , E [ f |A ] = 1 (cid:9) (2.14) with q = p ( p − − and x ∗ ( E (cid:2) f · |A (cid:3) ) := esssup X ∈ L n E (cid:2) f X |A (cid:3) − x ( X ) o , f ∈ D . Proof.
From [2, Theorem 1.1], for the case p ∈ [1 , ∞ ), and [9, Theorem3.3 and Proposition 3.14], for p = ∞ , we know that there is a one-to-onerelationship between V and D . Then the results follow directly from therepresentation (2.4). 10he representations of convex functionals were studied in the recent litera-ture of risk measures in the case when the σ -algebra A is trivial. In [18] therepresentation is studied for convex risk measures (i.e. convex, monotone,lower semicontinuous, and translation invariant functionals) defined on the whole L p ( B ) with p ∈ [1 , ∞ ] with values in ( −∞ , ∞ ). In [17] the study iscarried on for p = ∞ . In both cases it is crucial that the functionals aredefined on the whole space.In [4], the representation is studied for convex, monotone, order continuousfunctionals defined on Fr´echet lattices and taking values in ( −∞ , ∞ ]. Thisallows for a general setup, however the vector space L on which the mapis defined needs to be a vector lattice. This is not the case in the presentpaper. The lattice property in Lemma 2.1 and 2.2 is a lattice property inthe arrival space L p ( A ), not in L . Another important difference is that in[4] the sigma-algebra A is trivial.If A is non-trivial, then we can refer to [5] and [14] for studies on the rep-resentation of convex operators in the context of conditional risk measures(i.e. convex, monotone, lower semicontinuous, translation invariant opera-tors) defined on the whole L ∞ ( B ).In conclusion, our contribution to this area provides a representation ofconvex operators defined on a subspace L ⊆ L p ( B ) with values in L p ( A ), p ∈ [1 , ∞ ], with very mild hypothesis on L . In the sequel we consider a criterion for the existence of an extension ¯ x of theconvex operator x to the whole L p ( B ). The given x lies within two operators m and M . This extension is sandwich preserving. There is no uniqueness ofsuch sandwich preserving extension, but our approach allows for an explicitrepresentation of at least one of them, denoted ˆ x , which turns out to be themaximal.First of all we introduce the minorant as a superlinear operator: m : L p ( B ) + −→ L p ( A ) + , i.e., m ( X + Y ) ≥ m ( X ) + m ( Y ) , X, Y ∈ L p ( B ) + ,m ( λX ) = λm ( X ) , X ∈ L p ( B ) + , λ ≥ , and the majorant as a sublinear operator: M : L p ( B ) + −→ L p ( A ) + , i.e., M ( X + Y ) ≤ M ( X ) + M ( Y ) , X, Y ∈ L p ( B ) + ,M ( λX ) = λM ( X ) , X ∈ L p ( B ) + , λ ≥ . We remark that sublinearity implies M (0) = 0.11oreover, in the case p = ∞ , we say that the map M : L ∞ ( B ) + −→ L ∞ ( A ) + is regular if for every decreasing sequence ( X n ) n in L ∞ ( B ) with X n ↓ n → ∞ P -a.s, we have M ( X n ) → , n → ∞ P a.s. (3.1)
Theorem 3.1.
Fix p ∈ [1 , ∞ ] . Let x : L −→ L p ( A ) be of type (2.1) .Consider the weak A -homogeneous operators m, M : L p ( B ) + → L p ( A ) + such that m is superlinear and M is sublinear and, if p = ∞ , M is alsoregular. Assume the sandwich condition: m ( Z ) + x ( X ) ≤ M ( Y ) (3.2) ∀ X ∈ L ∀ Y, Z ∈ L p ( B ) + : Z + X ≤ Y. Then x admits an extension ˆ x (to the whole L p ( B ) ), which is convex, mono-tone, lower-semicontinuous, weak A -homogeneous and satisfying the projec-tion property such that (3.2) is preserved, i.e. m ( Z ) + ˆ x ( X ) ≤ M ( Y ) (3.3) ∀ X ∈ L p ( B ) ∀ Y, Z ∈ L p ( B ) + : Z + X ≤ Y. In particular the operator ˆ x ( X ) := esssup V ∈V S (cid:8) V ( X ) − x ∗ ( V ) (cid:9) , X ∈ L p ( B ) , (3.4) with x ∗ ( V ) := esssup X ∈ L (cid:8) V ( X ) − x ( X ) (cid:9) , V ∈ V S , (3.5) is a sandwich preserving extension of x . Here above V S is the set of lin-ear, continuous, non-negative, A -homogeneous operators on L p ( B ) such that E [ x ∗ ( V )] < ∞ , and satisfying the sandwich condition: m ( X ) ≤ V ( X ) ≤ M ( X ) , X ∈ L p ( B ) + . Moreover, for any other such extension ¯ x , we have that ˆ x ( X ) ≥ ¯ x ( X ) , X ∈ L p ( B ) . We call ˆ x the maximal extension.Proof. From Theorem 2.1, for all X ∈ L , x ( X ) = esssup V ∈V (cid:8) V ( X ) − x ∗ ( V ) (cid:9) . Thus ∀ V ∈ V , the restriction of V to L satisfies: m ( Z ) + V ( X ) − x ∗ ( V ) ≤ M ( Y ) ∀ X ∈ L ∀ Y, Z ∈ L p ( B ) + : Z + X ≤ Y. α > αm ( Z ) + αV ( X ) − x ∗ ( V ) ≤ αM ( Y )Let A = { m ( Z ) + V ( X ) − M ( Y ) ≥ } ,0 ≤ E (1 A ( m ( Z ) + V ( X ) − M ( Y )) ≤ α E (1 A x ∗ ( V )) < ∞ . Let α → ∞ , it follows that 1 A ( m ( Z ) + V ( X ) − M ( Y )) = 0 P a.s. . Thus m ( Z ) + V ( X ) ≤ M ( Y ) (3.6)for all V ∈ V and ∀ X ∈ L, ∀ Y, Z ∈ L p ( B ) + : Z + X ≤ Y . From the sand-wich extension theorem for linear operators, [9, Proposition 3.11] in caseof L p spaces 1 ≤ p < ∞ , and [9, Theorem 3.9] in case of L ∞ spaces (seealso [2, Theorem 5.1]), every V ∈ V restricted to L , admits a sandwich pre-serving linear extension to the whole L p ( B ) denoted V S which is monotone,lower semi continuous, weak A -homogeneous, and satisfying the sandwichcondition: m ( Z ) + V S ( X ) ≤ M ( Y ) ∀ X ∈ L p ( B ) ∀ Y, Z ∈ L p ( B ) + : Z + X ≤ Y. Define ˆ x ( X ) := esssup V ∈V S (cid:8) V ( X ) − x ∗ ( V ) (cid:9) , X ∈ L p ( B ) , where V S is the set described in the statement of the theorem, and x ∗ ( V ) =esssup Y ∈ L ( V ( Y ) − x ( Y )). It follows that ˆ x extends x and it is lower semicontinuous, convex, monotone, weak A -homogeneous and it also satisfiesthe projection property. It remains to verify that ˆ x satisfies the sandwichcondition. Let Y, Z ∈ L p ( B ) + ∀ X ∈ L p ( B ) : Z + X ≤ Y , m ( Z ) + ˆ x ( X ) = esssup V ∈V S (cid:8) m ( Z ) + V ( X ) − x ∗ ( V ) (cid:9) ≤ M ( Y ) + esssup V ∈V S ( − x ∗ ( V ))= M ( Y ) + esssup V ∈V ( − x ∗ ( V )) = M ( Y ) . (3.7)Now consider any other convex, monotone, lower-semicontinuous, weak A -homogeneous extension ¯ x satisfying the sandwich condition. From Theorem2.1 we have that¯ x ( X ) = esssup V ∈V ¯ x (cid:8) V ( X ) − ¯ x ∗ ( V ) (cid:9) , X ∈ L p ( B ) , with ¯ x ∗ ( V ) = esssup X ∈ L p ( B ) (cid:8) V ( X ) − ¯ x ( X ) (cid:9) V ¯ x is given by the mentioned theorem with reference to the operator¯ x . Moreover, since ¯ x satisfies the sandwich condition we can see that (3.6)holds for V ¯ x and that¯ x ( X ) = esssup V ∈V S ¯ x (cid:8) V ( X ) − ¯ x ∗ ( V ) (cid:9) , X ∈ L p ( B ) . From the definition of ¯ x ∗ and of x ∗ with ¯ x ( X ) = x ( X ), X ∈ L , we can seethat ¯ x ∗ ( V ) ≥ x ∗ ( V ) is valid for all V ∈ V . Hence E [¯ x ∗ ( V )] ≥ E [ x ∗ ( V )], V ∈ V , and in particular V S ¯ x ⊆ V S . Then ¯ x ( X ) ≤ esssup V ∈V S (cid:8) V ( X ) − ¯ x ∗ ( V ) (cid:9) . On the other hand for every V ∈ V S and X ∈ L p ( B ) we have V ( X ) − ¯ x ( X ) ≤ ¯ x ∗ ( V ), hence V ( X ) − ¯ x ∗ ( X ) ≤ ¯ x ( X ). Thus we concludeesssup V ∈V S (cid:8) V ( X ) − ¯ x ∗ ( V ) (cid:9) ≤ ¯ x ( X ) and we have proved that:¯ x ( X ) = esssup V ∈V S (cid:8) V ( X ) − ¯ x ∗ ( V ) (cid:9) . Since ¯ x ∗ ( V ) ≥ x ∗ ( V ) for all V ∈ V S ¯ x , then ¯ x ( X ) ≤ ˆ x ( X ) for all X ∈ L p ( B ). Remark 3.1.
The above extension ˆ x (3.4) satisfies the following inequality: ∀ X ∈ L p ( B ) + m ( X ) ≤ − ˆ x ( − X ) ≤ ˆ x ( X ) ≤ M ( X ) . This inequality is in fact equivalent to (3.3) for every convex, monotone,lower semi continuous, weak A -homogeneous operator defined on the whole L p ( B ) . The first assertion follows from equation (3.3) applied one time with(
Z, X, Y ) = ( X, − X,
0) and the other time with (
Z, X, Y ) = (0 , X, X ). Thesecond assertion follows from the convexity of ˆ x . Corollary 3.2.
For every V ∈ V S , the penalty (3.5) in the representation (3.4) of the extension ˆ x of the operator x , satisfies x ∗ ( V ) = ˜ x ( V ) , where ˜ x ( V ) := esssup X ∈ L p ( B ) (cid:8) V ( X ) − ˆ x ( X ) (cid:9) . (3.8) Moreover, define V S as the set of elements in V satisfying the sandwichcondition (3.6) . Then the extension (3.4) can be rewritten as: ˆ x ( X ) = esssup V ∈ V S (cid:8) V ( X ) − ˜ x ( V ) (cid:9) , X ∈ L p ( B ) . (3.9) Furthermore, we can also give the representation: ˆ x ( X ) = esssup f ∈ D S (cid:8) E [ f X |A ] − ˜ x ( E [ f · |A ]) (cid:9) , X ∈ L p ( B ) , (3.10) with D S := n f ∈ D : m ( X ) ≤ E [ f X |A ] ≤ M ( X ) , ∀ X ∈ L p ( B ) o . (3.11) The penalty ˜ x is called minimal penalty following the terminology of riskmeasures. roof. Fix V ∈ V S . From the definition we have x ∗ ( V ) ≤ ˜ x ( V ). On theother hand, from (3.4), we have ˆ x ( X ) ≥ V ( X ) − x ∗ ( V ), for all X ∈ L p ( B ).Hence, x ∗ ( V ) ≥ V ( X ) − ˆ x ( X ), for all X ∈ L p ( B ). Thus x ∗ ( V ) ≥ ˜ x ( V ).This proves (3.8). For the proof of (3.9) we apply the same arguments asfor the proof of (2.4) in Theorem 2.1. And for the proof of (3.10) we applythe same arguments as for the proof of (2.13) in Corollary 2.5. Definition 3.1.
The operator m is non degenerate if it satisfies E ( m (1 B )) > for all B ∈ B such that P ( B ) > . Lemma 3.3.
Assume that m is non degenerate. Every f ∈ D S such that E (˜ x ( E ( f · |A )) < ∞ belongs to D e := { f ∈ D | f > P a.s. } . Proof.
Let B ∈ B such that P ( B ) >
0. It follows from the Remark 3.1 thatfor all real λ >
0, ˆ x ( − λ B ) ≤ − m ( λ B ). From the representation (3.10) ofˆ x ( − λ B ), we get ˆ x ( − λ B ) ≥ E ( − λ B f ) − ˜ x ( E ( f · |A )). It follows that forall λ > E (1 B f ) ≥ E ( m (1 B )) − E (˜ x ( E ( f. |A )) λ . Letting λ → ∞ , the resultfollows from E ( m (1 B )) >
0, being m non degenerate.We deduce the following result from Corollary 3.2 and Lemma 3.3. Corollary 3.4.
Assume that m is non degenerate, then ˆ x admits the fol-lowing representation ˆ x ( X ) = esssup f ∈ D S,e (cid:8) E [ f X |A ] − ˜ x ( E [ f · |A ]) (cid:9) , X ∈ L p ( B ) , (3.12) with D S,e := D S ∩ D e . (3.13)The following result can be regarded as an extension of [2, Theorem 5.2] tothe case of convex operators. Corollary 3.5.
If the minorant m and the majorant M in Theorem 3.1 arelinear operators: m ( X ) = E (cid:2) m X |A (cid:3) , X ∈ L p ( B ) + ,M ( X ) = E (cid:2) M X |A (cid:3) , X ∈ L p ( B ) + for some random variables m , M ∈ L q ( B ) + : q = p ( p − − such that ≤ m ≤ M . The extension (3.4) ˆ x can be written as: ˆ x ( X ) = esssup f ∈D (cid:8) E [ f X |A ] − x ∗ ( E [ f · |A ]) (cid:9) , X ∈ L p ( B ) , (3.14) where D := n f ∈ L q ( B ) : 0 ≤ m ≤ f ≤ M , E [ f |A ] = 1 o . roof. This is a direct application of Corollary 3.2.We now prove that under the sandwich condition the esssup in (3.4) isattained. This will be a consequence of the following compactness result.
Lemma 3.6.
Let M be sublinear, monotone, weak A -homogeneous, and, if p = ∞ , regular. Let K = { f ∈ D : 0 ≤ E ( f · |A ) ≤ M } . Identifying f ∈ K with the linear form E ( f · ) on L p ( B ) , K is a compact subset of the ball ofradius E ( M (1) q ) q of L ′ p ( B ) , ≤ p ≤ ∞ equiped with the weak* topology σ ( L ′ p , L p ) . In case p = ∞ , K is furthermore contained in L ( B ) .(Notice thatif p < ∞ , L ′ p = L q with q = p ( p − − .)Moreover, with the notations of Theorem 3.1, the set D S is compact for theweak* topology .Proof. K is a subset of the ball of radius E ( M (1) q ) q of L ′ p ( B ). As thisbounded ball is compact for the weak* topology (Banach Alaoglu theorem),it is enough to prove that K is closed for the weak* topology. Denote K theweak* closure of K . Let Ψ ∈ K . Ψ is a positive continuous linear form on L p ( B ).In case p ∈ [1 , ∞ ), Ψ is represented by an element of L q ( B ) for q = p ( p − − (Riesz representation theorem).We detail the case p = ∞ . We first prove that Ψ defines a measure on(Ω , B ). Let ( X n ) n be any sequence of elements of L ∞ ( B ) decreasing to 0 P a.s . From the regularity of M , ∀ ǫ >
0, there is n such that ∀ n ≥ n , E ( M ( X n )) ≤ ǫ . Denote U the neighborhood of Ψ defined as U = { φ ∈ L ′∞ ( B ) , | Ψ( X n ) − φ ( X n ) | ≤ ǫ } . Since Ψ ∈ K , there is φ ∈ U ∩ K . Forsuch φ , 0 ≤ φ ( X n ) ≤ E ( M ( X n )) ≤ ǫ . It follows that | Ψ( X n ) | ≤ ǫ . As Ψis a non negative linear functional and the sequence ( X n ) n is decreasing to0, it follows that 0 ≤ Ψ( X n ) ≤ ǫ for every n ≥ n . From Daniell StoneTheorem, see e.g. [17, Theorem A48], it follows that Ψ defines a probabilitymeasure on (Ω , B ). This probability measure is absolutely continuous withrespect to P and this gives the existence of some g ∈ L ( B ) such thatΨ = E ( g · ) (Radon Nikodym Theorem). For all X ∈ L ∞ ( A ), the equalityΨ( X ) = E ( X ) is obtained similarly making use of the neighborhood of Ψ: U X = { φ ∈ L ′∞ ( B ) , | Ψ( X ) − φ ( X ) | ≤ ǫ } . It follows that E ( g |A ) = 1. Theinequality E ( f X A ) ≤ E ( M ( X )1 A ) for X ∈ L p ( B ) and A ∈ A is obtainedsimilarly and hence Ψ = E ( g. ) where g belongs to K . This proves thecompactness of K for the weak* topology. D S is equal to { f ∈ K : E ( m ( X )1 A ) ≤ E ( f X A ) , ∀ X ∈ L p ( B ) , ∀ A ∈ A} .Thus D S is a closed subset of K for the weak* topology. Proposition 3.7.
Assume the hypothesis of Theorem 3.1. For every X ∈ L p ( B ) , there is some f X in D S (depending on X ) such that ˆ x ( X ) = E ( f X X |A ) − ˜ x ( E ( f X · |A )) . (3.15)16 roof. We start from the representation (3.10) given in Corollary 3.2:ˆ x ( X ) = esssup f ∈ D S (cid:8) E [ f X |A ] − ˜ x ( E [ f · |A ]) (cid:9) , X ∈ L p ( B ) . (3.16)From the lattice property proved in Lemma 2.2, it follows that E (ˆ x ( X )) =sup f ∈ D S [ E ( f X ) − E (˜ x ( E [ f · |A ])]). From the definition of ˜ x and the latticeproperty proved in Lemma 2.3, it follows that E (˜ x ( E [ f · |A ])) is a lower semicontinuous function of f ∈ D S for the weak* topology and thus we deducefrom the compactness of D S (see Lemma 3.6) that the upper semi continuousfunction E ( f X ) − E (˜ x ( E [ f ·|A ])), f ∈ D S , has a maximum attained for some f X (which may not be unique). From equation (3.16) it then follows that f X satifies (3.15). We equip the probability space (Ω , B , P ) with the right-continuous P - aug-mented filtration ( F t ) t ∈ [0 ,T ] . We assume that, for all t , F t is generated bya countable family of events, by which we mean that F t is the smallest σ -algebra containing the countable family and all P -null events.Let p ∈ [1 , ∞ ]. For any time t ∈ [0 , T ] ( T > L t ⊆ L p ( F t ) , L t ⊆ L T . (4.1)Let T ⊆ [0 , T ] such that 0 , T ∈ T . In the sequel we denote ( x s,t ) s,t ∈T on ( L t ) t ∈T the system of operators x s,t : L t −→ L s of the type (2.1), for s, t ∈ T : s ≤ t .In financial applications these operators represent a time-consistent systemfor ask prices in a market with friction. The time s is the price evaluationtime of an asset which has payoff at t and the prices are defined on thedomain L t of purchasable assets. Note that, in general, L t ⊂ L p ( F t ) forsome t ∈ [0 , T ] and L t = L p ( F t ) for all t ∈ [0 , T ] in a complete market. Definition 4.1.
The system ( x s,t ) s,t ∈T , is time-consistent (or T time-consistent ) if for all s, t, u ∈ T : s ≤ t ≤ ux s,u ( X ) = x s,t (cid:0) x t,u ( X ) (cid:1) , (4.2) for all X ∈ L u . Time-consistency is a natural assumption for such system of operators rep-resenting, e.g., price processes. This concept models the reasonable equiv-alence of the price evaluation for an asset with payoff at time u , say, whenthe evaluation is performed either in one step, i.e. the straight evaluation ofthe asset at time s , or in two steps, i.e. first an evaluation at time t : t ≤ u and then at s : s ≤ t ≤ u . This concept is also proper of a consistent riskmeasurements and it is studied for dynamic risk measures (where it is called strong time-consistency in [1]), see e.g. [11], [6].17 emark 4.1. For any s ≤ t ≤ T , x st is the restriction to L t of x sT . Indeed let X ∈ L t , then x tT ( X ) = X , by the projection property. Thus bytime-consistency we have x sT ( X ) = x st ( x tT ( X )) = x st ( X ), for all X ∈ L t .In the sequel we discuss extension of dynamic systems of operators which willbe sandwich preserving. We deal with systems of superlinear and sublinearoperators: each one representing the minorant and majorant of one of theoperators to be extended. Motivated by applications, a modification of theconcept of time-consistency is also necessary. Examples of studies of suchminorants and majorants are found in [2], [16], and [9]. It is in this lastpaper that the general concept of weak time-consistency is introduced for thefirst time in connection with no-good deal bounds. We are now consideringagain this general definition in this context of convex operators also in viewof upcoming applications to the study of ask prices in the context of risk-indifference pricing. Definition 4.2. • The family ( m s,t ) s,t ∈T of weak F s -homogeneous, su-perlinear operators m s,t : L p ( F t ) + → L p ( F s ) + is weak time-consistent if, for every X ∈ L p ( F t ) + , m r,s ( m s,t ( X )) ≥ m r,t ( X ) , ∀ r ≤ s ≤ t. (4.3) • The family ( M s,t ) s,t ∈T of weak F s -homogeneous, sublinear operators M s,t : L p ( F t ) + → L p ( F s ) + is weak time-consistent if, for every X ∈ L p ( F t ) + , M r,s ( M s,t ( X )) ≤ M r,t ( X ) , ∀ r ≤ s ≤ t. (4.4)Note that the operators m s,t , M s,t are not required to satisfy the projectionproperty. Definition 4.3.
We say that the family ( m s,t , M s,t ) s,t ∈T satisfies the mM1-condition if they are weak time-consistent families of superlinear, respectivelysublinear, weak F s -homogeneous operators such that m s,t , M s,t : L p ( F t ) + −→ L p ( F s ) + , m ,T is non degenerate, and M s,t is also regular if p = ∞ . Definition 4.4.
We say that the system of operators ( x s,t ) s,t ∈T satisfies the sandwich condition when m s,t ( Z ) + x s,t ( X ) ≤ M s,t ( Y ) (4.5) ∀ X ∈ L t ∀ Y, Z ∈ L p ( F t ) + : Z + X ≤ Y, for some families of operators ( m s,t ) s,t ∈T and ( M s,t ) s,t ∈T with m s,t , M s,t : L p ( F t ) + −→ L p ( F s ) + . .1 Finite discrete time systems First of all we consider a finite set T := { s , ..., s K : 0 = s ≤ ... ≤ s K } .For s ≤ t , denoted D Ss,t the set (3.11) corresponding to A = F s , B = F t , andto the minorant m s,t and majorant M s,t . Analogously for D S,es,t := D Ss,t ∩ D e ,cf. (3.13). Proposition 4.1.
Let us consider the time-consistent system (cid:0) x s,t (cid:1) s,t ∈T on ( L t ) t ∈T satisfying the sandwich condition (4.5) with ( m s,t , M s,t ) s,t ∈T fulfill-ing mM1. For any i < j , consider the operators: ˆ x s i ,s j ( X ) := esssup f ∈Q i,j (cid:8) E [ f X |F s i ] − α s i ,s j ( f ) (cid:9) , X ∈ L p ( F s j ) , (4.6) with the penalty α s i ,s j ( f ) := l = j − X l = i E [ α s l ,s l +1 ( g l +1 ) |F i ] (4.7) where α s l ,s l +1 ( g l +1 ) := esssup X ∈ L sl +1 n E [ g l +1 X |F s l ] − x s l ,s l +1 ( X ) o and Q i,j := { f ∈ L q ( F s j ) + : f = Π j − l = i g l +1 , g l +1 ∈ D S,es l ,s l +1 } with q = p ( p − − . For all s ≤ t in T , the operator ˆ x s,t extends x s,t on L p ( F t ) . This family of operators (cid:0) ˆ x s,t (cid:1) s,t ∈T is a time-consistent sandwichpreserving extension. Moreover (cid:0) ˆ x s,t (cid:1) s,t ∈T is maximal, in the sense that, if (cid:0) ¯ x s,t (cid:1) s,t ∈T is another such family we have that: for all i < j , ˆ x s i ,s j ( X ) ≥ ¯ x s i ,s j ( X ) , X ∈ L p ( F s j ) . Note that from Corollary 3.2, α s l ,s l +1 ( g l +1 ) = ˜ x s l ,s l +1 (cid:0) E [ g l +1 · |F s l ] (cid:1) , where˜ x s s ,s l +1 is the minimal penalty, see (3.8). Proof.
From Theorem 3.1, for every i ≤ K −
1, we consider the maximalextension ˆ x s i ,s i +1 of x s i ,s i +1 . The operator m ,T is non degenerate. It followsfrom the weak time-consistency of ( m s,t ) s,t ∈T that for all 0 ≤ s ≤ t ≤ T , theoperator m s,t is also non degenerate. From Corollary 3.4, ˆ x s i ,s i +1 admits arepresentationˆ x s i ,s i +1 ( X ) = esssup g ∈ D S,esi,si +1 (cid:8) E [ gX |F s i ] − α s i ,s i +1 ( g ) (cid:9) (4.8) D S,es i ,s i +1 = { g ∈ L q ( F s i +1 ) + : E [ g |F s i ] = 1 , g > P a.s.m s i ,s i +1 ( X ) ≤ E [ gX |F s i ] ≤ M s i ,s i +1 ( X ) , ∀ X ∈ L p ( F s i +1 ) + } α s i ,s i +1 ( g ) := ˜ x s i ,s i +1 ( E [ g · |F s i ])= esssup Y ∈ L si +1 (cid:8) E [ gY |F s i ] − x s i s i +1 ( Y ) (cid:9) . (4.9)For any i < j defineˆ x s i ,s j ( X ) := esssup f ∈Q i,j (cid:8) E [ f X |F s i ] − α s i ,s j ( f ) o with the penalty α s i ,s j ( f ) := l = j − X l = i E [ α s l ,s l +1 ( g l +1 ) |F s i ] , for f = g i +1 g i +2 · · · g j and Q i,j as in the statement. Note that for any f ∈ Q i,j and any set A ∈ F s i we have Q ( A ) := E [ f A ] = P ( A ). We remarkthat the penalties ( α i,j ) i
20e proceed then by induction on h such that j = i + h . Let i < l < j ˆ x s i ,s j ( X ) =ˆ x s i ,s l (ˆ x s l ,s j ( X )) ≥ ¯ x s i ,s l (ˆ x s l ,s j ( X )) ≥ ¯ x s i ,s l (¯ x s l ,s j ( X )) = ¯ x s i ,s j ( X ) , X ∈ L p ( B ) . By this we end the proof.
Corollary 4.2.
For each X ∈ L p ( F s j ) , there exists f X in Q i,j such that ˆ x s i ,s j ( X ) = E (cid:2) f X X |F s i (cid:3) − α s i ,s j ( f X ) . Proof.
For i = j − x s j − ,s j ( X ) = E (cid:2) f X,j X |F s j − (cid:3) − α s j − ,s j ( f X,j ) . From Lemma 3.3, f X,j belongs to D S,es i ,s i +1 . From the time-consistency of (cid:0) ˆ x s,t (cid:1) s,t ∈T and the definition of α s i ,s j in (4.7) we have f X = j − Y l = i f X,l +1 . By this we end the proof.
Let us now consider a countable set
T ⊂ [0 , T ], with 0 , T ∈ T , and asequence of finite sets ( T n ) ∞ n =1 : T n ⊆ T n +1 , such that T = ∪ ∞ n =1 T n . Letus consider the time-consistent system (cid:0) x s,t (cid:1) s,t ∈T on ( L t ) t ∈T satisfying thesandwich condition (4.5) with ( m s,t , M s,t ) s,t ∈T fulfilling mM1. Lemma 4.3.
For any n , let ( x ns,t ) s,t ∈T n be the maximal time-consistent sand-wich preserving extensions of ( x s,t ) s,t ∈T n . Now consider s, t, ∈ T . Let n ∈ N such that s, t ∈ T n . Then, for any n > n and X ∈ L p ( F t ) , the sequence ( x ns,t ( X )) n>n is non increasing P a.s.
Hence it admits a limit ˆ x s,t ( X ) := lim n →∞ x ns,t ( X ) . (4.12) Moreover, for n > n , let α ns,t be the minimal penalty associated to x ns,t . Thispenalty has representation α ns,t ( Q ) := esssup X ∈ L p ( F s ) ( E Q ( X |F s ) − x ns,t ( X )) , (4.13) for all probabilility measure Q ∼ P , where α ns,t ( Q ) = α ns,t ( f ) with f = dQdP .Then, for all Q ∼ P , the sequence ( α ns,t ( Q )) n>n is non negative and nondecreasing P a.s. . Hence it admits a limit ˆ α s,t ( Q ) := lim n →∞ α ns,t ( Q ) . (4.14)21 roof. The extensions ( x ns,t ) s,t ∈T n are maximal over all other sandwich pre-serving extensions time-consistent on T n . Then, for s, t, ∈ T and n > n , wecan regard the extension x n +1 s,t as another sandwich preserving extension of x s,t , ( x n +1 s,t ) s,t ∈T n is time-consistent on T n . Thus x ns,t ( X ) ≥ x n +1 s,t ( X ).From Corollary 4.2, x ns,t admits a representation with equivalent probabilitymeasures. The result for α ns,t ( Q ), Q ∼ P , is then an immediate consequenceof equation (4.13). Theorem 4.4.
Let us consider the discrete time-consistent system (cid:0) x s,t (cid:1) s,t ∈T on ( L t ) t ∈T satisfying the sandwich condition (4.5) with mM1. Then eachoperator in this family admits an extension to the whole L p ( F t ) with valuesin L p ( F s ) satisfying the sandwich condition and such that the family of ex-tensions is time-consistent. In particular, the family of operators (ˆ x s,t ) s,t ∈T given in Lemma 4.3 is a time-consistent and sandwich preserving extensionof (cid:0) x s,t (cid:1) s,t ∈T . Moreover, for any s ≤ t , the operators ˆ x s,t (4.12) and ˆ α s,t (4.14) satisfy the relationship: ˆ x s,t ( X ) = esssup Q ∼ P ( E Q [ X |F s ] − ˆ α s,t ( Q ))= esssup f ∈ D S,es,t ( E [ f X |F s ] − ˆ α s,t ( f )) , X ∈ L p ( F t ) . (4.15) Moreover, for all X there is f X ∈ D S,es,t such that ˆ x s,t ( X ) = E ( f X X |F s ) − ˆ α s,t ( f X ) . (4.16) This extension is maximal, in the sense that, for any other such extension (cid:0) ¯ x s,t (cid:1) s,t ∈T we have that: for all s < t ∈ T , ˆ x s,t ( X ) ≥ ¯ x s,t ( X ) , X ∈ L p ( F t ) . Also for all s, t ∈ T , ˆ α s,t is the minimal penalty associated to ˆ x s,t .Proof. In Lemma 4.3 we have defined, for all s, t ∈ T ,ˆ x s,t ( X ) := lim k →∞ x ks,t ( X ) , X ∈ L p ( F t ) , where x ks,t is the maximal extension of x s,t on T k and for f ∈ D S,es,t with s, t ∈T k , we have set ˆ α s,t ( f ) := ˆ α s,t ( Q ), α ks,t ( f ) = α ks,t ( Q ), where f = dQ/dP ,with ˆ α s,t ( f ) := lim k →∞ α ks,t ( f ) , f ∈ D S,es,t
Step 1: Proof of the representations (4.15) , (4.16) and the sandwich property. Let s, t ∈ T . Fix X ∈ L p ( F t ). For every k , from Corollary 4.2, there is f X,k ∈ D S,es,t such that x ks,t ( X ) = E ( f X,k X |F s ) − α ks,t ( f X,k ) . (4.17)22rom Lemma 3.6 the set D Ss,t is compact for the weak* topology, thus there isa subsequence of ( f X,k ) k converging to f X ∈ D Ss,t . Without loss of generalitywe can assume that the sequence ( f X,k ) k itself has the limit f X (for theweak* topology). Fix n > n . From equation (4.13), α ns,t is lower semicontinuous for the weak* topology thus α ns,t ( f X ) ≤ lim inf k →∞ α ns,t ( f X,k ) . From Lemma 3.3 it follows that f X ∈ D S,es,t . From Lemma 4.3, for given k , the sequence ( α ns,t ( f X,k )) n is non decreasing. Therefore for every k ≥ n , α ns,t ( f X,k ) ≤ α ks,t ( f X,k ). Thus by (4.17), α ns,t ( f X ) ≤ lim inf k →∞ (cid:16) E ( f X,k X |F s )) − x ks,t ( X ) (cid:17) . Passing to the limit as k → ∞ , we get the inequality α ns,t ( f X ) ≤ E ( f X X |F s )) − ˆ x s,t ( X ) . Letting n → ∞ ˆ α s,t ( f X ) ≤ E ( f X X |F s )) − ˆ x s,t ( X ) . (4.18)On the other hand, for every Q ∼ P , and Y ∈ L p ( F t ), for every n , α ns,t ( Q ) ≥ E Q ( Y |F s )) − x ns,t ( Y ). Passing to the limit this gives:ˆ α s,t ( Q ) ≥ E Q ( Y |F s )) − ˆ x s,t ( Y ) . (4.19)It follows that ˆ x s,t ( X ) = E ( f X X |F s )) − ˆ α s,t ( f X ) . Then from the above equation and (4.19) we have proved the representations(4.15) and (4.16).Notice that the sandwich condition follows from the sandwich condition for x ns,t passing to the limit for n → ∞ , see (4.12). Step 2: Time-consistency.
From (4.15), ˆ x s,t is lower semi continuous. From the definition of ˆ x s,t as thelimit of x ns,t it follows that ˆ x s,t extends x s,t for every s, t ∈ T . On the otherhand for every r ≤ s ≤ t in T and every n large enough such that r, s, t be-long to T n , we already know that ( x ns,t ) s,t ∈T is T n time-consistent. We recallthat the minimal penalty of a time-consistent family of operators satisfiesthe local property [6, Lemma 2.3] and the cocycle condition [6, Theorem2.5]. The family of penalties ( α ns,t ) s,t ∈T satisfies the cocycle condition: α nr,t ( Q ) = α nr,s ( Q ) + E Q ( α ns,t ( Q ) |F r ) , Q ∼ P. Hence, passing to the limit for the non decreasing sequence ( α ns,t ) s,t ∈T weget the cocycle condition:ˆ α r,t ( Q ) = ˆ α r,s ( Q ) + E Q ( ˆ α s,t ( Q ) |F r ) , Q ∼ P. x s,t ) s,t ∈T is time-consistent. Step 3: Maximality of ˆ x s,t . Notice that if another family x s,t satisfies all the above properties. Necessar-ily for all s, t ∈ T and n large enough such that s, t ∈ T n , from the maximalproperty of x ns,t it follows that x s,t ( X ) ≤ x ns,t ( X ). Thus passing to the limitwe get the maximality for ˆ x s,t . Step 4: Minimality of ˆ α s,t . To see that ˆ α s,t is the minimal penalty associated to ˆ x s,t , we proceed as fol-lows. From Lemma 4.3, the sequence ( x ns,t ( X )) n>n is non increasing P a.s. .We already know that α ns,t is the minimal penalty associated to ( x ns,t ( X )).Thus α nst ( Q ) = esssup X ∈ L p ( F t ) (cid:0) E Q ( X |F s ) − x ns,t ( X ) (cid:1) ≤ esssup X ∈ L p ( F t ) (cid:0) E Q ( X |F s ) − ˆ x s,t ( X ) (cid:1) . Passing to the limit, we haveˆ α st ( Q ) ≤ esssup X ∈ L p ( F t ) (cid:0) E Q ( X |F s ) − ˆ x s,t ( X ) (cid:1) . On the other hand, for all X we have α nst ( Q ) ≥ E Q ( X |F s ) − x ns,t ( X ) . Passing to the limit, we haveˆ α st ( Q ) ≥ E Q ( X |F s ) − ˆ x s,t ( X ) , ∀ X ∈ L p ( F t ) . Hence ˆ α st ( Q ) = esssup X ∈ L p ( F t ) (cid:0) E Q ( X |F s ) − ˆ x s,t ( X ) (cid:1) . In this section we study sandwich preserving extensions for a system ofoperators ( x s,t ) s,t ∈ [0 ,T ] . These extensions are time-consistent. We stressthat to obtain a time-consistent extension it is not enough to collect all theextensions of single operators in one family. Time-consistency is achievedwith some careful procedure of extension involving the representation of theoperators and an appropriate passage from discrete to continuous time. Forthis we first define the system of majorant and minorant operators servingas bounds in the sandwiches. Definition 5.1.
We say that the family ( m s,t , M s,t ) s,t ∈ [0 ,T ] satisfies themM2-condition if . mM1 is satisfied (Definition 4.3);2. esssup s ≤ T ( M s,T ( X )) belongs to L p ( F T ) + for all X ∈ L p ( F T ) + ;3. for every X ∈ L p ( F t ) + , m s,t ( X ) = lim t ′ >t,t ′ ↓ t m st ′ ( X ); (5.1)
4. for every X ∈ L p ( F t ) + , M s,t ( X ) = lim t ′ >t,t ′ ↓ t M st ′ ( X ); (5.2)
5. for every X ∈ L p ( F t ) + , m s,t ( X ) ≤ lim sup s ′ >s,s ′ ↓ s m s ′ t ( X ); M s,t ( X ) ≥ lim inf s ′ >s,s ′ ↓ s M s ′ t ( X ); (5.3)Let T be a countable dense subset of [0 , T ] containing 0 and T . Definition 5.2.
A system (cid:0) x s,t (cid:1) s,t ∈ [0 ,T ] on ( L t ) t ∈ [0 ,T ] is right-continuousif for all t , all X ∈ L t , and all sequences ( s n ) n , s < s n ≤ t , s n ↓ s , x s,t ( X ) = lim n →∞ x s n ,t ( X ) , where the convergence is P a.s. Lemma 5.1.
Assume mM2 condition. Let (cid:0) ˆ x s,t (cid:1) s,t ∈T and (cid:0) ˆ α s,t (cid:1) s,t ∈T be asin Lemma 4.3. There is a probability measure Q equivalent to P such thatfor all s, t ∈ T , ≤ s ≤ t ≤ T , ˆ α s,t ( Q ) = 0 .Proof. From Theorem 4.4, there is a probability measure Q such that 0 =ˆ x ,T (0) = − ˆ α ,T ( Q ). From Lemma 3.3, Q is equivalent to P . It followsfrom the T -cocycle condition and the non negativity of the penalty thatˆ α s,t ( Q ) = 0 for all s ≤ t in T . Proposition 5.2.
The notations are those of Lemma 5.1.1. For all X ∈ L p ( F T ) , (cid:0) ˆ x s,T ( X ) (cid:1) s ∈T is a Q -supermartingale.2. For every sequence ( s n ) n in T decreasing to s ∈ T , E Q (ˆ x s n ,T ( X )) hasthe limit E Q (ˆ x s,T ( X )) , for all X ∈ L p ( F T ) .Proof.
1. It follows easily from Lemma 5.1, the T time-consistency, andthe representation of ˆ x s,T ( X ) (4.15).25. The proof is inspired by the one of Lemma 4 in [8]. The main differ-ences are the facts that here ˆ x s,t is only defined for s, t in T and thatthe operator is defined on L ⊂ L p ( F t ) 1 ≤ p ≤ ∞ , while in [8] the dy-namic risk measure ρ s,t was defined on L ∞ ( F t ) and time-consistencywas considered for all real indexes.Let X ∈ L p ( F T ) and s ∈ T . From Theorem 4.4, there is an f X ∈ D S,es,T such that ˆ x s,T ( X ) = E ( f X X |F s ) − ˆ α s,T ( f X ). Let R X be the probabil-ity measure such that dR X dP = f X . It follows from the cocycle conditionthat ˆ x s,T ( X ) = E R X ( X |F s ) − ˆ α s,T ( R X )= E R X [ E R X ( X |F s n ) − ˆ α s n ,T ( R X )] |F s ) − ˆ α ss n ( R X ) . (5.4)Furthermore E R X ( X |F s n ) − ˆ α s n ,T ( R X ) ≤ ˆ x s n ,T ( X ) and the penaltiesare non negative. Hence, we have thatˆ x s,T ( X ) ≤ E R X (ˆ x s n ,T ( X ) |F s ) (5.5)andˆ x s,T ( X ) ≤ E R X (ˆ x s n ,T ( X ) |F s ) = E ( f X ˆ x s n ,T ( X ) |F s )= E ( E ( f X |F s n )ˆ x s n ,T ( X ) |F s ) . Let g ∈ L q ( F s ) be the Radon Nykodym derivative of the restrictionof Q to F s . Taking the Q expectation, g being F s -measurable, weobtain E Q (ˆ x s,T ( X )) ≤ E [ gE ( f X |F s n )ˆ x s n ,T ( X )] (5.6)= E ( g (ˆ x s n ,T ( X )) + E [ˆ x s n ,T ( X )( g ( E ( f X |F s n ) − f X belongs to D Ss,T and g belongs to D S ,s , so gf X belongsto L q ( F T ) and g ( E ( f X |F s n ) −
1) has limit 0 in L q ( F T ).For all X , | ˆ x s n T ( X ) | ≤ ˆ x s n ,T ( | X | ) ≤ M s n ,T ( | X | ). From property in Definition 5.1, sup n E (( | ˆ x s n ,T ( X ) | p ) < ∞ . It follows from H¨olderinequality that ǫ n ( X ) := E [ˆ x s n ,T ( X )( g ( E ( f X |F s n ) − δ n ( X ) := − E Q (ˆ x s n ,T ( X )) + E ( g ˆ x s n ,T ( X ))= − E (cid:0) [ E ( dQ dP |F s n ) + E ( dQ dP |F s ) (cid:3) ˆ x s n ,T ( X ) (cid:1) has limit 0. Observe that, from the Q -supermartingale property(point ), if follows that E Q (ˆ x s n ,T ( X )) ≤ E Q (ˆ x s,T ( X )) . E Q (ˆ x s,T ( X )) − ǫ n ( X ) − δ n ( X ) ≤ E Q (ˆ x s n ,T ( X )) ≤ E Q (ˆ x s,T ( X )) . This proves the result.
Theorem 5.3.
Let us consider a right-continuous time-consistent systemof operators (cid:0) x s,t (cid:1) s,t ∈ [0 ,T ] of type (2.1) defined on ( L t ) t ∈ [0 ,T ] satisfying thesandwich condition with mM2.Then there is a right-continuous, time-consistent, sandwich preserving ex-tension (cid:0) ˆ x s,t (cid:1) s,t ∈ [0 ,T ] defined on the whole ( L p ( F t )) t ∈ [0 ,T ] . One such exten-sion can be represented as ˆ x s,t ( X ) = esssup R ∈R [ E R ( X |F s ) − ˆ α s,t ( R )] , X ∈ L p ( F t ) , (5.7) with R := { R ∼ P : ˆ α ,T ( R ) < ∞} (5.8) and ˆ α ,T is the minimal penalty associated to ˆ x ,T as in Theorem 4.4. Alsofor any X ∈ L p ( F t ) , there exists R X ∈ R such that ˆ x s,t ( X ) = E R X ( X |F s ) − ˆ α s,t ( R X ) ∀ s ≤ t. Furthermore for all t > , and all X ∈ L p ( F t ) , ˆ x s,t ( X ) ≤ s ≤ t admits a c`adl`agversion.Proof. Step 1: Definition of the extension ˆ x s,t for indices in the whole [0 , T ] . Let T be a countable dense subset of [0 , T ] containing 0 and T . Let (cid:0) ˆ x s,t (cid:1) s,t ∈T be the time-consistent extension of (cid:0) x s,t (cid:1) s,t ∈T constructed in Section 4.2.Let X ∈ L p ( F t ). From Proposition 5.2 and from the Modification The-orem (Chapter VI Section 1 in [13]) applied to the Q -supermartingale(ˆ x s,T ( X )) s ∈T , it follows that (ˆ x s,T ( X )) s ∈T admits a modification which canbe extended into a c`adl`ag process (ˆ x s,T ( X )) s ∈ [0 ,T ] defined for all s ∈ [0 , T ].Notice that from Remark 4.1, x s,t coincides on L t with the restriction of x s,T . For 0 ≤ s ≤ t ≤ T , we define ˆ x s,t to be the restriction of ˆ x s,T on L p ( F t ). It follows that ˆ x s,t is an extension of x s,t to the whole L p ( F t ). If s, t ∈ T this extension coincides also with the construction of ˆ x st given inthe previous section. To see this, consider the T time-consistency of ˆ x st (ofthe previous section) and its projection property. Step 2: Extension of the penalty on a set R of probability measures andright-continuity. Consider the penalties ( ˆ α s,t ) s,t ∈T associated to (ˆ x s,t ) s,t ∈T given in Section4.2. Define R := { R ∼ P : ˆ α ,T ( R ) < ∞} X in L p ( F T ). From Theorem 4.4 (see (4.16)), there is a probabilitymeasure R X ∼ P such thatˆ x ,T ( X ) = E R X ( X ) − ˆ α ,T ( R X ) . (5.9)Thus R is non empty.Let s ∈ [0 , T ] and consider ( s n ) n ⊂ T , s n ↓ s . For every probability measure R ∈ R , the sequence E R ( ˆ α s n ,T ( R ) |F s ) is increasing. Thus it admits a limit.From Section 4.2 we can see that, if s belongs to T , this limit is equalto ˆ α sT ( R ). Indeed it is enough to exploit the representation as minimalpenalty. Then by right-continuity of the filtration and the c`adl`ag extensionof Step 1 we haveˆ α s,T ( R ) ≥ lim n →∞ E R [ ˆ α s n ,T ( R ) |F s ] ≥ E R [esssup X ∈ L p ( F T ) lim n →∞ ( E R [ X |F s n ] − ˆ x s n ,T ( R ) |F s ] = ˆ α s,T ( R ) . If s / ∈ T we can define ˆ α sT ( R ) as the limit of E R ( ˆ α s n ,T ( R ) |F s ) < ∞ R a.s. .Moreover, for r, s ∈ T , due to the T time-consistency proved in Section 4.2,ˆ α r,s ( R ) satisfies the cocycle condition on T . Note that sup s ≤ T E R ( ˆ α sT ( R )) =ˆ α T ( R ) < ∞ . Then we can define for all 0 ≤ r ≤ s ≤ T ˆ α r,s ( R ) := ˆ α rT ( R ) − E R ( ˆ α s,T ( R ) |F r ) (5.10)Thus ˆ α r,s ( R ) is now defined for all indices in [0 , T ] and all R ∈ R . More-over ˆ α r,s ( R ) satisfies the cocycle condition. It follows also from the right-continuity of ˆ α s,T ( R ) in s and from the cocycle condition that for all R ∈ R ,ˆ α r,s ( R ) is also right-continuous in s . Step 3: Representation of ˆ x r,t ( X ) . Let 0 ≤ r ≤ t ≤ T . Let X ∈ L p ( F t ). Let R X ∈ R such that (5.9) is satisfied.Now we prove that, for all r ≤ t ,ˆ x r,t ( X ) = E R X ( X |F r ) − ˆ α r,t ( R X ) . (5.11)In fact, making use of the cocycle condition and then of the T time-consistency,we get that for all r ∈ T ,ˆ x ,T ( X ) = E R X ( E R X ( X |F r ) − ˆ α r,T ( R X )) − ˆ α ,r ( R X ) ≤ E R X (ˆ x r,T ( X )) − ˆ α ,r ( R X ) ≤ ˆ x ,r (ˆ x r,T ( X )) = ˆ x ,T ( X ) . Thus every inequality in the expression above is an equality. In particularˆ x r,T ( X ) = E R X ( X |F r ) − ˆ α r,T ( R X ) , R X a.s. ∀ r ∈ T , (5.12)and ˆ x ,r (ˆ x r,T ( X )) = E R (ˆ x r,T ( X )) − ˆ α ,r ( R X ) , R X a.s. ∀ r ∈ T . (5.13)28onsider a sequence ( r n ) n ⊂ T , r n ↓ r , with r ∈ [0 , T ]. Passing to the limitin the corresponding equation to (5.12), we can see thatˆ x r,T ( X ) = E R X ( X |F r ) − ˆ α r,T ( R X ) , R X a.s. ∀ r ∈ [0 , T ] . (5.14)We have assumed that X ∈ L p ( F t ) thus ˆ x t,T ( X ) = X . It follows then from(5.14) applied with r = t , that ˆ α t,T ( R X ) = 0.Then, again from (5.14) and the cocycle condition for R X ∈ R , it followsthat, for all X ∈ L p ( F t ),ˆ x r,t ( X ) = ˆ x r,T ( X ) = E R X ( X |F r ) − ˆ α r,t ( R X ) R X a.s. ∀ r, t ∈ [0 , T ] . (5.15) Step 4: Another representation of ˆ x r,t for all ≤ r ≤ t ≤ T . Let X ∈ L p ( F t ). We will prove that ˆ x r,t ( X ) admits the following represen-tation: ˆ x r,t ( X ) =esssup R ∈R [ E R ( X |F r ) − ˆ α r,t ( R )]= E R X ( X |F r ) − ˆ α r,t ( R X ) , (5.16)where R X satisfies equation (5.9). For all r, t in T , being ˆ α r,t the minimalpenalty, we have that for all R ∈ R ,ˆ α r,t ( R ) ≥ E R ( X |F r ) − ˆ x r,t ( X ) , ∀ X ∈ L p ( F t ) . (5.17)The above equation can also be writtenˆ α r,t ( R ) ≥ E R ( X |F r ) − ˆ x r,T ( X ) , ∀ X ∈ L p ( F t ) . (5.18)Exploiting the right-continuity of the filtration and of ˆ α r,t and ˆ x r,t as dis-cussed in Steps 1 and 2, we can see that, passing to the limit in r and thento the limit in t , equation (5.18) and thus also (5.17) are satisfied for all r ≤ t in [0 , T ]. Equation (5.16) follows then from (5.15) and (5.17). Step 5 : Time consistency of ˆ x s,t . Set 0 ≤ r ≤ s ≤ t ≤ T . Let X ∈ L p ( F t ). Let R X ∈ R such that (5.9) issatisfied. From (5.16), for all Y ∈ L p ( F s ), ˆ x r,s ( Y ) ≥ E R X ( Y |F r ) − ˆ α r,s ( R X ).With Y = ˆ x s,t ( X ), making use of equation (5.15) and of the cocycle condi-tion, it follows that ˆ x r,s (ˆ x s,t ( X )) ≥ ˆ x r,t ( X ) , ∀ r ≤ s ≤ t. (5.19)On the other hand, let R Y ∈ R such thatˆ x ,T ( Y ) = E R Y ( Y ) − α ,T ( R Y ) (5.20)(cf. (5.9)). From equation (5.15), it follows that for all r, s : r ≤ s ,ˆ x r,s ( Y ) = E R Y ( Y |F r ) − ˆ α r,s ( R Y ) R Y a.s. (5.21)29e already know that (ˆ x s,t ) s,t ∈T satisfies T time consistency, and the sand-wich condition. From the right-continuity of (ˆ x s,t ( X )) s ∈ [0 ,t ] for all t ≥ s ≤ t | ˆ x s,t ( X ) | belongs to L p ( F t ) + . Let s n ∈ T such that s n ↓ s . From the right-continuityand the dominated convergence theorem, it follows that E R Y (ˆ x s,t ( X ) |F r ) =lim n →∞ E R Y (ˆ x s n ,t ( X ) |F r ). The probability measure R Y belongs to R thusfrom step 2, ˆ α r,s ( R Y ) = lim n →∞ ˆ α r,s n ( R Y ). From equation (5.21), we thenget thatˆ x r,s (ˆ x s,t ( X )) = ˆ x r,s ( Y ) = lim n →∞ [ E R Y (ˆ x s n ,t ( X ) |F r ) − ˆ α r,s n ( R Y )] . (5.22)From equation (5.16), for all n , E R Y (ˆ x s n ,t ( X ) |F r ) − ˆ α r,s n ( R Y ) ≤ ˆ x r,s n (ˆ x s n ,t ( X )) . In the case r belongs to T , applying the T time consistency, we have thatthe right-hand side of the above equation is equal to ˆ x r,t ( X ). Thus E R Y (ˆ x s n ,t ( X ) |F r ) − ˆ α r,s n ( R Y ) ≤ ˆ x r,t ( X ) . (5.23)For a general r ∈ [0 , T ], the corresponding equation to (5.23) is obtainedby right-continuity. Then from these equations together with (5.22), wededuce that ˆ x r,s (ˆ x s,t ( X )) ≤ ˆ x r,t ( X ). This, together with (5.19) gives thetime-consistency. Step 6: Sandwich and projection property.
When r, t ∈ T , it follows from Theorem 4.4 that ˆ x rt extends x rt and satisfiesthe sandwich condition. These properties extend to all r ∈ [0 , T ] makinguse of the right-continuity and condition mM2 in x r,t , m r,t and M r,t . Theyextend then to every t ∈ [0 , T ] using the right-continuity of m r,t and M r,t in t (see mM2), and the fact that x r,t is the restriction of x r,T .For all r ≤ t ≤ T , the projection property for ˆ x r,t follows from equation(5.15) Corollary 5.4.
The extension (ˆ x st ) s,t ∈ [0 ,T ] is maximal, in the sense that,for any other such extension (cid:0) ¯ x s,t (cid:1) s,t ∈ [0 ,T ] we have that: for all s < t , ˆ x s,t ( X ) ≥ ¯ x s,t ( X ) , X ∈ L p ( F t ) . Furthermore for all s, t ∈ [0 , T ] , and all R ∈ R , ˆ α s,t ( R ) is the minimalpenalty associated to ˆ x s,t , i.e. ˆ α s,t ( R ) = esssup X ∈ L p ( F t ) [ E R ( X |F s ) − ˆ x s,t ( X )] . (5.24) Define for all R ∼ P ˆ α s,t ( R ) by the formula (5.24). Then (ˆ x s,t ) s,t ∈ [0 ,T ] admits the following representation where ˆ α s,t is the minimal penalty: ˆ x s,t ( X ) = esssup Q ∼ P [ E Q ( X |F s ) − ˆ α s,t ( Q )] . (5.25)30 roof. The maximality of (ˆ x st ) s,t ∈ [0 ,T ] among all the extensions satisfyingthe required properties follows from Theorem 4.4 for s, t ∈ T . For s ∈ [0 , T ],we apply right-continuity. For t ∈ [0 , T ] we apply the fact that ˆ x s,t =ˆ x s,T and also ¯ x s,t = ¯ x s,T . (see Remark 4.1 and step 1 in the proof of theTheorem).Note that ˆ x s,t ( X + Y ) = ˆ x s,t ( X ) + Y for all X ∈ L p ( F t ) and Y ∈ L p ( F s ).This property is known as L p ( F s ) -translation invariance . Thus (ˆ x st ) s,t ∈ [0 ,T ] is up to a minus sign a time-consistent dynamic risk measure. Denote β s,t the minimal penalty associated to ˆ x s,t : β s,t ( Q ) = esssup X ∈ L p ( F t ) [ E Q ( X |F s ) − ˆ x s,t ( X )] . It follows from Delbaen et al [12] appendix, that the minimal penalty β s,t ( Q )is right-continuous both in s and t , for all Q ∼ P . Note that for all R ∈ R ,ˆ α s,t ( R ) is also right continuous both in s and t (Theorem 5.3). Furthermorewe know from Section 4.2 that ˆ α s,t ( R ) is the minimal penalty for ˆ x s,t forall s, t ∈ T . This implies that for all s, t ∈ [0 , T ], and all R ∈ R , ˆ α s,t ( R ) = β s,t ( R ). Acknowledgments.
The authors thank CMA, University of Oslo, andEcole Polytechnique, CMAP, Paris for the support in providing occasionsof research visits. Also the program SEFE at CAS - Centre of AdvancedStudy at the Norwegian Academy of Science and Letters is gratefully ac-knowledged.
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