Risk assessment for uncertain cash flows: Model ambiguity, discounting ambiguity, and the role of bubbles

Transcription

1 Finance and Sochasics manuscrip No. (will be insered by he edior) Risk assessmen for uncerain cash flows: Model ambiguiy, discouning ambiguiy, and he role of bubbles Bearice Acciaio Hans Föllmer Irina Penner Received: dae / Acceped: dae Absrac We sudy he risk assessmen of uncerain cash flows in erms of dynamic convex risk measures for processes as inroduced in Cheridio, Delbaen, and Kupper [10. These risk measures ake ino accoun no only he amouns bu also he iming of a cash flow. We discuss heir robus represenaion in erms of suiably penalized probabiliy measures on he opional σ-field. This yields an explici analysis boh of model and discouning ambiguiy. We focus on supermaringale crieria for ime consisency. In paricular we show how bubbles may appear in he dynamic penalizaion, and how hey cause a breakdown of asympoic safey of he risk assessmen procedure. Keywords Dynamic convex risk measures Cash flows Discouning ambiguiy Model ambiguiy Robus represenaion Time consisency Dynamic penalizaion Asympoic safey Bubbles Cash subaddiiviy Mahemaics Subjec Classificaion (2000): 60G35, 91B30, 91B16 JEL Classificaion: D81 B. Acciaio Deparmen of Economy, Finance and Saisics, Universiy of Perugia, Via A. Pascoli 20, Perugia, Ialy bearice.acciaio@sa.unipg.i H. Föllmer Humbold-Universiä zu Berlin, Insiu für Mahemaik, Uner den Linden 6, Berlin, Germany foellmer@mah.hu-berlin.de I. Penner Humbold-Universiä zu Berlin, Insiu für Mahemaik, Uner den Linden 6, Berlin, Germany penner@mah.hu-berlin.de

2 2 Bearice Acciaio e al. 1 Inroducion The classical assessmen of an uncerain cash flow akes he sum of he discouned fuure paymens and compues is expecaion wih respec o a given probabiliy measure. Boh he probabilisic model and he discouning facors are assumed o be known. In realiy, however, one is usually confroned boh wih model uncerainy and wih uncerainy abou he ime value of money. The purpose of his paper is o deal wih his problem by using conceps and mehods from he heory of convex risk measures. In a siuaion where financial posiions are described by random variables on some probabiliy space, a convex risk measure can usually be represened as he wors expeced loss over a class of suiably penalized probabilisic models; see Arzner, Delbaen, Eber, and Heah [2, 3, Delbaen [12, 13 for he coheren case, and Föllmer and Schied [22, 23, Frielli and Rosazza Gianin [24 for he general convex case. This can be seen as a robus mehod which deals explicily wih he problem of model uncerainy. In he dynamical seing of a filered probabiliy space, he risk assessmen a a given ime should depend on he available informaion. This is specified by a dynamic risk measure, i.e., by a sequence (ρ ) of condiional convex risk measures adaped o he filraion. On he level of random variables, and under an addiional requiremen of ime consisency, he srucure of such dynamic risk measures is now well undersood; cf., e.g, [4,37,16,14,40,30,6,20,11,39,34,15,1, and references herein. There is also a growing lieraure on dynamic risk measures applied o cash flows ha are described as adaped sochasic processes on he given filered probabiliy space; cf. Arzner, Delbaen, Eber, Heah, and Ku [4, Cheridio, Delbaen, and Kupper [10, and also [35,8,9,11,25,28. In his conex, no only he amoun of a paymen maers, bu also is iming. In paricular, he risk is reduced by having posiive paymens earlier and negaive ones laer. This is expressed by he propery of cash subaddiiviy, which was inroduced by El Karoui and Ravanelli [18 in he conex of risk measures for random variables in order o accoun for discouning ambiguiy. Convex risk measures for processes have ha propery, and so hey provide a naural framework o capure boh model uncerainy and uncerainy abou he ime value of money. In his paper we sudy dynamic convex risk measures for bounded adaped processes, as inroduced in [10. Any such process can be viewed as a bounded measurable funcion on he produc space Ω = Ω T endowed wih he opional σ-field. I is hus naural o use resuls from he heory of risk measures for random variables and o apply hem on produc space. This idea already appears in [4 in a saic seing, and even earlier in Epsein and Schneider [19 in he conex of dynamic preferences; see also Maccheroni, Marinacci, and Rusichini [31. Here we use i for dynamic risk measures, and we ake a more probabilisic approach. This involves a careful sudy of absoluely coninuous probabiliy measures Q on he opional σ-field. In paricular, we derive a decomposiion Q = Q D, where Q is a locally absoluely coninuous probabiliy measure on he original space, and D is a predicable discouning process. The

3 Risk assessmen for uncerain cash flows 3 probabilisic approach has wo advanages. In he firs place, i allows us o make explici he join role of model uncerainy, as expressed by he measures Q, and of discouning uncerainy, as described by he discouning processes D, in he robus represenaion of condiional risk measures. Moreover, i is crucial for our analysis of he supermaringale aspecs of ime consisency. A key issue in he dynamical framework is ime consisency of he risk assessmen; see [4,14,16,30,10,6,20,11,15 and references herein. We characerize ime consisency by supermaringale properies of he discouned penaly and risk processes, in analogy o various resuls for random variables from [4, 14,6,20,34,7. These characerizaions allow us o apply maringale argumens o prove maximal inequaliies and convergence resuls for he risk assessmen procedure. In paricular, we show ha he appearance of a maringale componen in he Riesz decomposiion of he discouned penaly process amouns o a breakdown of asympoic safey. Such a maringale can be seen as a bubble, which appears on he op of he fundamenal penalizaion and hus causes an excessive neglec of he model under consideraion. The paper is organized as follows. In Secion 3 we clarify he probabilisic srucure of condiional convex risk measures for processes. To his end, we inroduce he appropriae produc space in Subsecion 3.1 and sae a decomposiion heorem for probabiliy measures on he opional σ-field; is proof is given in Appendix B. In Subsecion 3.2 risk measures for processes are idenified wih risk measures for random variables on he produc space. Under an assumpion of global coninuiy from above, his allows us o obain a robus represenaion of risk measures for processes in Subsecion 3.3, which involves boh model ambiguiy and discouning ambiguiy. Secion 4 characerizes ime consisency of dynamic risk measures, wih special emphasis on he corresponding supermaringale properies. We focus on he srong noion of ime consisency inroduced in [4. In Subsecion 4.1 we sae several equivalen crieria, and use hem in Subsecion 4.2 o derive he Doob and he Riesz decomposiion of he penaly processes. In Subsecion 4.3 we discuss asympoic properies such as asympoic safey and asympoic precision, and we relae hem o he appearance of bubbles in he Riesz decomposiion. Subsecion 4.4 saes a maximal inequaliy for he excess of he capial requiremen over he penalized expeced loss compued for a specific model. The coheren case is discussed in Subsecion 4.5. In Secion 5 we discuss cash subaddiiviy of risk measures for processes, and we characerize heir calibraion wih respec o some numéraire following [18. If a ime consisen dynamic risk measure is calibraed o a erm srucure specified by he prices of zero coupon bonds, hen discouning ambiguiy is compleely resolved, and we are only lef wih model ambiguiy. In Secion 6 our analysis is illusraed by some examples, including enropic risk measures and varians of Average Value a Risk for processes.

4 4 Bearice Acciaio e al. 2 Preliminaries We consider a discree-ime seing wih ime horizon T N { }. We denoe by T he se of ime poins, i.e., T := {0,..., T } if T <, and in case T = we disinguish beween he wo cases T := N 0 and T := N 0 { }. We use he noaion T := {s T s } for T. We fix a filered probabiliy space (Ω, F, (F ) T N0, P ), wih F 0 = {, Ω}, and F = F := σ( N0 F ) for T =. For T, we use he noaion L := L (Ω, F, P ), L,+ := {X L X 0}, and L := L (Ω, F T, P ). All equaliies and inequaliies beween random variables and beween ses are undersood o hold P -almos surely, unless saed oherwise. We denoe by M(P ) (resp. by M loc (P )) he se of all probabiliy measures Q on (Ω, F) which are absoluely coninuous wih respec o P (resp. locally absoluely coninuous wih respec o P in he sense ha Q P on F for each T N 0 ), and by M e (P ) (resp. by M e loc (P )) he se of all probabiliy measures on (Ω, F) which are equivalen (resp. locally equivalen) o P. Noe ha M(P ) coincides wih M loc (P ) if T <. Le R denoe he space of adaped sochasic processes X = (X ) T on (Ω, F, (F ) T, P ) such ha { X := inf x R } sup X x <. (2.1) T For T = N 0 { } we also consider he subspace { X := X R } X = lim X P -a.s.. For 0 s T, we define he projecion π,s : R R as π,s (X) r = 1 { r} X r s, r T, and use he noaion R,s := π,s (R ) and R := π,t (R ). The spaces X,s and X are defined accordingly. On he one hand, a process X R can be inerpreed as a value process, which migh model he evoluion of some financial value such as he marke value of a firm s equiy or of an invesmen porfolio. On he oher hand, X can be seen as a cumulaed cash flow, as explained in Remark 2.1 and in Example 2.2. Remark 2.1 An adaped cash flow C = (C ) T N0 yielding an uncerain amoun C L a ime induces a cumulaed cash flow X = (X ) T N0 wih X = C s. s=0

5 Risk assessmen for uncerain cash flows 5 If T <, or if T = and T N 0 C <, he process X belongs o R, and even o X, wih X := =0 C. Conversely, each process X R induces an adaped cash flow C := X := X X 1, T N 0, (2.2) where we use he convenion X 1 := 0. Example 2.2 Assume ha here is a money marke accoun (B ) T N0 of he form B = (1 + r s ) s=1 wih some adaped (or even predicable) process (r ) T N0 of nonnegaive shor raes. For a given (undiscouned) adaped cash flow ( C ) T N0 R consider he discouned cash flow C = (C ) T N0 defined by C = B 1 C. If T = and he shor raes are bounded away from zero by some consan δ > 0, hen he cumulaed discouned cash flow X wih X := =0 C belongs o R, and even o X, since C 1 δ C <. =0 Here he norm in he firs erm is he usual essenial supremum norm on random variables, and ha in he second erm is he norm on processes defined in (2.1). In he preceding example, he value X arises naurally as he limiing value of a cumulaed cash flow. More generally, for X R wih T = N 0 { }, he value X can be seen as a erminal paymen. In his way, dynamic risk measures for random variables wih infinie ime horizon as considered in [20 can be included ino our framework; cf. Remark Considering he inerpreaion in erms of cash flows, our resuls will be formulaed boh for X and for he underlying cash flow C given by (2.2). On a echnical level, however, our main focus will be on cumulaed cash flows X R. This will allow us o apply in a sraighforward manner sandard resuls for convex risk measures defined on bounded random variables. 3 Condiional risk measures A each ime he risk of a fuure cumulaive cash flow will be assessed by a condiional risk measure based on he informaion available a ha ime. The following definiion was inroduced in [10. Definiion 3.1 A map ρ : R L for T N 0 is called a condiional convex risk measure (for processes) if i saisfies he following properies for all X, Y R :

6 6 Bearice Acciaio e al. Condiional cash invariance: for all m L, ρ (X + m1 T ) = ρ (X) m; Monooniciy: ρ (X) ρ (Y ) if X Y componenwise; Condiional convexiy: for all λ L wih 0 λ 1, Normalizaion: ρ (0) = 0. ρ (λx + (1 λ)y ) λρ (X) + (1 λ)ρ (Y ); A condiional convex risk measure is called a condiional coheren risk measure (for processes) if i has in addiion he following propery for all X R : Condiional posiive homogeneiy: for all λ L wih λ 0, ρ (λx) = λρ (X). A sequence (ρ ) T N0 is called a dynamic convex risk measure (for processes) if, for each, ρ : R L is a condiional convex risk measure (for processes). Definiion 3.1 is analogous o he definiion of a risk measure for random variables given in [16. Noe, however, ha condiional cash invariance in he conex of processes akes ino accoun he iming of he cash paymen; he consequences will be discussed in more deail in Secion 5. Condiional cash invariance and convexiy could also be formulaed in erms of cash flows C as in Remark 2.1 raher han in erms of cumulaed cash flows X. Noe, however, ha monooniciy wih respec o X is sronger han monooniciy wih respec o C. This sronger condiion is naural since i reflecs he ime value of money; cf. Secion Opional filraion and predicable discouning In heir sudy of dynamic preferences for consumpion processes, Epsein and Schneider [19 derive a numerical represenaion by applying resuls from Gilboa and Schmeidler [26 on he appropriae produc space endowed wih he opional filraion. In he same spiri, Arzner e al. [4 idenify saic risk measures for processes wih risk measures for random variables on produc space. Here we exend his idea o he dynamic seing, and we focus on he probabilisic srucure of he resuling robus represenaion in erms of probabiliy measures on he opional σ-field. Consider he produc space ( Ω, F, P ) defined by Ω = Ω T, F = σ({a {} A F, T), P = P µ, where µ = (µ ) T is some adaped reference process such ha T µ = 1 and µ > 0 T, and where [ E P µ [X := E P X µ T

7 Risk assessmen for uncerain cash flows 7 for any bounded measurable funcion X on ( Ω, F). Noe ha F coincides wih he opional σ-field generaed by all adaped processes. Every adaped process can be idenified wih a random variable on ( Ω, F, P ), and in paricular we have R = L := L ( Ω, F, P ). We also inroduce he opional filraion ( F ) T on ( Ω, F) given by F = σ ( {A j {j}, A T Aj F j, j <, A F } ), T. A random variable X = (X s ) s T on ( Ω, F, P ) is F -measurable if and only if X s is F s -measurable for all s = 0,..., and X s = X s >. In paricular, R 0, = L := L ( Ω, F, P ). The se R 0,0 of all consan processes will be idenified wih R. For T = we will use he Lebesgue decomposiion of a measure Q M loc (P ) wih respec o P. Le M = (M ) N0 denoe he densiy process of Q wih respec o P. The limi M := lim M exiss P -a.s., since M is a nonnegaive P -maringale. By [38, Theorem VII.6.1 M exiss also Q-a.s., and Q admis he Lebesgue decomposiion Q[A = E P [1 A M + Q[A {M = }, A F, (3.1) ino he absoluely coninuous and he singular par wih respec o P on (Ω, F ). For a measure Q M loc (P ) we inroduce he se Γ (Q) of opional random measures γ = (γ ) T on T which are normalized wih respec o Q. More precisely, γ Γ (Q) is a nonnegaive adaped process, such ha γ = 1 T wih he addiional propery ha Q-a.s., γ = 0 Q-a.s. on {M = }, if T = N 0 { }. We also consider he following se D(Q) of predicable discouning processes: D = (D ) T D(Q) is a predicable non-increasing process wih D 0 = 1, and D = lim D Q-a.s. for T =, where and D = 0 Q-a.s. for T = N 0, D = 0 Q-a.s. on {M = } for T = N 0 { }. For T < we define D T +1 := 0.

8 8 Bearice Acciaio e al. Lemma 3.2 For any probabiliy measure Q M loc (P ), he se Γ (Q) can be idenified wih D(Q). More precisely, o each γ in Γ (Q) we can associae a process D D(Q) given by 1 D := 1 γ s, T N 0, and D := γ for T = N 0 { }. (3.2) s=0 In paricular we have D = s T γ s Q-a.s. T. (3.3) Conversely, every process D D(Q) defines an opional random measure γ Γ (Q) via γ := D D +1, T N 0, and γ := D for T = N 0 { }. (3.4) Moreover, for any pair γ Γ (Q) and D D(Q) relaed o each oher via (3.3) and (3.4), he inegraion by pars formula s T γ s X s = T D s (X s X s 1 ) Q-a.s., T, (3.5) s= holds for any X R if T < or if T = N 0, and for X X if T = N 0 { }. Proof I is obvious ha he process D defined by (3.2) belongs o D(Q) and saisfies (3.3), and ha γ defined by (3.4) belongs o Γ (Q). To prove (3.5), noe ha γ s X s = D s (X s X s 1 ) D +1 X (3.6) s=0 s=0 for all T N 0. Thus (3.5) is obvious for T <, and i also holds if T = N 0 for all X R, since X is bounded and D 0 Q-a.s.. For T = N 0 { } and for any X X, he limi D X = lim D +1 X exiss Q-a.s., since D 0 Q-a.s. on he singular par of Q wih respec o P, and so (3.5) follows again from (3.6). From now on we use he following assumpion which allows us o apply an exension resul of Parhasarahy [33 for consisen sequences of measures. This will be needed in he proof of Theorem 3.4. Assumpion 3.3 In he case T =, we assume ha for each T N 0 he σ-field F is σ-isomorphic o he Borel σ-field on some complee separable meric space, and ha n A n for any decreasing sequence (A n ) n N0 such ha A n is an aom of F n.

9 Risk assessmen for uncerain cash flows 9 We denoe by M( P ) he se of all probabiliy measures on ( Ω, F) which are absoluely coninuous wih respec o P. The nex heorem shows ha each probabiliy measure Q in M( P ) admis a decomposiion Q(dω, d) = Q(dw) γ(w, d) for some probabiliy measure Q on (Ω, F T ) and some opional random measure γ on T such ha Q M loc (P ) and γ Γ (Q). Theorem 3.4 For any probabiliy measure Q M( P ) here exis a probabiliy measure Q M loc (P ) and an opional random measure γ Γ (Q) (resp. a predicable discouning facor D D(Q)) such ha [ E Q[X = E Q γ X (3.7) T [ T = E Q D (X X 1 ), (3.8) =0 where (3.7) holds for all X R, whereas (3.8) holds for all X R if T < or if T = N 0, and only for X X if T = N 0 { }. Conversely, any Q M loc (P ) and any γ Γ (Q) (resp. any D D(Q)) define a probabiliy measure Q M( P ) such ha (3.7) and (3.8) hold. We wrie Q = Q γ = Q D o denoe he decomposiion of Q in he sense of (3.7) and (3.8). The proof is posponed o Appendix B. Remark 3.5 A coninuous ime analogue o Theorem 3.4 appears independenly in Kardaras [29, Theorem 2.1. While we make use of he Iô-Waanabe decomposiion (in discree ime, cf. Proposiion A.1) and of a measure heoreic exension, [29, Theorem 2.1 gives a direc consrucion of a discouning process and a local maringale, wihou relaing he laer o a probabiliy measure Q in he general case. 3.2 Condiional risk measures viewed on he opional filraion In he previous secion we have idenified processes in R wih random variables in L. This induces a one-o-one correspondence beween condiional risk measures for processes and condiional risk measures for random variables on he opional σ-field: Proposiion 3.6 Any condiional convex risk measure for processes ρ : R L for T N 0 defines a condiional convex risk measure for random variables ρ : L L via ρ (X) = X 0 1 {0}... X 1 1 { 1} + ρ (X)1 T, X R, (3.9)

10 10 Bearice Acciaio e al. where we use he noaion ρ (X) := ρ π,t (X) for X R. Conversely, any condiional convex risk measure on random variables ρ : L L is of he form (3.9) wih some condiional convex risk measure on processes ρ : R L. Proof Clearly, ρ defined via (3.9) is a condiional convex risk measure in he sense of [16. To see, e.g., condiional cash invariance, le m L, i.e. m = (m 0,..., m 1, m, m,...) wih m i L i for i = 0,...,. Then ρ (X + m) = ( X 0 m 0,..., X 1 m 1, ρ (X + m), ρ (X + m),...) = ρ (X) m by condiional cash invariance of ρ. To prove he converse implicaion, le ρ : L L be a condiional convex risk measure for random variables. Since A := Ω {0,..., 1} F, he local propery (cf., e.g., [16, Proposiion 2), condiional cash invariance and normalizaion of ρ imply ρ (X) = 1 A ρ (1 A X) + 1 A c ρ (1 A c X) = X 0 1 {0} X 1 1 { 1} + ρ (X1 T )1 T. Finally, i is easy o see ha ρ : R L defined by ρ (X) := ( ρ (X)) is a condiional convex risk measure for processes in he sense of Definiion 3.1. Le ρ : R L be a condiional convex risk measure for processes, and consider he corresponding accepance se A = {X R ρ (X) 0}. Then he accepance se of ρ relaed o ρ via (3.9) is given by Ā = { X L ρ (X) 0 P -a.s. } = { X R Xs 0 s = 0,..., 1, ρ (X) 0 P -a.s. } = A + (L 0,+... L 1,+ {0}...). (3.10) For each Q M( P ), he minimal penaly funcion of ρ is given by ᾱ ( Q) = Q-ess sup X Ā E Q[ X F. Due o (3.10) and Corollary B.3, his akes he form ᾱ ( Q) = α ( Q)1 T, (3.11)

11 Risk assessmen for uncerain cash flows 11 where α ( Q) denoes he minimal penaly funcion of ρ and is given by α (Q γ) = α (Q D) = Q-ess sup E Q X A [ (E Q = Q-ess sup X R [ s T s T γ s D X s F γ s D X s F ρ (X) (3.12) ). Here Q D = Q γ denoes he decomposiion of he measure Q in he sense of Theorem 3.4. Noe ha α (Q γ) is well defined Q-a.s. on {D > 0}; cf. Corollary B Robus represenaions In his secion we derive a robus represenaion of a condiional convex risk measure for processes which expresses explicily he combined role of model ambiguiy and discouning ambiguiy. Our proof will consis in combining he robus represenaion of risk measures for random variables as saed in [16, [5, [7, [30, [20, and [1, wih our Decomposiion Theorem 3.4 for measures on he opional σ-field. The following coninuiy propery was inroduced in [10, Definiion Definiion 3.7 A condiional convex risk measure ρ : R L for processes is called coninuous from above if ρ (X n ) ρ (X) P -a.s wih n for any decreasing sequence (X n ) n R and X R such ha Xs n X s P -a.s. for all s T. Theorem 3.8 A condiional convex risk measure for processes ρ is coninuous from above if and only if i admis he following robus represenaion: ρ (X) = ess sup Q Q loc ess sup γ Γ (Q) where α is defined in (3.12), (E Q [ s T γ s X s F α (Q γ) Q loc := { Q M loc (P ) Q = P on F }, ), X R, (3.13) and Γ (Q) := { γ Γ (Q) γs = 0 s < }.

12 12 Bearice Acciaio e al. Proof I is easy o check ha ρ is coninuous from above if and only if he condiional risk measure ρ defined in (3.9) is coninuous from above. By [16, Theorem 1, coninuiy from above of ρ is equivalen o he robus represenaion ( [ ρ (X) = ess sup E Q X F ᾱ ( Q) ), Q Q where Using Corollary B.3, his akes he form Q := { Q M( P ) Q = P on F }. (3.14) ρ (X) = X 0 1 {0}... X 1 1 { 1} [ (E Q + ess sup Q γ Q s T γ s D X s F α (Q γ) ) 1 T, (3.15) where D is relaed o γ via (3.2). Lemma B.5 implies ha Q γ Q if and only if Q Q loc, and γ s = µ s for s = 0,..., 1; in paricular D = s T µ s > 0. For each Q Q loc we can idenify he se {( γs D ) s T Q γ Q } wih Γ (Q), and so he represenaion (3.13) follows from (3.15) due o (3.9). Using he inegraion by pars formula (3.5) we can rewrie (3.13) as follows. Corollary 3.9 In erms of discouning facors, he represenaion (3.13) akes he following form for X R if T < or if T = N 0, and for X X if T = N 0 { }: [ ) (E Q ρ (X) = ess sup Q Q loc where ess sup D D (Q) T D s X s F α (Q D) s= D (Q) = { D D(Q) Ds = 1 s }., (3.16) Remark 3.10 In [10 Cheridio, Delbaen, and Kupper consider he cases T < and T = N 0. They work on he space R equipped wih he dual space { } A 1 := a = (a ) T a adaped, EP [ T a a 1 <, where a 1 := 0. The robus represenaion of condiional convex risk measures in [10 is formulaed in erms of he se { D 0,T := a A [ } 1 a a 1 for all T, E P a 1 ) = 1 ; T(a cf. [10, Theorem Noe ha D 0,T can be idenified wih he se M( P ). Indeed, every a D 0,T defines a densiy Z of Q M( P ) via Z µ = a a 1, T,

13 Risk assessmen for uncerain cash flows 13 and vice versa. By emphasizing M( P ) raher han D 0,T we ake a more probabilisic approach. In paricular, we exploi he decomposiion Q = Q γ = Q D of probabiliy measures in M( P ). We have [ [ [ E P X s a s F = E Q X s γ s F = E Q X s D s F s T s T s T for all T and all X R. The represenaion on he righ hand side has wo advanages. In he firs place i allows us o make explici he join role of model uncerainy, as expressed by he measures Q M loc (P ), and of discouning uncerainy, as described by he discouning processes D D(Q). Moreover, he probabilisic approach allows us o discuss he case T = in erms of a measure heoreic exension problem, and i will be crucial for our analysis of he supermaringale aspecs of ime consisency. As a special case, our represenaion (3.16) applied for T = 1 a = 0 o he process (0, X T ) wih X T L, yields he represenaion (4.5) in [18, Corollary 4.4 in he saic conex of cash subaddiive risk measures for random variables; cf. also Remark 5.3. In analogy o he proof of Theorem 3.8, he resuls in [20, Corollary 2.4, [1, Corollary 11 and [20, Lemma 3.5 ranslae ino robus represenaions in our conex which use a smaller se of measures: Corollary 3.11 A condiional convex risk measure on processes ρ is coninuous from above if and only if any of he following represenaions holds: 1. ρ is of he form (3.13), where he essenial supremum is aken over he se { Q γ } [ Q Q loc ( ), γ Γ (Q), E Q µ s α (Q γ) <. s T 2. for all Q = Q D M( P ) and X R we have ( ) 1 ρ (X) = Q-ess sup E R [ s T ξ s X s F α (R ξ) R ξ Q ( Q) D Q-a.s. on {D > 0}, where Q ( Q) := { R M( P ) R = Q F }. Moreover, if here exiss a probabiliy measure P P on ( Ω, F) such ha α ( P ) <, hen coninuiy from above is also equivalen o a represenaion of he form (3.13) as an essenial supremum over he se {Q γ Q M e loc (P ), γ Γ e (Q)}, where Γ e (Q) := { γ Γ (Q) γ > 0 P -a.s. for all T }.

14 14 Bearice Acciaio e al. 4 Supermaringale crieria for ime consisency In his secion we assume ime consisency, derive corresponding crieria in erms of supermaringales, and discuss some of he consequences, in paricular condiions for asympoic safey. 4.1 Srong ime consisency and is characerizaion A srong noion of ime consisency for risk measures for processes was inroduced and characerized in [10 and [11. Here we adop he definiion from [10, cf. [10, Definiion 4.2, Proposiion 4.4, Proposiion 4.5. Definiion 4.1 A dynamic convex risk measure for processes (ρ ) T N0 on R is called (srongly) ime consisen if for all in T such ha < T and for all X, Y R X = Y and ρ +1 (X) ρ +1 (Y ) = ρ (X) ρ (Y ). (4.1) Noe ha a dynamic risk measure for processes (ρ ) T N0 is ime consisen if and only if he corresponding dynamic convex risk measure for random variables ( ρ ) T N0 on L defined by (3.9) is ime consisen, ha is, if ρ +1 (X) ρ +1 (Y ) implies ρ (X) ρ (Y ) for all X, Y L and all T, < T. Crieria for ime consisency of risk measures for random variables were sudied inensively in he lieraure, see, e.g., [16, [30, [4, [20, [6, [7, [1 and he references herein. Using Proposiion 3.6 we can ranslae hese crieria ino our presen framework. By [20, Proposiion 4.2 applied o ρ, ime consisency (4.1) of ρ is equivalen o recursiveness, ha is ρ (X) = ρ (X 1 {} ρ +1 (X)1 T+1 ) (4.2) = X + ρ ( ρ +1 (X X )1 T+1 ). If we resric he condiional convex risk measure ρ o he space L +1, he accepance se is given by Ā,+1 := { X L ( Ω, F +1, P ) ρ (X) 0 P -a.s. } where = A,+1 + (L 0,+... L 1,+ {0}...), A,+1 := {X R,+1 ρ (X) 0}, T, < T, denoes he accepance se of he risk measure for processes ρ resriced o R,+1. The corresponding one-sep minimal penaly funcion for ρ akes he form ᾱ,+1 ( Q) := Q-ess sup X Ā,+1 E Q[ X F = α,+1 ( Q)1 T, Q M( P ),

15 Risk assessmen for uncerain cash flows 15 where he funcion α,+1 ( Q) is given for Q = Q D = Q γ M( P ) by α,+1 (Q D) = 1 D Q-ess sup X A,+1 E Q [ γ X D +1 X +1 F, T, < T, due o Corollary B.3. Noe ha he penaly funcions α (Q D) and α,+1 (Q D) are only defined Q-a.s. on {D > 0}. In he following we define for Q D M( P ) α (Q D) :=, α,+s (Q D) := Q-a.s. on {D = 0} for all, s 0, and use henceforh he convenion 0 := 0. The following resul characerizes ime consisency in erms of a spliing propery of he accepance ses and in erms of supermaringale properies of he penaly process and he dynamic risk measure. I ranslaes [20, Theorem 4.5 and [1, Theorem 20 o our presen framework. Theorem 4.2 Le (ρ ) T N0 be a dynamic convex risk measure on R such ha each ρ is coninuous from above. Then he following condiions are equivalen: (i) (ρ ) T N0 is ime consisen; (ii) A = A,+1 + A +1 for all T, < T ; (iii) for all T, < T and Q = Q D M( P ) D α (Q D) = D α,+1 (Q D) + E Q [D +1 α +1 (Q D) F Q-a.s.; (iv) for all X R, T, < T, and Q = Q D M( P ) E Q [D +1 (X +ρ +1 (X)+α +1 (Q D)) F D (X +ρ (X)+α (Q D)) Q-a.s.. Moreover, if here exiss a probabiliy measure P P on ( Ω, F) such ha α 0 ( P ) <, condiion (iv) saed only for he measures Q := { Q M e ( P ) α 0 ( Q) < } (4.3) = { Q γ Q M e loc(p ), γ Γ e (Q), α 0 (Q γ) < } already implies ime consisency, and he robus represenaion (3.13) of ρ also holds if he essenial supremum is aken only over he se Q. Proof Follows from [1, Theorem 20 and [20, Theorem 4.5 applied o ρ defined in (3.9) using Corollary B.3. Remark 4.3 Equivalence of ime consisency and (ii) of Theorem 4.2 holds wihou assuming coninuiy from above and was already proved in [10, Theorem 4.6. Characerizaions of ime consisency in erms of penaly funcions as in condiion (iii) are given in [10, Theorem 4.19, Theorem However, he laer resuls use neiher he decomposiion of Q ino a measure Q and a discouning facor D, nor he one-sep penaly funcions α,+1. The role of

16 16 Bearice Acciaio e al. α,+1 in condiion (iii) is analogous o he corresponding characerizaion of ime consisency of risk measures for random variables in [6, Theorem 2.5 and [20, Theorem 4.5. In he same way, he supermaringale characerizaion (iv) of ime consisency ranslaes he corresponding crierion from [20, Theorem 4.5 ino our presen framework. In he following we use he noaion Q α 0 := { Q D M( P ) α0 (Q D) < }. Corollary 4.4 Le (ρ ) T N0 be a ime consisen dynamic convex risk measure on R such ha each ρ is coninuous from above. Then 1. For any Q = Q D Q α 0, he discouned penaly process (D α (Q D)) T N0 is a nonnegaive Q-supermaringale. Is Doob decomposiion is given by he predicable process i.e., 1 A Q,D := D k α k,k+1 (Q D), T N 0, M Q,D k=0 := D α (Q D) + A Q,D, T N 0, (4.4) is a Q-maringale. 2. For all X R and all Q Qα 0, he process W Q,D (X) := D ρ (X X 1 T )+ is a Q-supermaringale. D s ( X s )+D α (Q D), T N 0, s=0 (4.5) Remark 4.5 In he same way as in Theorem 4.2, we can ranslae he weaker conceps of ime consisency from [40,4,39,36,34,17,1 ino our presen framework, and obain resuls analogous o [1, Theorem 31, Proposiion 33, Proposiion Riesz decomposiion of he penaly process and he appearance of bubbles The following proposiion characerizes he maringale M Q,D in he Doob decomposiion of he Q-supermaringale (D α (Q D)) T N0 from Corollary 4.4; i ranslaes [1, Proposiion 24 and [34, Proposiion ino our presen conex. Proposiion 4.6 The maringale M Q,D in (4.4) is of he form [ T 1 = E Q D k α k,k+1 (Q D) F M Q,D k=0 + N Q,D Q-a.s., T N 0,

17 Risk assessmen for uncerain cash flows 17 where N Q,D := { 0 if T < lim E Q [D s α s (Q D) F if T = Q-a.s., T N 0, s is a nonnegaive Q-maringale. Thus he Riesz decomposiion of he Q-supermaringale (D α (Q D)) ino a poenial and a maringale akes he form [ T 1 D α (Q D) = E Q D k α k,k+1 (Q D) F +N Q,D Q-a.s., T N 0. k= Proof Propery (iii) of Theorem 4.2 yields [ +s 1 D α ( Q) = E Q k= (4.6) D k α k,k+1 ( Q) F + E Q [ D +s α +s ( Q) F Q-a.s. (4.7) for all, s N 0 s.. + s T and all Q M( P ). For T < he claim is obvious, since α T ( Q) = 0 P -a.s.. For T =, by monooniciy here exiss he limi [ s S Q,D = lim s E Q k= D k α k,k+1 ( Q) F [ = E Q D k α k,k+1 ( Q) F k= Q-a.s. for all T N 0, where we have used he monoone convergence heorem for he second equaliy. Thus (4.7) implies exisence of and N Q,D = lim s E Q[ D +s α +s ( Q) F Q-a.s., T N 0 D α ( Q) = S Q,D The process (S Q,D ) is a Q-poenial. Indeed, E Q [ S Q,D + N Q,D Q-a.s., T N 0. [ E Q D k α k,k+1 ( Q) F α 0 ( Q) < k=0 and E Q [ S Q,D +1 F S Q,D Q-a.s. for all T N 0 by definiion. Moreover, monoone convergence implies lim E Q[ S Q,D = E Q [ lim k= D k α k,k+1 ( Q) The process (N Q,D ) is a nonnegaive Q-maringale, since = 0 Q-a.s.. E Q [N Q,D +1 N Q,D F = D α,+1 ( Q) D α,+1 ( Q) = 0 Q-a.s. for all T N 0 by propery (iii) of Theorem 4.2 and he definiion of (S Q,D ).

18 18 Bearice Acciaio e al. The nonnegaive maringale N Q,D, which may appear in he decomposiion (4.6) of he penaly process for T =, plays he role of a bubble. Indeed, i appears on he op of he fundamenal componen which is given by he poenial S Q,D generaed by he one-sep penalies, and his addiional penalizaion causes an excessive neglec of he model Q D in assessing he risk. As a resul, asympoic safey breaks down under he model Q D, as explained in he nex secion. 4.3 Asympoic safey and asympoic precision In his secion we discuss he asympoic properies of dynamic convex risk measures for processes. Throughou his secion we consider he case T = N 0 { }. In he case T = N 0 our assumpion of global coninuiy from above implies ha here is no mass a infiniy, i.e., D = 0 Q-a.s. for all Q D M( P ), and he discussion below reduces o he rivial case. A sysemaic discussion of he case T = N 0, bu wihou assuming he exisence of a global reference measure P and global coninuiy from above, appears in Föllmer and Penner [21. Consider a ime consisen dynamic convex risk measure for processes (ρ ) N0. As before, ( ρ ) N0 denoes he corresponding ime consisen dynamic convex risk measure for random variables on produc space given by (3.9). Le Q = Q γ = Q D Q α 0, and le us focus on he behavior of ( ρ ) N0 under Q. The measure Q will now play he same role as he reference measure P in [20, Secion 5. In paricular, he assumpion Q from [20, Secion 5 is saisfied for Q, since Q Q α 0. The resuls in [20 imply he exisence of he limis ᾱ ( Q) := lim ᾱ ( Q) and ρ (X) := lim ρ (X) Q-a.s. for all X R. Due o (3.9) and (3.11), we have ρ (X) = X1 N0 +ρ (X)1 { } and ᾱ ( Q) = α ( Q)1 { } Q-a.s., (4.8) where ρ (X) := lim ρ (X) and α ( Q) = lim α ( Q) Q-a.s. on {D > 0} by 3 of Remark B.2. Definiion 4.7 We call a dynamic convex risk measure for processes (ρ ) N0 asympoically safe under he model Q = Q D if he limiing capial requiremen ρ (X) covers he final loss X, i.e. for any X R. ρ (X) X Q-a.s. on {D > 0}

19 Risk assessmen for uncerain cash flows 19 Noe ha due o (4.8) asympoic safey of (ρ ) N0 is equivalen o he condiion ρ (X) X Q-a.s., i.e., o asympoic safey of ( ρ ) N0 in he sense of [20, Definiion 5.2. The following resul ranslaes [20, Theorem 5.4 and [34, Corollary o our presen seing. I characerizes asympoic safey by he absence of bubbles in he penaly process. This is plausible since, as we saw in Subsecion 4.2, such bubbles reflec an excessive neglec of models which may be relevan for he risk assessmen. Theorem 4.8 Le (ρ ) N0 be a ime consisen dynamic convex risk measure such ha each ρ is coninuous from above. Then for any model Q = Q D Q α 0, he following condiions are equivalen: 1. (ρ ) is asympoically safe under he model Q; 2. he model Q has no bubble, i.e., he maringale N Q,D in he Riesz decomposiion (4.6) of he discouned penaly process (D α ( Q)) N0 vanishes; 3. he discouned penaly process (D α ( Q)) N0 is a Q-poenial; 4. no model R Q wih α0 ( R) < admis bubbles. Proof Properies 2 and 3 are equivalen by (4.6), and obviously 4 implies 2. To prove 1 2 we use [20, Theorem 5.4. There i was shown ha ( ρ ) is asympoically safe under Q if and only if ᾱ ( Q) = 0 Q-a.s. and in L 1 ( Q). By Corollary B.3, (3.11), and (3.3) we have E Q[ᾱ ( Q) = E Q [ s T γ s α ( Q) = E Q [ D α ( Q). Thus ᾱ ( Q) 0 in L 1 ( Q) if and only if D α ( Q) 0 in L 1 (Q). This is equivalen o N Q,D 0, since he bubble N Q,D = (N Q,D ) N0 is a nonnegaive Q-maringale wih N Q,D [ 0 = lim E Q D α ( Q). Due o (4.6), N Q,D 0 also implies α ( Q) = 0 Q-a.s. on {D > 0}, hus ᾱ ( Q) = 0 Q-a.s. by (4.8). To prove 2 4 noe ha asympoic safey under Q implies asympoic safey under any model R Q wih α0 ( R) <, hus no model R admis bubbles by he same reasoning as above. Definiion 4.9 We call a dynamic convex risk measure for processes (ρ ) N0 asympoically precise under he model Q = Q D Qα 0 if for any X R. ρ (X) = X Q-a.s. on {D > 0} By (4.8), asympoic precision of (ρ ) is equivalen o asympoic precision of ( ρ ) in he sense of [20, Definiion 5.9. The following corresponds o [32, Lemma 2.7.

20 20 Bearice Acciaio e al. Lemma 4.10 A ime consisen dynamic convex risk measure (ρ ) N0 such ha each ρ is coninuous from above is asympoically precise under he model Q = Q D Q α 0 if and only if ρ (X) X Q-a.s on {D > 0} for all X R. Proof By [20, Lemma 5.1 he funcional ρ is convex and normalized. This implies ρ (X) ρ ( X) for all X R. Thus we obain X ρ (X) ρ ( X) X Q-a.s for all X R, which is equivalen o ρ (X) = X Q-a.s. on {D > 0} by (4.8). The following resul ranslaes [20, Proposiion 5.11 o our presen seing. Proposiion 4.11 Le (ρ ) N0 be a ime consisen dynamic convex risk measure such ha each ρ is coninuous from above, and assume ha for each X R he supremum in he robus represenaion (3.13) of ρ 0 (X) is aained by some wors case measure Q X γ X = Q X, such ha Q X Q. Then (ρ ) N0 is asympoically precise under Q. Proof Since ρ 0 (X) = ρ 0 (X), QX is also a wors case measure for ρ 0 (X). By [1, Proposiion 21, he measure Q X is hen a wors case measure for X a all imes N 0, i.e., [ ρ (X) = E QX X F ᾱ ( Q X ) Q-a.s. N0, and in paricular Q X Q α 0. By maringale convergence, ρ (X) = X ᾱ ( Q X ) Q-a.s., which is equivalen o ρ (X) = X α ( Q X ) Q-a.s. on {D > 0} due o (4.8). Asympoic precision of (ρ ) now follows from Lemma 4.10, since α ( Q X ) 0 Q-a.s. on {D > 0}. 4.4 A maximal inequaliy for he capial requiremens For X R and Q D M( P ), we can inerpre [ F Q,D (X) := E Q γ s X s F α (Q γ) on {D > 0} D s T as a risk evaluaion of he cash flow X a ime T N 0, using he specific model Q and he specific discouning process D. The nex proposiion provides, from he poin of view of he model Q, a maximal inequaliy for he excess of he required capial ρ (X) over he risk evaluaion F Q,D (X).

21 Risk assessmen for uncerain cash flows 21 Proposiion 4.12 Le (ρ ) T N0 be a ime consisen dynamic convex risk measure such ha each ρ is coninuous from above. Then for Q D M( P ), X R, and c > 0 we have ( { ( )} ) Q sup D ρ (X) F Q,D (X) c ρ 0(X) F Q,D 0 (X). (4.9) T N 0 c Proof Fix Q D M( P ). If α 0 (Q D) =, hen he inequaliy (4.9) holds rivially. Assume ha α 0 (Q D) <. By 2 of Corollary 3.11 we have [ ρ (X) E Q γ s X s F α (Q γ) = F Q,D (X) Q-a.s. on {D > 0}. D s T Thus he Q-supermaringale W Q,D (X) defined in (4.5) saisfies [ W Q,D (X) E Q γ s X s F Q-a.s. on {D > 0}. s T On {D = 0} = {D s = 0 s T }, we have W Q,D (X) = 1 s=0 D s X s. Therefore, he process Y Q,D (X) := D ( ρ (X) F Q,D ) (X) [ = W Q,D (X) + E Q γ s X s F, T N 0, s T is a nonnegaive Q-supermaringale, and (4.9) follows by a classical maximal inequaliy; cf., e.g., [38, Theorem VII The coheren case Due o posiive homogeneiy of a coheren risk measure, he penaly funcion can only ake values 0 or, and hus a coheren risk measure for processes ρ is coninuous from above if and only if i admis he robus represenaion [ ρ (X) = ess sup E Q γ s X s F, X R Q γ Q, (4.10) 0 s T where Q 0 := { Q Q α ( Q) = 0 }. In his subsecion we reformulae properies (iii) and (iv) of Theorem 4.2 in he coheren case. This involves a ranslaion of he noions of pasing of measures and sabiliy of ses as used in [4, [14, [20 in conex of coheren risk measures for random variables o our presen framework.

22 22 Bearice Acciaio e al. For Q 1, Q 2 M( P ) such ha Q 1 Q 2 on F and for B F we denoe by Q 1 Q B 2 he pasing of Q 1 and Q 2 in via B, i.e., he probabiliy measure on ( Ω, F) defined by Q 1 B Q 2 (A) = E Q1 [ E Q2[1 A F 1 B + 1 B c1 A, A F. Theorem 3.4 yields he decomposiion Q i = Q i D i, i = 1, 2, wih Q 1 Q 2 on F. Then Q 1 B Q 2 = Q 0 D 0, where Q 0 = Q 1 B Q 2 wih B = {ω (ω, ) B} F, i.e. Q 0 (A) = E Q 1 [ EQ 2[1 A F 1 B + 1 B c 1 A, A FT, and γu 0 = D 1 γu 2 D 2 γ 1 u u = 0,..., 1 1 {D 2 >0}1 B + γ 1 u1 B c u T. Here γ i and D i are relaed o each oher via (3.2) and (3.4) for i = 0, 1, 2. Noe ha Q 0 M loc (P ) and D 0 D(Q 0 ), in oher words, he pasing of Q 1 D 1 wih Q 2 D 2 admis a decomposiion wih he pasing of Q 1 wih Q 2 and he pasing of D 1 wih D 2. Definiion 4.13 We call a se Q M( P ) sable if, whenever Q 1, Q 2 Q and Q 1 Q 2 on F, he pasing of Q1 and Q 2 in via B belongs o Q for every B F and all T N 0. We associae o any Q M( P ) he ses Q 0 ( Q) = { R M( P ) R = Q F, ᾱ ( R) = 0 Q-a.s. }, and Q 0,+s( Q) { = R P F+s R = Q F, ᾱ,+s ( R) = 0 Q-a.s. }. The noion of pasing corresponds o concaenaion defined in [10, Definiion 4.10 on A 1, and he following corollary is corresponds o [10, Theorem 4.13, Corollary Theorem 4.14 Suppose ha he dynamic risk measure (ρ ) T N0 is coheren, and ha each ρ is coninuous from above. Then he following condiions are equivalen: 1. (ρ ) T N0 is ime consisen. 2. For all T, < T and Q M( P ), Q 0 ( Q) = { Q1 +1 Ω Q 2 Q1 Q 0,+1( Q), Q2 Q 0 +1( Q 1 ) }.

23 Risk assessmen for uncerain cash flows For all T, < T, X R and Q = Q D M( P ) such ha α ( Q) = 0 Q-a.s. on {D > 0}, E Q [D +1 (X + ρ +1 (X)) F D (X + ρ (X)) and α +1 ( Q) = 0 Q-a.s. on {D +1 > 0}. Moreover, if he se Q defined in (4.3) is no empy, hen ime consisency is equivalen o each of he following condiions: 4. The se Q is sable, and ρ has he represenaion 1 ρ (X) = ess sup E Q [ s T γ s X s F Q γ Q D (4.11) for all X R and T N The represenaion (4.11) holds for all T N 0 and all X R, and he process D ρ (X X 1 T ) D s X s, T N 0, s=0 is a Q-supermaringale for all Q = Q D Q. Proof Follows by applying [1, Corollary 26 and [20, Corollary 4.12 o ρ defined in (3.9) and using Corollary B.3. Coherence implies ha he risk measure is asympoically safe under any model Q = Q D Q0 0. Indeed, by 1 of Corollary 4.4, (D α ( Q)) N0 is a nonnegaive Q-supermaringale beginning a 0, and hence i vanishes. In paricular, here are no bubbles in he coheren case, and so he asympoic safey follows form Theorem Cash subaddiiviy and calibraion o numéraires As noed afer Definiion 3.1, cash invariance of risk measures for processes differs from he corresponding propery of risk measures for random variables, since i akes ino accoun he iming of he paymen. This aspec can be made precise using he noion of cash subaddiiviy. Cash subaddiiviy was inroduced by El Karoui and Ravanelli [18 in he conex of risk measures for random variables in order o accoun for discouning ambiguiy. I will be shown in Proposiion 5.2, and i also follows from he robus represenaion given in Subsecion 3.3, ha every risk measure for processes is cash subaddiive. Thus risk measures for processes provide a naural framework o capure uncerainy abou he ime value of money, and a sysemaic approach o he issue of discouning ambiguiy.

24 24 Bearice Acciaio e al. 5.1 Cash subaddiiviy Definiion 5.1 A condiional convex risk measure for processes ρ is called cash subaddiive if ρ (X + m1 T+s ) ρ (X) m, s > 0, m L, m 0; (5.1) cash addiive a ime + s, wih s > 0 and + s T, if ρ (X + m1 T+s ) = ρ (X) m, m L, cash addiive if i is cash addiive a all imes s T +1. Noe ha (5.1) is equivalen o ρ (X + m1 T+s ) ρ (X) m, s > 0, m L, m 0, since ρ (X) = ρ (X + m1 T+s m1 T+s ). Cash subaddiive risk measures accoun for he iming of he paymen in he sense ha he risk is reduced by having posiive inflows earlier and negaive ones laer. Oher equivalen characerizaions of cash subaddiiviy can be found in [18, Secion 3.1. As noed in [11 in he ime consisen case, cash subaddiiviy is an immediae consequence of he basic properies of a condiional risk measure for processes. Proposiion 5.2 Every condiional convex risk measure for processes ρ is cash subaddiive. Proof Cash subaddiiviy follows sraighforward from monooniciy and cash invariance of ρ : ρ (X) m = ρ (X + m1 T ) ρ (X + m1 T+s ), s > 0, m L, m 0. Cash subaddiiviy of risk measures for processes is also apparen from he robus represenaion given in Subsecion 3.3 due o he appearance of he discouning facors. Remark 5.3 In paricular, for T < or T = N 0 { }, every risk measure for processes resriced o he space {X R X = 0, < T } defines a cash subaddiive risk measure on L in he sense of [18, Definiion 3.1. Remark 5.4 For T = N 0, a condiional convex risk measure for processes ρ ha is coninuous from above canno be cash addiive. Indeed, if ρ is cash addiive a + s for all s > 0, coninuiy from above implies for X R and m L, m > 0, m + ρ (X) = ρ (X + m1 T+s ) ρ (X) wih s, which is absurd. The inerpreaion of his resul is clear: If we are indifferen beween having an amoun of money oday or omorrow or a any fuure ime, hen any paymen can be shifed from one dae o he nex, and so i would never appear.

25 Risk assessmen for uncerain cash flows 25 The following proposiion describes he inerplay beween ime consisency and cash addiiviy. Proposiion 5.5 Le (ρ ) T N0 be a ime consisen dynamic convex risk measure on R such ha each ρ is cash addiive a ime + 1. Then each ρ is cash addiive. Proof Follows by inducion using one-sep cash addiiviy and recursiveness (4.2). In view of Proposiion 5.5 and Remark 5.4 we obain he following resul. Corollary 5.6 For T = N 0, a dynamic convex risk measure (ρ ) N0 on R such ha each ρ is coninuous from above and cash addiive a ime + 1 canno be ime consisen. Remark 5.7 Corollary 5.6 and Remark 5.4 heavily depend on he assumpion of coninuiy from above, which was formulaed as a global propery. For T = N 0, he corollary in fac suggess o replace global coninuiy from above by a local version; his is done in [ Calibraion o numéraires Cash addiiviy can be seen as addiiviy wih respec o he numéraire 1. In his secion we discuss addiiviy wih respec o oher possible numéraires. To his end we formulae condiional versions of some resuls from [18. As usual, we denoe by α he minimal penaly funcion of ρ, and for T N 0 we define where Q α := { Q Q α (Q) < }, Qα := { Q Q α ( Q) < }, Q := { Q M(P ) Q = P on F } and Q is defined in (3.14). The following lemma is a condiional version of [18, Lemma 2.3. Lemma 5.8 Le ρ : L L be a condiional convex risk measure for random variables ha is coninuous from above, and le N L. Then he following condiions are equivalen: (i) ρ (λ N) = λ ρ (N) for all λ L ; (ii) E Q [ N F = ρ (N) for all Q Q α ; (iii) ρ (X + λ N) = ρ (X) + λ ρ (N) for all X L and all λ L. Proof (i) (ii). (i) and [20, Corollary 2.4 imply for each λ L and Q Q λ ρ (N) = ρ (λ N) λ E Q [ N F α (Q). If α (Q) <, we have α (Q) λ (E Q [N F + ρ (N)) for any λ L, hus ρ (N) = E Q [ N F. (ii) (iii) follows from [20, Corollary 2.4, and (iii) (i) from normalizaion.

26 26 Bearice Acciaio e al. Due o (i) of Lemma 5.8, we can assume wihou loss of generaliy ha he random variable N saisfies he condiion ρ (N) = 1. Then condiion (ii) of Lemma 5.8 means ha he condiional expecaion of he numéraire N is unique under all relevan probabiliy measures, and condiion (iii) can be viewed as addiiviy wih respec o he numéraire N: ρ (X + λ N) = ρ (X) λ X L, λ L. The following lemma ranslaes Lemma 5.8 o he framework of risk measures for processes. Lemma 5.9 Le ρ : R L be a condiional convex risk measure for processes such ha each ρ is coninuous from above, and le N s L s for some s T +1. Then he following condiions are equivalen: (i) ρ (λ N s 1 Ts ) = λ ρ (N s 1 Ts ) for all λ L ; D (ii) E Q [ N s s D F = ρ (N s 1 Ts ) for all Q = Q D Qα ; (iii) for all X R and λ L ρ (X + λ N s 1 Ts ) = ρ (X) + λ ρ (N s 1 Ts ). Proof Consider he condiional convex risk measure ρ : L L associaed o ρ via (3.9). The lineariy condiion (i) for ρ is equivalen o ρ (λ N s 1 Ts ) = λ ρ (N s 1 Ts ) λ L, i.e., ρ is linear on {Λ N s 1 Ts Λ L }. By Lemma 5.8 and (3.9) his is equivalen o E Q[ N s 1 Ts F = ρ (N s 1 Ts )1 T Q-a.s. Q = Q D Qα, and his is equivalen o (ii) by Corollary B.3. In he same way, Lemma 5.8 and (3.9) imply ha (i) is equivalen o (iii). Since each D D (Q) is non-decreasing, Lemma 5.9 applied o N s = 1 for some s > yields he following characerizaion of cash addiiviy: Corollary 5.10 A condiional convex risk measure for processes ρ : R L such ha each ρ is coninuous from above, is cash addiive a ime s T +1 if and only if for all Q = Q D Qα. D = D +1 =... = D s Q-a.s. In oher words, cash addiiviy a ime s > means ha here is no discouning beween and s in all he relevan models. In paricular we have he following proposiion.

27 Risk assessmen for uncerain cash flows 27 Proposiion 5.11 A condiional convex risk measure for processes ρ is coninuous from above and cash addiive a ime s T +1 if and only if i admis he robus represenaion ) (E Q [ k Ts γ k X k F ρ (X) = ess sup ess sup α (Q γ), X R. Q Q loc γ Γ s(q) (5.2) In his case ρ is cash addiive up o s, i.e., a all imes + 1,..., s. In paricular, if T < or if T = N 0 { }, a risk measure for processes ρ ha is coninuous from above is cash addiive if and only if i reduces o a risk measure on L : ρ (X) = ess sup (E Q [ X T F β (Q)), (5.3) Q Q where β (Q) := α (Q δ {T } ), and δ {T } denoes he Dirac measure a T. Proof Obviously, represenaion (5.2) implies coninuiy from above and cash addiiviy up o ime s. The converse follows from 1 of Corollary 3.11 and Corollary To prove he las par of he asserion, noe ha Γ T (Q) = {δ {T } } if T < or T = N 0 { }. Moreover, we have Q P for any Q Q loc such ha Q δ {T } Q α. This is obvious for T <, and i follows from Lemma B.4 if T = N 0 { }, since γ = 1 Q-a.s. in his case. Thus he represenaion (5.3) follows from (5.2). Remark 5.12 In paricular, in he cash addiive case and for T < or T = N 0 { }, he resuls of Secion 4 reduce o he corresponding resuls for risk measures for random variables from [20, 1. The following example exends [18, Proposiion 2.4 o our presen framework. Example 5.13 Le ρ : R L be a condiional convex risk measure for processes ha is coninuous from above. Assume ha here is a money marke accoun (B ) T N0 as in Example 2.2, and ha zero coupon bonds for all mauriies k >, k T N 0 are available a prices B,k, respecively. Suppose ha ρ saisfies he following calibraion condiion: ( ) B ρ λ 1 Tk = λ B,k λ L, k T N 0. (5.4) B k Lemma 5.9 applied o N k = B implies ha he calibraion condiion (5.4) is B k equivalen o ( ) B ρ X + λ 1 Tk = ρ (X) λ B,k X R, λ L, k T N 0, B k

28 28 Bearice Acciaio e al. and also o E Q [ B B k D k D F = B,k k T N 0, Q = Q D Q α. (5.5) Using (5.5), he robus represenaion from par 1 of Corollary 3.11, and monoone convergence for T =, i can be seen ha he calibraion condiion (5.4) is equivalen o he following one, ha may seem sronger a firs sigh: ) ρ ( T k=+1 λ k B B k 1 Tk = T k=+1 λ k B,k λ k L. Moreover, if he shor rae process (r ) and hence also he money marke accoun (B s ) s T N0 is predicable, hen (5.5) implies B B +1 D +1 D = B,+1, and hus D +1 = D for all Q = Q D Qα, since B,+1 = (1 + r +1 ) 1 by a sandard no arbirage argumen. Hence ρ is cash addiive a ime + 1 by Corollary In paricular, if a dynamic convex risk measure (ρ ) is ime consisen, and if each ρ is coninuous from above and saisfies he calibraion condiion (5.4) wih a predicable money marke accoun, hen each ρ is cash addiive by Proposiion 5.5. In view of Remark 5.4, a ime consisen dynamic convex risk measure ha is coninuous from above canno saisfy condiion (5.4) for all T if T = N 0. 6 Examples In his secion we illusrae our analysis by discussing some examples, in paricular analogues o classical risk measures for random variables such as he enropic risk measure and Average Value a Risk. Anoher class of examples is obained by separaing model and discouning ambiguiy in he robus represenaions of Subsecion Enropic risk measures In his secion we inroduce enropic risk measures for processes. As a firs varian we simply ake he usual condiional enropic risk measure on produc space, ha is he map ρ : L L defined by ρ (X) = 1 [ log E P e R X F R wih risk aversion parameer R = (r 0,..., r 1, r, r,...) L, where r s > 0 and r 1 s L s for all s = 0,...,, and e RX = (e rsxs ) s T.