Scalar Multivariate Risk Measures with a Single Eligible Asset

arXiv:1807.10694v5 [q-fin.RM] 4 Feb 2021 U,brudloff@wu.ac.at. AUT, clrmliait ikmaue ihasnl eligible single a with measures risk multivariate Scalar ∗ † tvn nttt fTcnlg,Sho fBsns,Hoboken, Business, of School Technology, of Institute Stevens inaUiest fEooisadBsns,IsiuefrStatist for Institute Business, and Economics of University Vienna ihrnnuiu ripiil eursasseai incr systematic a portf a requires frictio of implicitly reduction the or the is, non-unique therefore pa That and either this possible costs. In not is impact assets market manager. measure and risk risk r transaction multivariate a capital of for to minimal properties acceptable consistency the investment time provide by an functions encoded make Such are to nex principles the measures. these at maker management, risk decision risk same guara the of by and context reversed problems be not optimization will dynamic made are of consistency time computation and the principle programming dynamic The Introduction 1 emphasis special a with with given, measure are measure. risk examples underlying Several the process. to an respect corre consistency W with time prici which of consistent recursion notion defined, weaker fixed satisfied. this any be of not equivalence under the can but and consistency consistency strong time time too scalar of is 69 notion [50, consistency in weaker time as kn costs, well of transaction is concept with It markets frictions. in with measure markets p risk this in of measures m results risk main market the scalar the obtain repr to with used r dual consistent are dua representations the capital prices of on and space minimal the provided measures the on martingale are given as emphasis results particular defined some with measures, are First, measures asset. risk cash Such costs. nti ae epeetrslso clrrs esrsi ma in measures risk scalar on results present we paper this In eray5 01(rgnl uy2,2018) 27, July (Original: 2021 5, February ahr Feinstein Zachary Abstract asset ∗ 1 igtRudloff Birgit J000 S,[email protected]. USA, 07030, NJ oalwfrtepossibility the for allow to s aei ikns.A ilbe will As riskiness. in ease c n ahmtc,Ven A-1020, Vienna Mathematics, and ics eti yeo backward of type certain a d pro iecnitnyfor consistency time on aper lot igenuméraire is single a to olio w rmtesuperhedging the from own 2,ta h sa scalar usual the that 62], , gpoes ewl prove will We process. ng aibe s(equivalent) as variables l ntespregn risk superhedging the on ls ovrinbetween conversion nless xdcnitn pricing consistent fixed dl hn hs dual these Then, odel. snaino uhrisk such of esentation rpriso dynamic on properties ia ocpsbt for both concepts vital qieet necessary equirements ten htdecisions that nteeing on ntm.I the In time. in point t kt ihtransaction with rkets pnst h usual the to sponds ilso hta that show will e qieet nthe in equirements e einvestigate we per † detailed later in this work, our approach reduces this problem to the time consistency of the dynamic risk measures under all possible no-arbitrage pricing processes. The organization of this paper is as follows. In Section 1.1, we provide a literature review and in Section 1.2 an overview of the main results along with our motivation for their study. In Section 2 we present background material on risk measures and notation that will be used throughout this paper. Much of this notation is comparable to that utilized in the set-valued risk measure literature. In Section 3.1 we present the definition of the dynamic scalar risk measures we will consider in this paper and basic results on boundedness properties. In Section 3.2 we present dual representations for the scalar risk measures with a single eligible asset. Special emphasis is placed on a representation akin to (1.1) below considered in [50], but for general convex and coherent dynamic risk measures in markets with transaction costs, which relates those risk measures to the set of no-arbitrage pricing processes. In Section 3.3 we present results on relevance or sensitivity of the scalar risk measures which provide a condition for the dual representation to be with respect to equivalent probability measures only. In Section 4 we introduce a new notion of time consistency for scalar risk measures with a single eligible asset. This notion coincides with time consistency of the corresponding univariate scalar risk measure under any market-consistent frictionless (and no-arbitrage) price process. As such, unlike much of the prior literature, we can and do demonstrate that such a property is satisfied by the usual examples; in particular we give details on the superhedging risk measure and composed risk measures in Section 5.

1.1 Literature review In their seminal paper, Artzner, Delbaen, Eber, and Heath [2] introduced the concept of coherent risk measures in a static univariate setting which provides the minimal capital necessary to compensate for the risk of a contingent claim. Coherent risk measures were further studied in [22]. Such risk measures were generalized to the convex case in [38, 40] while retaining the same financial interpretation and a notion of diversification. T When a filtration (Ft)t=0 is introduced, it is natural to consider dynamic risk measures, i.e., the minimal capital necessary to compensate for the risk of a contingent claim conditionally on the information at time t. In this time-dynamic setting, the manner in which the risk of a contingent claim propagates through time is of great importance, as it has significant implications on risk management. One such condition, called (strong) time consistency, consists of the condition that if one portfolio is riskier than another in the future, that same ordering must hold at all prior times as well. This property is studied in the univariate setting in [3, 66, 25, 18, 70, 9, 10, 11, 37, 20, 19, 1, 39] in discrete time and [41, 23, 24] in continuous time. Other, weaker, forms of time consistency have been presented for dynamic risk measure in, e.g., [61, 60, 67, 74, 75]. In this work, we will consider multivariate risk measures. The multivariate setting arises from markets with frictions, as the liquidation of a portfolio into some numéraire does not allow for repurchasing the same asset. This setting was studied in a set- valued static (one period) framework in [51, 45, 47, 44, 46]. Set-valued risk measures, in the one period framework, have recently been applied to studying systemic risk in [35, 8]. In a set-valued time-dynamic framework a new notion of time consistency, called multiportfolio time consistency, was introduced in [29, 6]. This property was

2 further studied in [31, 30, 33, 34, 17]. Computation of such set-valued risk measures was studied in [32, 62]. The focus of this work is on dynamic scalar multivariate risk measures with a single eligible asset. Some results on dynamic scalar multivariate risk measures with multiple eligible assets have been discussed previously in [30, 33, 34]. Scalar risk measures in frictionless markets with either a single eligible asset [38, 2] or multiple eligible assets [28, 5, 77, 4, 42, 56, 71] can be considered as scalar multivariate risk measures (see [30, Example 2.26] for such a comparison). Additionally, such functions have been considered in the context of systemic risk in, e.g., [59, 15]. Our focus in this paper is of the dynamic version of the multivariate risk measures with a single eligible asset as presented in [16, 76] (see [30, Example 2.27 and 2.28] for a brief discussion). Of course time (in-)consistency has not just been studied in the context of risk measurement as considered here. There has been substantial development and inves- tigation of time (in-)consistency for stochastic control and stopping problems coming, e.g., from behavioral economics where one considers the (in-)consistency of an agent’s control (i.e., the choice of strategies or stopping times) over time. Time (in-)consistency was surveyed for utility maximization problems by [73]. We wish to highlight the recent works of [27, 13, 12] on time consistency for stochastic control and [49, 21] on time consistency for optimal stopping problems; these works utilize a standard equilibrium or consistent planning approach proposed in [73]. We also wish to highlight [54] which very succinctly deﬁnes such problems of time (in-)consistency for control problems; that work proposes and uses several diﬀerent approaches distinct from the equilibrium approach of the prior references. Such problems in time (in-)consistency have drawn increased attention since the seminal work of Ekeland and Lazrak [26]. In fact, such time (in-)consistency questions are intimately related with the same notions for risk measures. This control viewpoint can be seen through the dual representation, which is a maximization problem over probability measures. This is made explicit in the coherent case via stability [3], i.e., the pasting of these probability measures through time. Indeed, we can view time consistency of risk measures as the recursion of the value function from the control literature. The situation is similar in the set-valued framework. Stochastic control problems in a multivariate setting, where the value function is now a set-valued function, have been studied in [57, 36]; the corresponding backward recursion of the value function in the time consistent case is in analogy to the backward recursion of set-valued risk measures in the multi-portfolio time consistent case, as the value function (i.e., the risk measure itself) recurses backwards in time. Of course stochastic control problems can have a much more complicated structure as, e.g., dependencies on initial capital levels can complicate the numerical computations in the backward recursion, see [57], compared to risk measures, where the input is the random vector at terminal time.

1.2 Motivation In this work we consider risk measurement in markets with transaction costs. These market frictions result in the need to consider multivariate risk measures (i.e., risk measures for a vector-valued portfolio of “physical units” – the number of units of the d available assets or securities held – rather than the value of that portfolio in a single num´eraire) because no single value of a portfolio or investment can be deﬁned. For instance, a mark-to-market value of the portfolio using the current mid-price between

3 the bid and ask prices is strictly larger than the liquidation value of that same portfolio; both such values would result in different capital requirements and neither, necessarily, encodes the true risk of the portfolio. Set-valued risk measures (see, e.g., [51, 46]) were introduced in order to handle this problem by considering the set of all capital requirements, possibly held in multiple different assets, that make the portfolio acceptable. This eliminates the need to first convert the d-dimensional portfolio vector into a one-dimensional quantity and thus avoids the problems related to that. We are focused on dynamic multivariate risk measures and questions surrounding time consistency. For set-valued risk measures, the typical dynamic programming principle is called multiportfolio time consistency (see, e.g., [29, 31]). Such a property relates the risk between vector-valued portfolios over time by comparing the full set of acceptable capital allocations. The usual risk measures, e.g., the superhedging risk measure, satisfy this time consistency property when the full space of assets are eligible to be held as capital. However, typically, capital is required to be held in a numéraire, or cash, asset only. In such a setting, multiportfolio time consistency may no longer hold for the corresponding set-valued risk measure (this is the case, e.g., for superhedging). Furthermore, for a single eligible asset, the minimal capital requirement (often called the scalarization of the set-valued risk measure) becomes the logical object to consider, but it typically does not satisfy the scalar concept of time consistency either. So the motivating questions of this paper are the following. If the eligible asset is just the cash asset, the set-valued risk measure does not satisfy multiportfolio time consistency, and the scalarization does not satisfy the usual time consistency, is there a more appropriate, weaker form of time consistency for this scalar risk measure? Can one use the fact that the set-valued risk measure, for the full set of eligible assets, is multiportfolio time consistent? The answer to both questions is yes and will be answered in this work, see in particular Lemma 4.20 below. We are specifically motivated to study the problem of scalar multivariate risk measures with a single eligible asset from the many works that consider the scalar superhedging price in a market with frictions. We refer to, e.g., [7, 14, 65, 50, 68] for studies of the scalar superhedging price in a market with transaction costs and two assets. This has been extended to a general number of assets in [69, 62]. The d dimensional T version of the scalar superhedging price given by [50] under a sequence (Kt)t=0 of solvency cones (for a market with proportional transaction costs, see Section 3.1 for further discussion) is as follows. Under the appropriate no-arbitrage argument, the scalar superhedging price ρSHP (−X) in units of the first (cash) asset at time t =0 is given by

SHP Q T ρ (−X)= sup E ST X (1.1) (Q,S)∈Q h i ∞ d for X ∈ L (Ω, FT , P; R ) which is the payoff in physical units. The set of dual variables T is given by Q which denotes the set of all processes S = (St)t=0 along with their equiv- 1 dQ ∞ + alent martingale measures Q such that St ≡ 1 and E dP Ft St ∈ L (Ω, Ft, P; Kt ) for all t. For the context of this paper, we note that Theoremh i 6.1 of [62] relates (1.1) to the scalarization of the set-valued superhedging risk measure under a single eligible asset (the numéraire asset). We will present a similar dual representation for general convex risk measures in a dynamic framework. We then use this representation to define a new time consistency property, as detailed below, which is satisfied by, e.g., the

4 superhedging risk measure. This is in contrast to prior works on time consistency for scalar multivariate measures [48, 58] whose time consistency property does not include the superhedging risk measure as an example. This agrees also with recent research on scalarizations of multivariate problems, as in [54, 57], indicating that the usual scalar concept of time consistency is way too strong to be satisfied for scalarizations of multivariate risk measures. Though weaker forms of time consistency (as in, e.g., [75]) may be satisfied for the scalarization, we are interested in a weaker version that is strong enough to still allow a relation to the dynamic programming principle. We call the weaker time consistency property defined in this work π-time consistency S as it corresponds to the usual scalar time consistency for all the risk measure πt appearing when fixing a pricing process S. That is, the risk measure is time consistent in the usual way when the market model follows any (frictionless) no-arbitrage pricing process S. We will prove the equivalence of this weaker notion of time consistency S and a certain type of backward recursion with respect to πt . We will show that the superhedging risk measure in markets with transaction costs (expressed in the cash asset) as in [50, 69, 62] while not satisfying the usual scalar time consistency property, which also means it is not recursive with itself, will satisfy π-time consistency and thus S a backward recursion with respect to πt .

2 Setup

T Consider a filtered probability space Ω, F, (Ft)t=0 , P satisfying the usual conditions with F0 the trivial sigma algebra and F = FT . Let | · |n be an arbitrary norm in n R for n ∈ N; note that | · |1 is equivalent to the absolute value operator. Denote 0 0 by Lt (D) = L (Ω, Ft, P; D) the set of equivalence classes of Ft-measurable functions n p 0 X :Ω → D ⊆ R for some n ∈ N. Additionally, denote by Lt (D) ⊆ Lt (D) those 1 0 p p random variables X ∈ Lt (D) such that kXkp = Ω |X(ω)|ndP < +∞ for p ∈ (0, +∞) and kXk∞ = esssupω∈Ω |X(ω)|n < +∞ for p = +∞. We further note that p p R 0 d L (D) := LT (D) by definition. In particular, if p = 0, Lt (R ) is the linear space of the d p d equivalence classes of Ft-measurable functions X :Ω → R . For p> 0, Lt (R ) denotes d the linear space of Ft-measurable functions X :Ω → R such that kXkp < +∞ p d for p ∈ (0, +∞]. Note that an element X ∈ Lt (R ) has components X1, ..., Xd in p Lt (R) for any choice of p ∈ [0, +∞]. For p ∈ [1, +∞] we will consider the dual pair p d q d 1 1 Lt (R ),Lt (R ) , where p + q = 1 (with q = +∞ when p = 1 and q = 1 when ∞ Rd 1 Rd p =+∞), and endow it with the norm topology, respectively the σ Lt ( ),Lt ( ) - topology on L∞(Rd) in the case p =+∞. Throughout most of this paper we will focus t on the case that p =+∞. Throughout this work we will make use of the indicator functions which we will denote by 1D :Ω → {0, 1} for some D ∈ F. These are defined so that 1D(ω)=1if ω ∈ D and 0 otherwise. Additionally, throughout we will consider the summation of sets by p d Minkowski addition. A set B ⊆ L (R ) is called Ft-decomposable if 1DB + 1Dc B ⊆ B for any measurable D ∈ Ft. As in [52] and discussed in [72, 53], the portfolios in this paper are in “physical units” of an asset (in a market with d assets) rather than the value in a fixed numéraire, ∞ d except where otherwise mentioned. That is, for a portfolio X ∈ Lt (R ), the values of Xi (for 1 ≤ i ≤ d) are the number of units of asset i in the portfolio at time t.

5 Let M = R × {0}d−1 denote the set of eligible portfolios, i.e., those portfolios which can be used to compensate for the risk of a portfolio. That is, for the purpose of this paper, we will assume that the first asset is the only asset available to compensate for risk (in typical examples this will be the cash asset). For notational simplicity we ∞ ∞ d−1 ∞ d define Mt = Lt (M)= Lt (R)×{0} , a weak* closed linear subspace of Lt (R ) (see d d−1 Section 5.4 and Proposition 5.5.1 in [53]). We will denote M+ := M ∩R+ = R+×{0} to be the nonnegative elements of M. We will additionally denote Mt,+ := Mt ∩ ∞ d ∞ ∞ d−1 Lt (R+)= Lt (M+)= Lt (R+) × {0} to be the nonnegative elements of Mt. Denote the upper sets by P (Mt; Mt,+) where P (Z; C) := {D ⊆ Z | D = D + C} for some vector space Z and an ordering cone C ⊂ Z. Additionally, let G(Z; C) := {D ⊆ Z | D = cl co (D + C)} ⊆ P(Z; C) be the upper closed convex subsets. As noted previously, for the duality results below we will consider the weak* topology for p =+∞. We will briefly describe the set of dual variables from the set-valued biconjugation theory as utilized in, e.g., [29, 31], and first defined in [43]. First, define the space of probability measures absolutely continuous with respect to P as M and those that are equivalent probability measures as Me. The space of d-dimensional probability measures absolutely continuous with respect to P is thus denoted by Md. Throughout we will use a P-almost sure version of the Q-conditional expectation where Q ∈ M. This is defined in, e.g., [19, 29]. Briefly, let X ∈ L∞(R), then define the conditional expectation Q E [X| Ft] := E ξ¯t,T (Q)X Ft , where dQ E[ dP |Fσ](ω) dQ dQ if E Fs (ω) > 0 E F (ω) dP ξ¯s,σ(Q)[ω] := [ dP | s]  h i 1 else  for every ω ∈ Ω. For a vector-valued probability measure Q ∈ Md the result is  T Q Q1 Q defined component-wise, i.e., E [X| Ft] = E [X1| Ft] ,..., E d [Xd| Ft] for any T ∞ d ¯ ¯ X ∈ L (R ), and define ξs,σ(Q) := ξs,σ(Q1),..., ξs,σ(Qd) . Note that, for any Q ∈ M and any times 0 ≤ t ≤ s ≤ σ ≤ T , dQ = ξ¯ (Q) and ξ¯ (Q) = ξ¯ (Q)ξ¯ (Q) dP 0,T t,σ t,s s,σ almost surely. From this representation we can define the set of dual variables from the set-valued biconjugation theory [43] to be

d + ⊥ T 1 d Wt := (Q, w) ∈ M × Mt,+\Mt | wt (Q, w) ∈ L (R+) . (2.1) n o In this representation we deﬁne

s wt (Q, w) = diag (w) ξt,s(Q) for any times 0 ≤ t ≤ s ≤ T and where diag (x) denotes the diagonal matrix with the components of x on the main diagonal. Additionally, we employ the notation ⊥ 1 d T Mt = v ∈ Lt (R ) | E v u = 0 ∀u ∈ Mt to denote the orthogonal space of Mt and C+ = v ∈ L1(Rd) | E vTu ≥ 0 ∀u ∈ C to denote the positive dual cone of a cone t C ⊆ L∞(Rd). t

6 3 Scalarizations

In this section we introduce the scalar multivariate risk measures of interest in this paper. We provide, in Section 3.1, an axiomatic definition of these risk measures akin to those in [2, 38, 40] and, further, relate these scalar risk measures to the set-valued risk measures of [29, 31, 30]. In Section 3.2, we consider the dual representation of these scalar risk measures producing two, equivalent, representations. The first is with respect to the dual variables considered in the set-valued literature, the second providing a representation generalizing the dual representation (1.1) of the scalar multivariate superhedging risk measure from [50]. We conclude with Section 3.3 to extend the dual representation results in order to determine a sufficient condition for such a representation to be over the space of equivalent probability measures only.

3.1 Definition First, we wish to consider an axiomatic definition for the scalar conditional convex risk measures which will be the subject of this paper. These properties provide the classical interpretations that the risk measure is a capital requirement, that having more in each asset corresponds with lower risk, that diversification reduces risk, and that not being invested in the market carries with it a finite risk (possibly 0). Throughout T d this paper we will consider the unit vector e1 := (1, 0, ..., 0) ∈ R and the zero vector T 0 := (0, 0, ..., 0) ∈ Rd.

∞ d ∞ Deﬁnition 3.1. A mapping ρt : L (R ) → Lt (R) is called a scalarized conditional convex risk measure if it satisﬁes the following properties for all X,Y ∈ L∞(Rd) ∞ and m ∈ Lt (R):

• Conditional cash invariance: ρt(X + me1)= ρt(X) − m; • ∞ d Monotonicity: Y − X ∈ L (R+) ⇒ ρt(X) ≥ ρt(Y );

• Conditional convexity: ρt(λX + (1 − λ)Y ) ≤ λρt(X) + (1 − λ)ρt(Y ) for every ∞ λ ∈ Lt ([0, 1]); • ∞ Finite at zero: ρt(0) ∈ Lt (R). A scalarized conditional convex risk measure is called coherent if it additionally satis- ﬁes: • ∞ Conditional positive homogeneity: ρt(λX)= λρt(X) for every λ ∈ Lt (R+).

A scalarized conditional convex risk measure is called Kt-compatible if it additionally satisﬁes: • ∞ d T ∞ ρt(X) = essinfk∈Kt ρt(X − k) for every X ∈ L (R ), where Kt = s=t Ls (Ks) for Fs-measurable random convex cones Ks. P Kt-compatibility is used to allow the possibility of trading at time points s ∈ {t, ..., T } according to, e.g., the bid-ask prices modeled by a sequence of solvency cones T (Kt)t=0. The following two results relate the scalarized conditional risk measures ρt with set-valued risk measures Rt over a single eligible asset (provided by Mt). Brieﬂy, a set-valued risk measure is a mapping Rt from the space of contingent claims into sets of eligible portfolios, i.e., capital allocations, that make the initial claim acceptable

7 to a risk manager or regulator. These risk measures satisfy cash invariance (Rt(X + ∞ d m) = Rt(X) − m for X ∈ L (R ) and m ∈ Mt), monotonicity (Rt(X) ⊆ Rt(Y ) if ∞ d Y −X ∈ L (R+)), conditional convexity (Rt(λX +(1−λ)Y ) ⊇ λRt(X)+(1−λ)Rt(Y ) ∞ d ∞ for X,Y ∈ L (R ) and λ ∈ Lt ([0, 1])), and finiteness at zeros. For more details on the set-valued conditional risk measure we refer to [29, 31, 30]. ∞ d Theorem 3.2. Let Rt : L (R ) → G(Mt; Mt,+) := {D ⊆ Mt | D = cl co (D + Mt,+)} be a set-valued conditional convex (coherent) risk measure that is Kt-compatible (i.e., R X R X k X L∞ Rd ρ t( )= k∈Kt t( − ) for every ∈ ( )), then the mapping t defined by S ∞ ∞ d ρt(X) := ess inf{m ∈ Lt (R) | me1 ∈ Rt(X)} ∀X ∈ L (R ) (3.1) is a scalarized conditional convex (resp. coherent) Kt-compatible risk measure and, conversely, ∞ ∞ d Rt(X) = (ρt(X)+ Lt (R+))e1 ∀X ∈ L (R ). ∞ ∞ d Proof. First we will show that ρt(X) ∈ Lt (R) for any X ∈ L (R ). By assumption − + ∞ − + there exists some Z ,Z ∈ L (R) such that Z e1 ∈ X − Kt and Z e1 ∈ X + Kt. − + This implies Rt(Z e1) ⊆ Rt(X) ⊆ Rt(Z e1) by Kt-compatibility. This implies + + + ρt(X) ≥ ρt(Z e1) ≥ ρt(kZ k∞e1) ≥−(kρt(0)k∞ + kZ k∞) > −∞ − − − ρt(X) ≤ ρt(Z e1) ≤ ρt(−kZ k∞e1) ≤ kρt(0)k∞ + kZ k∞ < +∞. The remainder of the properties are proven in Theorem 3.15 and Corollary 3.17 of [30]. ∞ Second, consider the converse statement Rt(X) = (ρt(X)+ Lt (R+))e1 for every ∞ d X ∈ L (R ). If ue1 ∈ Rt(X) then u ≥ ρt(X) by the definition of the scalarization. The other direction follows if ρt(X)e1 ∈ Rt(X). To show this we use the result from [30, Lemma 3.18] that

M,w T Rt(X)= u ∈ Mt | ρt (X) ≤ w u w M + M ⊥ ∈ \t,+\ t n o M,w M,w T where ρt denotes the scalarization ρt (X) := ess infm∈Rt(X) w m. Then we note that

T ∞ w (ρt(X)e1)= w1ρt(X)= w1 ess inf {u ∈ Lt (R) | ue1 ∈ Rt(X)} T M,w = essinf w m | m ∈ Mt, m ∈ Rt(X) = ρt (X), n o which implies the result. Assumption 3.3. For the remainder of this paper we will assume that the conditions of Theorem 3.2 are satisfied and that ρt is constructed as in (3.1). We will additionally T ¯ assume that Kt is defined w.r.t. solvency cones (Kt)t=0 with KT ⊇ K almost surely ¯ d ¯ d for some convex cone K ⊆ R with int K ⊇ R+\{0}. ∞ d The acceptance set At ⊆ L (R ) is defined via the set-valued risk measure Rt, i.e., ∞ d At = {X ∈ L (R ) | 0 ∈ Rt(X)}. From the acceptance set we can also define ρt as

∞ ∞ d ρt(X) := ess inf{m ∈ Lt (R) | X + me1 ∈ At} ∀X ∈ L (R ). (3.2)

Under this setting ρt defines the minimal capital (in the first asset, e.g., Euros) necessary to hedge the risk of X when trades defined by a market model Kt are allowed.

8 ∞ d ∞ Proposition 3.4. Let ρt : L (R ) → Lt (R) be a scalarized conditional convex risk measure. Then it also satisﬁes the following two properties:

1. Ft-Lipschitz continuity: |ρt(X) − ρt(Y )| ≤ kX − Y kKt,t almost surely for every X,Y ∈ L∞(Rd), and ∞ d 2. Local property: 1Bρt(X) = 1Bρt(1BX) for every X ∈ L (R ) and B ∈ Ft where, as deﬁned in [6], ∞ kXkKt,s := ess inf{c ∈ Ls (R) | ce1 ≥Kt X ≥Kt −ce1} (3.3) for 0 ≤ s ≤ t ≤ T and with Y ≥Kt X if Y − X ∈ Kt. ∞ d Proof. Let X,Y ∈ L (R ) and B ∈ Ft.

1. By deﬁnition kX − Y kKt,te1 ≥Kt X − Y . By Kt-compatibility this implies

ρt(X) ≥ ρt(Y + kX − Y kKt,te1)= ρt(Y ) − kX − Y kKt,t. This provides the desired result by symmetry of X and Y . 2. By conditional convexity we recover

ρt(1B X)= ρt(1B X + 1Bc 0) ≤ 1Bρt(X) + 1Bc ρt(0),

which implies 1Bρt(1BX) ≤ 1Bρt(X). To prove the converse

ρt(X)= ρt(1B[1BX] + 1Bc X) ≤ 1Bρt(1B X) + 1Bc ρt(X),

which implies 1Bρt(X) ≤ 1Bρt(1BX).

3.2 Dual representation We will now consider the dual, or robust, representation for the scalar conditional risk measures. We present two equivalent formulations, both reliant on a lower semicon- tinuity property encoded in the following Fatou property. The ﬁrst of these two dual representations is intimately related to the dual representation of the underlying set- valued risk measures as discussed in Theorem 3.2. See [29, 31, 33] for discussion on the dual representation for set-valued risk measures. The second of these dual representations is a generalization of representation (1.1) of the multivariate superhedging price derived in [50, 62].

Deﬁnition 3.5. Let ρt be a scalarized conditional convex risk measure. ρt is said to satisfy the Fatou property if

ρt(X) ≤ lim inf ρt(Xn) n→∞ for any k · k∞-bounded sequence Xn converging to X almost surely. We will now provide the ﬁrst dual representation for these scalarized conditional convex risk measures. This representation is similar to the results in [33, Appendix A.2]. Recall the set of dual variables Wt deﬁned in (2.1). Additionally, we will make s extensive use of the notation introduced in Section 2 that wt (Q, w) = diag (w) ξt,s(Q) for (Q, w) ∈ Wt with ξt,s(Q) the vector of conditional Radon-Nikodym derivatives of Q w.r.t. P.

9 Proposition 3.6. Let ρt be a scalarized conditional convex risk measure. Then ρt satisﬁes the Fatou property if and only if for any X ∈ L∞(Rd)

T Q ρt(X)= esssup −αt(Q,m⊥) − (e1 + m⊥) E [X| Ft] , (3.4) (Q,m⊥)∈Wt(e1) where

d 1 d−1 T + Wt(e1) := (Q,m⊥) ∈ M × {0}× Lt (R+ ) | (Q, e1 + m⊥) ∈Wt, wt (Q, e1 + m⊥) ∈ Kt n o and T Q αt(Q,m⊥) := esssup (e1 + m⊥) E [−Z| Ft] . Z∈At

If ρt is additionally a conditional coherent risk measure then (3.4) can be reduced to

T Q ρt(X)= esssup − (e1 + m⊥) E [X| Ft] , (3.5) ρ (Q,m⊥)∈Wt (e1)

∞ d ρ for every X ∈ L (R ) where Wt (e1) := {(Q,m⊥) | αt(Q,m⊥) = 0 P-a.s.}. ∞ d Proof. 1. Assume the dual representation (3.4) holds. Let X ∈ L (R ) and (Xn)n∈N ⊆ ∞ d L (R ) be a bounded sequence such that X = limn→∞ Xn almost surely. There- T T 1 1 d fore, Y Xn → Y X a.s. and in L (R) for any Y ∈ L (R ) (by the dominated con- T T T T vergence theorem). Thus E −wt (Q, e1 + m⊥) Xn Ft → E −wt (Q, e1 + m⊥) X Ft a.s. and in L1(R) for any (Q,m ) ∈ W (e ) as well. Thus we can see (as in the ⊥ t 1 static setting, e.g., in [39])

T Q ρt(X)= esssup −αt(Q,m⊥) + (e1 + m⊥) E [−X| Ft] (Q,m⊥)∈Wt(e1) T Q = esssup −αt(Q,m⊥) + lim (e1 + m⊥) E [−Xn| Ft] n→∞ (Q,m⊥)∈Wt(e1) T Q = esssup lim −αt(Q,m⊥) + (e1 + m⊥) E [−Xn| Ft] n→∞ (Q,m⊥)∈Wt(e1) T Q ≤ lim inf ess sup −αt(Q,m⊥) + (e1 + m⊥) E [−Xn| Ft] n→∞ (Q,m⊥)∈Wt(e1) = lim inf ρt(Xn). n→∞

2. Assume that the “Fatou property” is satisfied. To prove the dual representation (3.4) holds, we only need to prove that E [ρt(·)] is proper and lower semicontinuous from the same logic as in Proposition A.1.7 of [33]. This modification to the conditions in [33, Proposition A.1.7] can be accomplished because the requirements on the set-valued risk measures in that result are utilized, via [33, Proposition A.1.1] and its proof, to guarantee that E [ρt(·)] is proper and lower semicontin- ∞ ∞ d uous. By ρt(X) ∈ Lt (R) for every X ∈ L (R ), we can conclude E [ρt(·)] ∞ d is proper. Define Cz := Z ∈ L (R ) | E [ρt(Z)] ≤ z for any z ∈ R. Define Cr := C ∩ Z ∈ L∞(Rd) | kZk ≤ r for any r > 0 (in this case we will take z z ∞ kZk := esssupmax |Z |). Take (X ) ⊆ Cr converging to X ∈ L∞(Rd) ∞ i=1,...,d i n n∈N z in probability, and there exists a bounded subsequence (Xnm )m∈N which converges

to X almost surely. Therefore ρt(X) ≤ lim infm→∞ ρt(Xnm ), and thus (by Fa- r r tou’s lemma) X ∈ Cz . That is Cz is closed in probability for any r > 0. By [53,

10 ∞ d Proposition 5.5.1], Cz is weak* closed. Now consider a net (Xi)i∈I ⊆ L (R ) ∞ d which converges to X ∈ L (R ) in the weak* topology. Let zǫ := E [ρt(X)] − ǫ c ∞ d for any ǫ> 0. Then (Czǫ ) = Z ∈ L (R ) | E [ρt(Z)] > zǫ is an open neighbor- hood of X for any ǫ> 0. Since Xi → X, for any ǫ> 0 there exists a jǫ ∈ I such c that for any i ≥ jǫ we have Xi ∈ (Czǫ ) , i.e., E [ρt(Xi)] > E [ρt(X)] − ǫ. Therefore lim infi∈I E [ρt(Xi)] ≥ E [ρt(X)], and the proof is complete. Finally, the coherent case follows identically to Corollary 2.5 of [37].

Remark 3.7. For the reader who is interested in the relation to set-valued risk measures, let us note that Proposition A.1.3 of [33] provides a suﬃcient condition on the underlying set-valued risk measure Rt (with any choice of eligible space M˜ ⊇ M) to guarantee that ρt satisﬁes the Fatou property.

With the dual representation from Proposition 3.6, we construct a second dual representation with a single probability measure and with respect to vectors of consistent prices. This is a generalization of the dual representation of the superhedging risk measure from [50]. This new dual representation is valuable for two primary reasons: it provides a clear interpretation as the supremum over all frictionless pricing processes consistent with the market model and, as a consequence, it allows for a decomposition of the risk measure into frictionless markets that allows for our new time consistency notion discussed in Section 4. For this representation, recall the deﬁnition of k · kKT ,0 from (3.3).

Theorem 3.8. Let ρt be a scalarized conditional convex risk measure. Then ρt satisﬁes the Fatou property if and only if for any X ∈ L∞(Rd)

Q T ρt(X) = esssup −βt(Q,S) − E ST X Ft (3.6) (Q,S)∈Qt h i where Q T βt(Q,S) = esssup −E ST Z Ft Z∈At h i and

T ∀s = t, ..., T, ∀i = 1, ..., d : T ∞ d Qt = (Q, (Ss)s=t) ∈M× Ls (R ) Ss,1 ≡ 1, kSs,ik∞ ≤ max{keikKT ,0, 1}, .  Q  s=t S = EQ [S | F ] , d S ∈ K+  Y s T s dP T T 

Proof. By Proposition 3.6 the result trivially follows if 

1. for every (R,m⊥) ∈Wt(e1) with E [αt(R,m⊥)] < +∞ there exists a (Q,S) ∈ Qt s ¯ such that wt (R, e1 + m⊥)= ξt,s(Q)Ss for every time s ≥ t, and s 2. for every (Q,S) ∈ Qt there exists a (R,m⊥) ∈Wt(e1) such that wt (R, e1 +m⊥)= ξ¯t,s(Q)Ss for every time s.

Let us show that the first property holds. Let (R,m⊥) ∈Wt(e1) with E [αt(R,m⊥)] < s T +∞. Let Zs := wt (R, e1 + m⊥); note that (Zs)s=t is a P-martingale. Then we will de- Zs,i on {Z > 0} Zs,1 s,1 fine Ss by Ss,i = for all indices i = 1, ..., d. Note that Ss,1 ≡ 1 for (1 else all times s and S = e + m ∈ L∞(Rd ). Further, define Q ∈ M by dQ = ZT,1 = Z t 1 ⊥ t + dP Zt,1 T,1

11 (since Zt = e1 + m⊥). Notice that {Zs,1 = 0} ⊆ {ZT,1 = 0} and by property of Kt and + Assumption 3.3 we have that {ZT,1 = 0} = {ZT,i = 0} (as ZT ∈ Kt ) for any choice of index i, therefore we can see that

dQ Q dP E E dQ dQ [ST,i| Fs]= 1{E F >0} ST,i + 1{E F =0}ST,i Fs  [ dP | s] dQ [ dP | s]  E Fs dP   ZT,1 h i ZT,i 1{Z 1>0} 1{Z >0} + 1{Z =0} s, Zs,1 T, 1 ZT,1 T,1 = E Fs  +1 1 ZT,i + 1  {Zs,1=0} {ZT,1>0} ZT,1 {ZT,1=0}

 ZT,i ZT,1 1 Z > 1 Z > + 1 Z > 1 Z E { s,1 0} { T,1 0} Zs,1 { s,1 0} { T,1=0} Zs,1 = ZT,i Fs " +1{Zs,1=0}1{ZT,1>0} + 1{Zs,1=0}1{ZT,1=0} # ZT,1 1 E Z = 1{Zs,1>0} 1{ZT,1>0} T,i Fs + 1{Zs,1=0} Zs,1 h i 1 = 1{Zs,1>0} E 1{ZT,i>0}ZT,i Fs + 1{Zs,1=0} Zs,1 h i 1 = 1{Zs,1>0} E [ZT,i| Fs] + 1{Z s,1=0} Zs,1 Zs,i = 1{Zs,1>0} + 1{Zs,1=0} = Ss,i Zs,1 for any s ∈ {t, ..., T } and any i = 1, ..., d. Finally

Z ξ¯ (Q)S = Z 1 s + 1 = 1 Z t,s s s,1 {Zs,1>0} Z {Zs,1=0} {Zs,1>0} s s,1 s R K+ = 1{ξ¯t,s(R1)>0}wt ( , e1 + m⊥) ∈ t .

∞ d It remains to show that ST ∈ L (R+) (the case for s < T would then follow trivially): by definition keikKT ,0e1 ≥KT ei ≥KT −keikKT ,0e1 for any index i and, by the definition dQ dQ dQ of Kt, we know keikKT ,0 < ∞. This implies dP keikKT ,0 ≥ dP ST,i ≥ − dP keikKT ,0 + + 1 d T ∞ (noting that Kt ⊆ KT = Z ∈ L (R ) | Z K ≥ 0 P-a.s. ∀K ∈ L (KT ) ), by dividing by dQ on the set { dQ > 0} recovers S ∈ L∞(Ω, F , Q; R) with kS kQ ≤ ke k dP dP T,i s T,i ∞ i KT ,0 (i.e., the ∞-norm of ST,i under measure Q is bounded). Finally, by construction dQ ST,i = 1 on { dP = 0}, thus kST,ik∞ ≤ max{keikKT ,0, 1}. Since S is a Q-martingale, the bound kSs,ik∞ ≤ max{keikKT ,0, 1} holds as well. To show the converse, i.e., the second property, let (Q,S) ∈ Qt. Define m⊥ = S dQ T,i on {S > 0} 1 d−1 dRi dP St,i t,i s St − e1 ∈ {0}× Lt (R ). And define dP = . Then wt (R, e1 + (1 else S m ) = S E dRi F = S E dQ T,i 1 + 1 F = E dQ S F = ⊥ i t,i dP s t,i dP St,i {St,i>0} {St,i=0} s dP T,i s ξ¯ (Q)EQ [S h| F ] = iξ¯ (Q)S h for every index i = 1, ..., d i(becauseh {S = 0}i ⊆ t,s T,i s t,s s,i t,i dQ dP ST,i = 0 by S a Q-martingale). n o Remark 3.9. For every choice of dual variables (Q,S) ∈ Qt we find that the dual norm k dQ S k∗ := esssup{|EQ STX F | | kXk ≤ 1 P-a.s.} = 1 almost surely. dP T KT ,t T t KT ,t

12 Corollary 3.10. Let ρt be a scalarized conditional coherent risk measure. Then ρt satisﬁes the Fatou property if and only if for any X ∈ L∞(Rd)

Q T ρt(X) = esssup −E ST X Ft (Q,S)∈Qρ t h i

ρ where Qt = {(Q,S) ∈ Qt | βt(Q,S) = 0 P-a.s.}. Proof. This follows directly from Theorem 3.8 and Proposition 3.6.

Remark 3.11. The d dimensional version of the dual representation of the scalar superhedging price (1.1) given in Jouini, Kallal [50] is with respect to dual variables 0 d (Q,S) ∈M× L (R+) with Ss ≡ 1 for every s, Q is a martingale measure for S, and dQ + dP ST ∈ K0 . This set of dual variables is a superset of Qt from Theorem 3.8. From the proof of Theorem 3.8, it is clear that this larger set of dual variables could be used within this work as well. However, for reasons that will be clear in subsequent results, e.g., Proposition 4.7, which is used extensively after, we restrict ourselves to the bounded prices as deﬁned in Qt.

3.3 Relevance In the below proposition we will deﬁne a property so that the dual representation (3.6) can be deﬁned w.r.t. the dual variables

e e Qt := {(Q,S) ∈ Qt | Q ∈ M }, i.e., the set of equivalent martingale measures and their associated price processes S. We will then demonstrate that this necessary and sufficient condition is satisfied under a version of relevance or sensitivity (see the usual definition in, e.g., [37]).

e e Remark 3.12. Before continuing, note that (Q,S) ∈ Qt if and only if Q ∈ M is a T ∞ + martingale measure for S and S ∈ s=t Ls (Ks ) such that Ss,1 ≡ 1 and kSs,ik∞ ≤ dQ + max{keikK , , 1}. This is in contrast to the deﬁnition of Qt in which ST ∈ K . This T 0 Q dP t modiﬁcation can be accomplished as we are now only using equivalent measures Q and + T 1 d T ∞ since Kt = s=t Y ∈ L (R ) | E [Y | Fs] ks ≥ 0 P-a.s. ∀ks ∈ Ls (Ks) . n o PropositionT 3.13. Let ρt be a scalarized conditional convex risk measure satisfying the Fatou property. Then

Q T ρt(X) = esssup −βt(Q,S) − E ST X Ft (3.7) (Q,S)∈Qe t h i

∞ d Q T e for every X ∈ L (R ) if and only if infZ∈At E ST Z > −∞ for some (Q,S) ∈ Qt . Q T ∞ d Proof. If ρ (X) = esssup e −β (Q,S) − E S X F for every X ∈ L (R ) t (Q,S)∈Qt t T t dQ T Q e then, by Ft-decomposability of { dP ST Z | ( ,S) ∈ Qt ,Z ∈ At} (recall from Section 2 that a set B is called Ft-decomposable if 1DB + 1Dc B ⊆ B for any D ∈ Ft) and the dQ T definition of the penalty function β , E [ρ (0)] = sup e inf E S Z > −∞ t t (Q,S)∈Qt Z∈At dP T ∞ e by ρt(0) ∈ Lt (R). In particular this implies that there exists some (hQ,S) ∈i Qt such Q T that infZ∈At E ST Z > −∞. 13 ˜ ˜ e Q˜ ˜T e Conversely, let (Q, S) ∈ Qt such that infZ∈At E ST Z > −∞. Since Qt ⊆ ∞ d Qt, the dual representation in Theorem 3.8 implies thath fori every X ∈ L (R ), Q T ρ (X) ≥ ess sup e −β (Q,S) − E S X F . As in the proof of Proposition t (Q,S)∈Qt t T t A.1.7 of [33] we will show the reverse inequality by considering the expectation. Since Q T {−βt(Q,S) − E ST X Ft | (Q,S) ∈ Qt} is Ft-decomposable we are able to in- terchange the expectation and essential supremum. As done in [37, Lemma 3.5], for ˜ ˜ e any ǫ ∈ (0, 1) and (Q,S ) ∈ Qt, define (Qǫ,Sǫ) := (1 − ǫ)(Q,S) + ǫ(Q, S) ∈ Qt . Additionally, by definition of the penalty function, we can conclude βt(Qǫ,Sǫ) ≤ ∞ d (1 − ǫ)βt(Q,S)+ ǫβt(Q˜ , S˜). Choose any X ∈ L (R )

dQ T E [ρ (X)]= sup E −β (Q,S) − S X t t dP T (Q,S)∈Qt dQ T ≥ sup E −βt(Q,S) − ST X Q e dP ( ,S)∈Qt dQ T ≥ sup E −β (Q ,S ) − ǫ S X t ǫ ǫ dP ǫ,T (Q,S)∈Qt dQ T ≥ sup (1 − ǫ)E −β (Q,S) − S X t dP T (Q,S)∈Qt Q˜ ˜ ˜ d ˜T +ǫE −βt(Q, S) − ST X " dP #! Q˜ ˜ ˜ d ˜T = (1 − ǫ)E [ρt(X)] + ǫE −βt(Q, S) − ST X . " dP #

Since E [ρt(X)] ∈ R then we take the limit as ǫ tends to 0 to recover that the ﬁnal line above is equivalent to the ﬁrst line, and the result is shown.

Corollary 3.14. Let ρt be a scalarized conditional coherent risk measure satisfying the Fatou property. Then

Q T ρt(X)= esssup −E ST X Ft (3.8) (Q,S)∈Qρ,e t h i

∞ d ρ,e e ρ for every X ∈ L (R ) if and only if Qt := Qt ∩ Qt 6= ∅, i.e., there exists some e (Q,S) ∈ Qt such that βt(Q,S) = 0 almost surely.

Proof. Recall from the logic of the proof of Corollary 2.5 of [37], for any (Q,S) ∈ Qt ρ we have that E [βt(Q,S)] < +∞ if and only if (Q,S) ∈ Qt . The results follows from Q T Proposition 3.13 above. If ρt(X) = esssup ρ,e −E S X Ft then there exists (Q,S)∈Qt T some (Q,S) ∈ Qe such that inf EQ STZ > −∞ (else E [ρ (·)] ≡ −∞). Conversely, t Z∈At T t if inf E STZ > −∞ for some (Q,S) ∈ Qe then we have Z∈At T t Q T ρt(X)= esssup −βt(Q,S) − E ST X Ft (Q,S)∈Qe t h i E[βt(Q,S)]<+∞ Q T = esssup −E ST X Ft . (Q,S)∈Qρ,e t h i

14 As in the literature for scalar risk measures, we will now introduce relevance (or e sensitivity) and demonstrate that it implies the existence of (Q,S) ∈ Qt such that Q T infZ∈At E ST Z > −∞. See, e.g., [37, 18, 55] for previous literature on relevance of univariate conditional risk measures.

Deﬁnition 3.15. A risk measure ρt is called relevant if

P(ρt(−ǫ1De1) > ρt(0)) > 0 for every ǫ> 0 and every D ∈ F with P(D) > 0.

Lemma 3.16. Let ρt be a scalarized conditional convex risk measure satisfying the Fa- e Q T tou property. If ρt is relevant then there exists (Q,S) ∈ Qt such that infZ∈At E ST Z > −∞.

Proof. First, {βt(Q,S) | (Q,S) ∈ Qt} is Ft-decomposable (by the deﬁnition of Qt and

βt). Therefore R ∋ E [−ρt(0)] = E ess inf(Q,S)∈Qt βt(Q,S) = inf(Q,S)∈Qt E [βt(Q,S)]. For any ǫ> 0 let the set Qǫ ⊆ Q be deﬁned by t t ǫ Qt := {(Q,S) ∈ Qt | βt(Q,S) < −ρt(0)+ ǫ P-a.s.} .

Now we will show that for any ǫ > 0 there exists a pair of dual variables (Q˜ , S˜) ∈ e ǫ Q˜ ˜T Qt ∩ Qt (and in particular this implies infZ∈At E ST Z > −∞). ǫ h i 1. We will show that Qt 6= ∅ for any ǫ > 0. Let (Qn,Sn)n∈N ⊆ Qt such that βt(Qn,Sn) ց−ρt(0) almost surely. This sequence exists because {βt(Q,S) | (Q,S) ∈ Qt} is Ft-decomposable, thus there exists a sequence (Qn,Sn)n∈N ⊆ Qt such that

βt(Qn,Sn) ց ess inf βt(Q,S)= −ρt(0) P-a.s. (Q,S)∈Qt

Deﬁne the Ft-measurable random variable τǫ by

τǫ := min {n | βt(Qn,Sn) < −ρt(0)+ ǫ} .

It holds τǫ < +∞ almost surely by construction of the sequence (Qn,Sn)n∈N, and thus ({τǫ = n})n∈N ⊆ Ft defines a partition of Ω. We can define (Q¯ ǫ, S¯ǫ) := ∞ ¯ ¯ ∞ n=1 1{τǫ=n}(Qn,Sn) ∈ Qt (trivially) with βt(Qǫ, Sǫ) ≤ n=1 1{τǫ=n}βt(Qn,Sn) < ǫ −ρt(0)+ ǫ, i.e., (Q¯ ǫ, S¯ǫ) ∈ Q . P t P 2. We will show that if ρt is relevant then for every ǫ > 0 there exists an element ˜ ˜ ǫ dQ˜ dQ ǫ (Q, S) ∈ Qt such that dP > 0 a.s. Fix ǫ> 0 and let cǫ := sup P( dP > 0) | (Q,S) ∈ Qt . ǫ dQnn o Take a sequence (Qn,Sn)n∈N ⊆ Qt such that limn→∞ P( dP > 0) = cǫ. Define ˜ ˜ ∞ 1 ˜ ˜ ǫ (Q, S) := n=1 2n (Qn,Sn). It holds (Q, S) ∈ Qt by convexity of βt. In fact, ˜ ˜ dQ ∞ dQn P dQ dP > 0 P= ∪n=1 dP > 0 . Therefore ( dP > 0) = cǫ. It remains to show e thatn cǫ =o 1, and thusn Q˜ ∈ Mo. dQ˜ Assume cǫ < 1, and let D := dP = 0 . Relevance implies n o P(ρt(−ǫ1De1) > ρt(0)) > 0,

15 and by construction we have

Q ρt(−ǫ1De1) = esssup −βt(Q,S)+ ǫE [1D| Ft] . (Q,S)∈Qt Therefore, there exists (Q¯ , S¯) ∈ Qt such that the set

Q¯ B := βt(Q¯ , S¯) < −ρt(0)+ ǫE [1D| Ft] ∈ Ft n o has positive probability, i.e., βt(Q¯ , S¯) < −ρt(0)+ ǫ on B. ˆ ˆ ǫ ¯ ¯ ǫ Now we will deﬁne (Q, S) ∈ Qt which is equal to (Q, S) on B. Let (Q,S) ∈ Qt ˆ ˆ ¯ ¯ ǫ ˆ ˆ (arbitrary) and let (Q, S) := 1B (Q, S) + 1Bc (Q,S) ∈ Qt (because βt(Q, S) = 1B βt(Q¯ , S¯) + 1Bc βt(Q,S)). By deﬁnition of B and since βt(Q¯ , S¯) ≥ −ρt(0) (by the dual representation of ρt(0)), dQˆ dQˆ E 1B1D = E 1BE 1D Ft > 0. " dP # " " dP ##

ˆ ˆ P dQ > D P dQ > D B > This implies that dP 0 ∩ ≥ dP 0 ∩ ∩ 0. Therefore 1 ˜ ˜ 1 ˆ ˆ nǫ o ǫn o 2 (Q, S)+ 2 (Q, S) ∈ Qt by convexity of Qt, and we have found a contradiction.

We now show that, in the coherent case, relevance is actually necessary and suﬃcient for a dual representation as in (3.8).

Proposition 3.17. Let ρt be a scalarized conditional coherent risk measure satisfying ∞ d the Fatou property. ρt is relevant if and only if (3.8) holds for every X ∈ L (R ).

Proof. As shown above in Corollary 3.14 and Lemma 3.16, if ρt is relevant then (3.8) ∞ d holds for every X ∈ L (R ). Conversely assume that ρt has dual representation with ρ,e respect to the dual variables Qt and assume ρt is not relevant, i.e., there exists some ǫ > 0 and D ∈ F with positive probability such that ρt(−ǫ1De1) ≤ ρt(0) = 0 almost Q ρ,e surely. In particular this implies ǫE [1D| Ft] ≤ ρt(−ǫ1De1) ≤ 0 for any (Q,S) ∈ Qt . ρ,e Therefore ǫQ(D) ≤ 0 for every (Q,S) ∈ Qt . However, this can only be true if Q(D) = 0, which implies P(D) = 0 since Q ∈ Me. This is a contradiction.

4 Time consistency

In this section we will study a modiﬁed version of the traditional time consistency property. Typically, as studied in e.g., [48], the time consistency of scalarized risk measures is considered for ρ directly, i.e.,

ρs(X) ≤ ρs(Y ) ⇒ ρt(X) ≤ ρt(Y ) (4.1) for X,Y ∈ L∞(Rd) and t ≤ s. As we will demonstrate in Section 5.1 the superhedging portfolios do not satisfy this ρ-time consistency. Instead we will consider a decomposition of the scalarizations via their dual representations as deﬁned in Theorem 3.8. These details are provided in Section 4.2. Then in Section 4.4, we consider a new time consistency property. In particular, with this new property we determine the typical equivalence of time consistency and a recursive relation.

16 Assumption 4.1. For the remainder of this paper we will assume that ρt is a relevant scalarized conditional convex risk measure satisfying the Fatou property, and corresponds with the acceptance set At as given in (3.2).

4.1 Overview Before continuing to the main results of this work in regards to time consistency of the dynamic multivariate risk measures ρ, we will provide a brief overview over the topic, illustrate the importance of the derived results, set them in relation to existing results in the literature, and provide an economic interpretation. As mentioned previously, the usual time consistency property, ρ-time consistency, is in general not satisﬁed for dynamic multivariate risk measures ρ given by (3.1). In particular, it does not hold for the prototypical case of the superhedging risk measure as detailed in, e.g., Section 5.1 below. This is because of the frictions inherent to the risk measure in covering all risks with the cash asset alone, i.e., by considering a market with transaction costs. In contrast, many prior works (e.g., [29, 31]) on time consistency of multivariate risk measures have considered set-valued risk measures and a property called multiportfolio time consistency. A set-valued risk measure is said to be multiportfolio time consistent if

Rs(X) ⊇ Rs(Y ) ⇒ Rt(X) ⊇ Rt(Y ) (4.2) X X [∈X [∈X for all X ⊆ L∞(Rd), Y ∈ L∞(Rd), and t ≤ s. This property relies on the choice of eli- ∞ d gible assets M˜ taken in the image space for the set-valued risk measure Rt : L (R ) → G(M˜t; M˜ t,+). For instance, the set-valued superhedging risk measure is multiportfolio time consistent w.r.t. the full space of eligible assets (i.e., when one chooses M˜ = Rd) but not with the cash asset only (i.e., when M˜ = M = R × {0}d−1 as assumed within this paper), see Section 5.1 below for details. As with ρ-time consistency, this is due to the frictions inherent to the risk measure when restricting to covering risk with only the cash asset. However, in Section 4.2, we will introduce a decomposition of the conditional risk measure ρt into a collection of multivariate risk measures in frictionless markets, each of which is defined by a price process satisfying a no-arbitrage condition. This decomposition allows us to recover the desired risk measure as the pointwise supremum over T this collection of frictionless risk measures. Given a price process S = (St)t=0 ∈ S (defined in (4.3) below), we denote these frictionless multivariate risk measures as πS. The relationship between these frictionless multivariate risk measures and the typical (univariate) risk measures of, e.g., [9, 25, 37, 1], is presented in Section 4.3. These new risk measures allow us to define a new notion of time consistency, called π-time consistency and provided in Definition 4.13 below, which overcomes the obstacles related to market frictions in both ρ-time consistency and multiportfolio time consistency. Briefly, if every no-arbitrage price S consistent with the market model K induces a risk measure πS that is time consistent in the usual way, then we say the original risk measure ρ is π-time consistent. Notably, the superhedging risk measure is π-time consistent. Results on π-time consistency and its relation to multiportfolio time consistency of the underlying set-valued risk measure (appearing in (3.1) and with possibly different choices of eligible assets) are provided in Sections 4.4 and 4.5. Though the proposed

17 time consistency notion is weaker than ρ-time consistency, it is stronger than the weak notion of ρ-time consistency (i.e., where ρs(X) ≤ ρs(0) implies ρt(X) ≤ ρt(0) for every portfolio X ∈ L∞(Rd)) as studied for univariate risk measures in, e.g., [75]. Section 4.5 also contains a result on a backward recursion for π-time consistent scalar multivariate S risk measures ρt with respect to the underlying risk measure π .

4.2 Preliminaries For notational simplicity we will introduce the set of eligible pricing processes:

T ∞ + ∀t = 0, ..., T, ∀i ∈ {2, 3, ..., d} : S := S ∈ Lt (Kt ) . (4.3) St,1 = 1, kSt,ik∞ ≤ max{keikK ,0, 1} ( t=0 T ) Y

In the following deﬁnition we introduce a decomposition of the risk measures ρt. This section will introduce some simple properties of these functions, which will then be used to deﬁne a new property for time consistency.

S ∞ d 0 Deﬁnition 4.2. Let S ∈ S. Deﬁne πt : L (R ) → Lt (R) ∪ {−∞} by

S Q T πt (X) := esssup −βt(Q,S) − E ST X Ft (4.4) Q∈Qe(S) h i

∞ d for any X ∈ L (R ) where βt is defined as in Theorem 3.8 dependent on the acceptance set At and e e Q (S) := {Q ∈ M | (Q,S) ∈ Q0} . S 0 ∞ Note that πt (X) ∈ Lt (R)∪{−∞} in general even though ρt(X) ∈ Lt (R). Though S πt (X) is trivially bounded from above by ρt(X), it is not integrable in general. Propo- S ∞ ∞ d sition 4.7 provides a simple condition so that πt (X) ∈ Lt (R) for any X ∈ L (R ). Definition 4.3. A sequence of prices S ∈ S satisfies the no-arbitrage condition if

T Γs(Ss) ∩ [−ΓT (ST )] ⊆ ΓT (ST ) (NA) "s # X=0 ∞ d T where Γt(St) := {Z ∈ Lt (R ) | St Z ≥ 0 a.s.}. The motivation for calling this property “no-arbitrage” comes from the following proposition which relates the condition to the fundamental theorem of asset pricing. Proposition 4.4. Let S ∈ S, then S satisfies the no-arbitrage condition if and only if Qe(S) 6= ∅. Proof. This follows directly from the fundamental theorem of asset pricing by noting that our definition of no-arbitrage is equivalent to the usual one in frictionless markets. This can be seen by: T T 1. A predictable process (ηt)t=0 is self-financing if and only if St (ηt+1 − ηt) ≤ 0 almost surely for every time t = 0, 1, ..., T − 1 with η0 = 0. Equivalently this can be written as ηt+1 − ηt ∈ −Γt(St) for every time t = 0, 1, ..., T − 1 with η0 = 0. Through summations, ηt is attainable through self-financing trading strategy if t−1 and only if ηt ∈− s=0 Γs(Ss). P 18 T 2. The wealth of a portfolio ηt at time t is given by Vt := St ηt. Therefore, in our notation, wealth satisfies the conditions Vt ≥ 0 and P(Vt > 0) > 0 if and only if ηt ∈ Γt(St) and ηt 6∈ −Γt(St) respectively.

Therefore it is immediately clear that (NA) is equivalent to stating that V0 = 0, VT ≥ 0 almost surely implies VT = 0 almost surely. Finally, we can replace the self-ﬁnancing T −1 condition − s=0 Γs(Ss) with the summation up to time T as these are equivalent conditions: P T 1. Assume s=0 Γs(Ss) ∩ [−ΓT (ST )] ⊆ ΓT (ST ) then immediately by 0 ∈ ΓT (ST ) h T −i1 we can concludeP s=0 Γs(Ss) ∩ [−ΓT (ST )] ⊆ ΓT (ST ). hP T −1 i T 2. Conversely, assume s=0 Γs(Ss) ∩[−ΓT (ST )] ⊆ ΓT (ST ) and let X ∈ s=0 Γs(Ss) ∩ T T [−ΓT (ST )]. We canh decomposeP Xi= s=0 Xs where Ss Xs ≥ 0 almosth surelyP and i STX ≤ 0 almost surely. Then we can easily see that T P T −1 T −1 T T T T 0 ≥ ST X = ST Xs + ST XT ≥ ST Xs. (4.5) s=0 s=0 X X T −1 T −1 T T −1 Therefore s=0 Xs ∈ s=0 Γs(Ss) ∩[−ΓT (ST )] ⊆ ΓT (ST ), i.e., ST s=0 Xs = T 0 almost surely.P Thish immediatelyP impliesi that ST X = 0 almost surelyP as well from (4.5), i.e., X ∈ ΓT (ST ).

Proposition 4.5. Let S ∈ S then the following are equivalent: S ∞ 1. πt (0) ∈ Lt (R), e ∞ 2. there exists some Q ∈ Q (S) such that βt(Q,S) ∈ Lt (R), and ∞ d ˜π 3. S satisfies (NA) and there exists some X ∈ Lt (R ) such that P X ∈ At = 0, π ∞ d S T ˜π where At := X ∈ L (R ) | πt (X) ≤ 0 = cl At + s=t Γs(Ss) and At is a π ∞ d ∞ ˜π Ft-measurable random set such that At ∩ Lt (R )= LPt (At ). Proof. First we will show that (1.) is equivalent to (2.), then we will show this is equivalent to (3.). e ∞ 1. First we will assume there exists some Q ∈ Q (S) such that βt(Q,S) ∈ Lt (R). For such a Q and for ρt defined by the same acceptance set At, ∞ S ∞ Lt (R) ∋−βt(Q,S) ≤ πt (0) ≤ ρt(0) ∈ Lt (R). Now we wish to prove the converse. We can trivially deduce a lower bound S ∞ e βt(Q,S) ≥ −πt (0) ∈ Lt (R) for every Q ∈ Q (S). To prove the upper bound e we first note that {βt(Q,S) | Q ∈ Q (S)} is an Ft-decomposable set. This e S implies there exists a sequence (Qn)n∈N ⊆ Q (S) such that −βt(Qn,S) ր πt (0) ǫ S almost surely. Let us define τ := min{n ∈ N | − βt(Qn,S)+ ǫ>πt (0)} < ∞ ˜ ∞ e almost surely. Then Q := n=1 1{τ ǫ=n}Qn ∈ Q (S) (trivially utilizing monotone convergence) and P ∞ ˜ S ∞ βt(Q,S) ≤ 1{τ ǫ=n}βt(Qn,S) < −πt (0)+ ǫ ∈ Lt (R). n X=1 19 ˜π π 2. First, At exists by [63, Theorem 2.1.6] since At is a nonempty, closed, and Ft- decomposable. ˜π Given the existence of At , first we will assume that S satisfies (NA) and there ∞ d ˜π ∞ d exists some X ∈ Lt (R ) such that P X ∈ At = 0. Fix X ∈ Lt (R ) such that ˜π e e P X ∈ At = 0, then there exists some Q ∈ Q (S) (where Q (S) 6= ∅ by Propo- Q T T ∞ sition 4.4) such that β(Q,S) = esssupZ∈At −E ST Z Ft < −St X ∈ Lt (R) by a conditional separating hyperplane argument. Additionally, βt(Q,S) ≥−ρt(0) ∈ ∞ e e Lt (R) for any Q ∈ Q (S) by construction. Conversely, Q (S) 6= ∅ implies the Q T prices S satisfy (NA) by Proposition 4.4. And given βt(Q,S) = esssupZ∈At −E ST Z Ft ∈ L∞(R), we can immediately conclude that − (β (Q,S)+ ǫ) e ∈ L∞(Rd) for any t t 1 t ˜π ǫ> 0 and P − (βt(Q,S)+ ǫ) e1 ∈ At = 0.

S Proposition 4.6. Fix S ∈ S satisfying (NA), the following properties hold for πt : S S T ∞ d 1. Conditional cash invariance: πt (X + m) = πt (X) − St m for any X ∈ L (R ) ∞ d and m ∈ Lt (R ); S S T T 2. Monotonicity: πt (X) ≤ πt (Y ) if ST X ≥ ST Y almost surely; S S S 3. Conditional convexity: πt (λX + (1 − λ)Y ) ≤ λπt (X) + (1 − λ)πt (Y ) for every ∞ λ ∈ Lt ([0, 1]); S S T ∞ d 4. Value consistent: πt (X)= πt ([ST X]e1) for any X ∈ L (R ). S If At is additionally a conditional cone then πt is conditional positive homogeneous. S Q e Proof. Trivially from the deﬁnition of πt and that St = E [ST | Ft] for any Q ∈ Q (S) and ST,1 = 1 almost surely.

S The following proposition provides a necessary and suﬃcient condition for πt (X) ∈ ∞ ∞ d Lt (R) for every X ∈ L (R ). Proposition 4.5 relates this condition to the no-arbitrage condition (NA) for the frictionless prices S.

∞ d S ∞ Proposition 4.7. Let S ∈ S and X ∈ L (R ) then, for any time t, πt (X) ∈ Lt (R) S ∞ if and only if πt (0) ∈ Lt (R) for any time t. T S ∞ Proof. Let kXk = (kX1k∞,..., kXdk∞) and assume πt (0) ∈ Lt (R).

S S S T ∞ πt (X) ≤ πt (−kXk)= πt (0)+ St kXk∈ Lt (R) S S S T ∞ πt (X) ≥ πt (kXk)= πt (0) − St kXk∈ Lt (R). The converse direction is trivial from the same inequalities.

For the remainder of this work, we will use the notation

S ∞ St := {S ∈ S | πt (0) ∈ Lt (R)}, i.e., S ∈ St if and only if S satisﬁes the no-arbitrage condition (NA) and there exists ∞ d ˜π some X ∈ L (R ) such that P X ∈ At = 0. The following proposition shows the decomposition of the dual representation (3.6) S of ρt into two components based on the two dual variables (Q,S). πt (X), as deﬁned in

20 (4.4), runs over the dual variables Q only. But, it is shown in Proposition 4.8 below, it is enough to take the essential supremum with respect to S over the set St only. This S means it is enough to consider in the dual representation of ρt just those πt (X) that ∞ map into Lt (R), i.e., to consider those frictionless prices that satisfy the no-arbitrage condition (NA). This will be of particular importance when considering the new time consistency concept presented in Sections 4.4 and 4.5.

Proposition 4.8. For any time t and any X ∈ L∞(Rd)

S ρt(X) = esssup πt (X). S∈St

e t T e dQt ¯ Proof. Let S ∈ S then Q ∈ Q (S) implies (Q , (Ss)s=t) ∈ Qt where dP := ξt,T (Q). By the definition of Qt and the P-almost sure definition of the conditional expectation, Q T t Qt T we recover that −βt(Q,S) − E ST X Ft = −βt(Q ,S) − E ST X Ft for any X ∈ L∞(Rd). Conversely, let (Q, (S )T ) ∈ Qe and define Sˆ ∈ S by Sˆ := EQ [S | F ] = s s=t t s T s

Ss if s ≥ t . Immediately it is clear that Q ∈ Qe(Sˆ). By deﬁnition of Sˆ, (E [St| Fs] if s

Q T ρt(X) = esssup −βt(Q,S) − E ST X Ft (Q,S)∈Qe t h i Q T = esssup ess sup −βt(Q,S) − E ST X Ft S∈S Q∈Qe(S) h i S = esssup πt (X). S∈S

From this and Proposition 4.5 we immediately ﬁnd

S Q T ρt(X) ≥ ess sup πt (X) ≥ ess sup −βt(Q,S) − E ST X Ft . S∈S (Q,S)∈Qe ∞ t πS(0)∈L (R) Q ∞ R h i t t βt( ,S)∈Lt ( )

Q T We now wish to show the reverse inequality. Since {−βt(Q,S)−E ST X Ft | (Q,S) ∈ Qe} is F -decomposable we can ﬁnd a sequence (Q ,S ) ⊆ Qe such that −β (Q ,S )− t t n n n∈N t t n n Qn T ∞ ǫ E Sn,T X Ft ր ρt(X) ∈ Lt (R) almost surely. Deﬁne τ := min{n ∈ N | βt(Qn,Sn)+ T Qn h i ˜ ˜ E Sn,T X Ft < −ρt(X)+ ǫ} < ∞ almost surely for any ǫ > 0, and let (Q, S) := ∞ e h 1 ǫ (Qin,Sn) ∈ Q (trivially using monotone convergence). Then n=1 {τ =n } t

P ∞ ˜ ˜ Q˜ ˜T ∞ Lt (R) ∋−ρt(0) ≤ βt(Q, S) ≤−ρt(X) − E ST X Ft + ǫ ∈ Lt (R). h i ˜ ˜ e Since such a (Q, S) ∈ Qt can be found for any ǫ> 0, a sequence of those can be used to get convergence as ǫ → 0 and the proof is complete.

21 4.3 Relation to univariate scalar risk measures S T In this section we wish to relate the terms πt t=0 for a ﬁxed S ∈ S introduced in the prior section to the univariate dynamic risk measures as studied in, e.g., [9, 25, 37, 1]. S T This relationship will then be used later to directly prove some properties for πt t=0. S ∞ ∞ Proposition 4.9. Let S ∈ St, then φt : L (R) → Lt (R) deﬁned by

S S φt (Z) := πt (Ze1) (4.6) for any Z ∈ L∞(R) is a convex (coherent) univariate scalar risk measure. Additionally, S S T ∞ d πt (X)= φt (ST X) for any X ∈ L (R ). S Proof. First we will show that φt is a traditional scalar risk measure: S S ∞ 1. φt (0) = πt (0) ∈ Lt (R). ∞ ∞ S S S 2. Let Z ∈ L (R) and m ∈ Lt (R), then φt (Z + m)= πt ([Z + m]e1)= πt (Ze1) − T S (St e1)m = φt (Z) − m. ∞ 3. Let Z1,Z2 ∈ L (R) where Z1 ≥ Z2 almost surely. Then Z1e1 ≥ Z2e1 almost S S S S surely, and by deﬁnition φt (Z1)= πt (Z1e1) ≤ πt (Z2e1)= φt (Z2). 4. Convexity and positive homogeneity follow immediately. S S T S T ∞ d The ﬁnal statement follows by πt (X)= πt ([ST X]e1)= φt (ST X) for any X ∈ L (R ).

S We will now consider the primal and dual representations for φt , deﬁned in (4.6). S Afterwards we will consider the representations for the stepped risk measures φt,s for t

S Corollary 4.10. Let S ∈ St, then the primal and dual representations for φt , deﬁned in (4.6), are given by: S ∞ T π ∞ π 1. φt (Z) = essinf m ∈ Lt (R) | Z + m ∈ ST At for any Z ∈ L (R) where At = {X ∈ L∞(Rd) | πS(X) ≤ 0}; t S Q ∞ 2. φt (Z) = esssupQ∈Qe(S) −βt(Q,S) − E [Z| Ft] for any Z ∈ L (R).

Proof. First, since φt is a traditional scalar risk measure, we immediately know a primal and dual representation exist: for any Z ∈ L∞(R)

S ∞ φ φt (Z) = essinf m ∈ Lt (R) | Z + m ∈ At

n S Q o = esssup −βt (Q) − E [Z| Ft] . Q∈M φ ∞ S S Q where At = Z ∈ L (R) | φt (Z) ≤ 0 and βt (Q) = esssup φ −E [Z| Ft] for every Z∈At φ T π Q ∈ M. Therefore showing At = ST A t is suﬃcient to prove this result. We note that π T At = cl At + s=t Γs(Ss) in order to ﬁnd h i P φ ∞ S At = Z ∈ L (R) | φt (Z) ≤ 0 ∞ S = Z ∈ L (R) | πt (Ze1) ≤ 0 22 T ∞ = Z ∈ L (R) | Ze1 ∈ cl At + Γs(Ss) ( " s=t #) X T T = ST X | X ∈ cl At + Γs(Ss) ∩ MT (4.7) ( " s=t # ) X T T = ST X | X ∈ cl At + Γs(Ss) (4.8) ( " s=t #) T π X = ST At .

Equation (4.7) follows from ST,1 = 1 almost surely for any S ∈ St. Equation (4.8) π T π follows since X ∈ At if and only if [ST X]e1 ∈ At . We conclude this proof with a consideration of the structure of the penalty function S βt . Let Q ∈ M, then

S Q βt (Q) = esssup −E [Z| Ft] φ Z∈At Q T = esssup −E ST X Ft X∈Aπ t h i T Q T Q T = esssup −E ST X Ft + ess sup −E ST ks Ft X∈At s=t ks∈Γs(Ss) h i X h i e βt(Q,S) on {Q ∈ Q (S)} = (∞ else

S Since the dual representation is taken on the set with E βt (Q) < ∞ this implies

S Q φt (Z) = esssup −βt(Q,S) − E [Z| Ft] Q∈Qe(S) for any Z ∈ L∞(R).

S ∞ ∞ Corollary 4.11. Let S ∈ St and t< s. Deﬁne φt,s : Ls (R) → Lt (R) as the S ∞ restriction of the domain of φt to Ls (R). Then primal and dual representations are deﬁned via the acceptance set and penalty function for primal and dual representations given in: φ T π π π ∞ d 1. At,s = Ss At,s where At,s := At ∩ Ls (R ); S Q T e 2. β (Q) = esssup π −E S X F for any Q ∈ Q (S). t,s X∈At,s s t S Proof. Since φt,s is a univariate stepped risk measure, if we prove the result for the acceptance set then the penalty function follows immediately. First we will show that φ φ ∞ T π π π ∞ d φ At,s := At ∩ Ls (R) ⊆ Ss At,s where At,s := At ∩ Ls (R ). Take Z ∈ At,s, then S S π T πt (Ze1) = φt (Z) ≤ 0 thus Ze1 ∈ At,s. By Ss,1 = 1 almost surely, Z = Ss (Ze1) ∈ T π φ T π π Ss At,s. Now we will show that At,s ⊇ Ss At,s. Let X ∈ At,s, then

S T S T S T S φt,s(Ss X)= φt (Ss X)= πt ([Ss X]e1)= πt (X) ≤ 0

S ∞ d where the last equality follows directly from the deﬁnition of πt and X ∈ Ls (R ).

23 S Remark 4.12. We would like to note that, unlike in the full case φt , the stepped risk measure need not satisfy the dual representation with respect to the penalty function from the scalarized conditional convex risk measure ρt. That is, it is possible that S Q ∞ φt,s(Z) 6= esssupQ∈Qe(S) −βt,s(Q,S) − E [Z| Ft] for some Z ∈ Ls (R). 4.4 π-time consistency

S T We will now introduce a new notion of time consistency for πt t=0. Using the above mentioned relations to univariate risk measures, we can deduce many properties for this time consistency property directly.

Deﬁnition 4.13. S T time consistent Fix S ∈ S. We say πt t=0 is if for all times t

T -time consistent S T The dynamic risk measure (ρt)t=0 is called π if πt t=0 is time consistent for every price sequence S ∈ S∗ := S . t≥0 t Proposition 4.14. ∗ S T T S T Fix S ∈ S , then πt t=0 is time consistent if and only if φt t=0 is time consistent (deﬁned in the usual way for univariate scalar risk measures). S T ∞ R S Proof. Let πt t=0 be time consistent and let Z1,Z2 ∈ L ( ) such that φs (Z1) ≥ φS(Z ) for times t

S T ∞ Rd Conversely, let φt t=0 be time consistent and let X1, X2 ∈ L ( ) such that πS(X ) ≥ πS(X ): s 1 s 2 S S S T S T πs (X1) ≥ πs (X2) ⇒ φs (ST X1) ≥ φs (ST X2) S T S T S S ⇒ φt (ST X1) ≥ φt (ST X2) ⇒ πt (X1) ≥ πt (X2).

S T We will now use the equivalence between the time consistency of πt t=0 and φS T πS T t t=0 to determine equivalent properties for time consistency of t t=0 . Theorem 4.15. Fix S ∈ S∗, then the following are equivalent for a normalized and S T S 0 convex πt t=0 (i.e., πt ( ) = 0 for all times t ≥ 0). S T 1. πt t=0 is time consistent; S S S ∞ d 2. π t (X )= πt (−πs (X)e1) for all t

~~∗ e Q (S) := {Q ∈ Q (S) | β0(Q,S) < ∞} ; (4.9)~~

~~24 5. For all Q ∈ Q∗(S) and all X ∈ L∞(Rd), the process~~

~~Q S Vt (X) := πt (X)+ βt(Q,S), t ≥ 0 is a Q-supermartingale. In each case the dynamic risk measure admits a robust representation in terms of the set Q∗(S) deﬁned in (4.9), i.e.,~~

~~S Q T πt (X) = esssup −βt(Q,S) − E ST X Ft Q∈Q∗(S) h i for all X ∈ L∞(Rd) and all times t ≥ 0.~~

S T Proof. These results all follow immediately from the equivalent properties of φt t=0- πS T time consistency and its equivalence to t t=0-time consistency (see Proposition 4.14). To show the equivalence of the recursive formulation and the summation of acceptance sets we consider Lemma 4.18 below as the equivalence between acceptance set versions S T S T of time consistency for φt t=0 and πt t=0 is less direct. Remark 4.16. S T Consider the setting of Theorem 4.15 but for a non-normalized πt t=0. πS T πS X πS πS 0 πS X e Then t t=0 is time consistent if and only if t ( )= t ([ s ( ) − s ( )] 1) for all t

~~∗ e Q (S)= {Q ∈ Q (S) | β0(Q,S) = 0} .~~

~~Corollary 4.17. ∗ S T Fix S ∈ S , then the following are equivalent for a coherent πt t=0. S T 1. πt t=0 is time consistent; S S S ∞ d 2. π t (X )= πt (−πs (X)e1) for all t~~

25 φ φ T π π 2. At,s + As = ST At,s + As π Proof. 1. First let X ∈ D + As . Immediately we can decompose X = XD + Xs such π S S S T that XD ∈ D and Xs ∈ As . Then πs (X) = πs (XD + Xs)= πs (Xs) − Ss XD ≤ T S T −Ss XD. That is, −πs (X) ≥ Ss XD and by property of D +Γs(Ss) ⊆ D this S S T S implies −πs (X)e1 ∈ D or −πs (X) ∈ Ss D by Ss,1 = 1. Conversely let −πs (X) ∈ T ∞ d S S Ss D for some X ∈ L (R ). We can then deﬁne X = −πs (X)e1 +(X +πs (X)e1). S We conclude that −πs (X)e1 ∈ D and by deﬁnition of the acceptance set we see S π that X + πs (X)e1 ∈ As , thus the proof is complete. φ φ S 2. First consider Z ∈ At,s + As . By [37, Lemma 4.6], this implies −πs (Ze1) = S φ T π π π T π π −φs (Z) ∈ At,s = Ss At,s. Therefore, Ze1 ∈ At,s+As and thus Z ∈ ST At,s + As by S = 1. Conversely, X ∈ Aπ + Aπ if and only if −φS(STX) = −πS(X) ∈ T,1 t,s s s T s T π φ T φ φ Ss At,s = At,s. By [37, Lemma 4.6] this is equivalent to ST X ∈ At,s + As , thus the proof is complete.

We conclude this section by considering the backwards composition to construct a π-time consistent risk measure. Proposition 4.19. Let S ∈ S∗ and consider a discrete time setting. Deﬁne the backwards composition of the portfolio returns Z ∈ L∞(R) by ˜S ˜S S ˜S φT (Z)= −Z, φt (Z)= φt (−φt+1(Z)) ∀t

~~S T S S S π˜T (X)= −ST X, π˜t (X)= πt (−π˜t+1(X)e1) ∀t~~

~~n T o be multiportfolio time consistent, then (ρt)t=0 deﬁned as in (3.1) is π-time consistent.~~

26 T We want to emphasize the following point about Lemma 4.20. Note that (ρt)t=0 is defined with respect to the single eligible asset (M = R × {0}d−1) by its definition via (3.1), or equivalently (3.2) as it is the essential infimum of the cash asset making T the position acceptable. However, we now consider (Rt)t=0 with any choice of eligible d space M˜ ⊆ R (which actually does not have an impact on the definition of ρt via (3.1) or (3.2), nor does it effect At). The remarkable power of Lemma 4.20 is the T fact that if (Rt)t=0 is multiportfolio time consistent with some choice of eligible space ˜ d T M ⊆ R , then (ρt)t=0 defined as in (3.1), and thus with the single eligible asset, is π-time consistent. This result will prove to be very useful in the example section.

~~Proof of Lemma 4.20. We will prove the result with full eligible space M˜ = Rd, the general case follows by the same logic. Let S ∈ S∗, t~~

S S πs (X) ≤ πs (Y ) T ∞ d π T ∞ d π ⇒ Ss u | u ∈ Ls (R ), X + u ∈ As ⊇ Ss u | u ∈ Ls (R ), Y + u ∈ As n oT n o ∞ d ⇒ u ∈ Ls (R ) | X + u ∈ cl As + Γr(Sr) ( " r=s #) X T ∞ d ⊇ u ∈ Ls (R ) | Y + u ∈ cl As + Γr(Sr) ( " r=s #) T X T ∞ d ∞ d ⇒ u ∈ Ls (R ) | X + u ∈ As + Γr(Sr) ⊇ u ∈ Ls (R ) | Y + u ∈ As + Γr(Sr) ( r=s ) ( r=s ) XT XT ∞ d ∞ d ⇒ u ∈ Ls (R ) | X + u ∈ As + Γr(Sr) ⊇ u ∈ Ls (R ) | Y + u ∈ As + Γr(Sr) ( r=t ) ( r=t ) X X ⇒ Rs(Xˆ ) ⊇ Rs(Yˆ ) Xˆ X T S Yˆ Y T S ∈ −P[r=t Γr( r) ∈ −P[r=t Γr( r)

⇒ Rt(Xˆ ) ⊇ Rt(Yˆ ) Xˆ X T S Yˆ Y T S ∈ −P[r=t Γr( r) ∈ −P[r=t Γr( r) T T ∞ d ∞ d ⇒ u ∈ Lt (R ) | X + u ∈ At + Γr(Sr) ⊇ u ∈ Lt (R ) | Y + u ∈ At + Γr(Sr) ( r=t ) ( r=t ) S S X X ⇒ πt (X) ≤ πt (Y ).

In fact, we will now use the above result relating multiportfolio time consistency to T π-time consistency in order to present a recursive deﬁnition of (ρt)t=0 with respect to S T the decomposed form πt t=0. Corollary 4.21. Consider the setting of Lemma 4.20 for normalized acceptance sets T ∞ d (At)t=0 (i.e., At = X ∈ L (R ) | 0 ∈ Rt(X) for set-valued risk measure such that R X R X R 0 X L∞ Rd t ρ T t( )= t( )+ t( ) for every ∈ ( ) for all times ). Then ( t)t=0 satisﬁes the recursive relation

~~S S S ρt(X) = esssup πt ([πs (0) − πs (X)]e1) (4.10) S∈St~~

~~27 for all portfolios X ∈ L∞(Rd) and times t~~

~~t t t t s s SX SX SX SX S0 SX ρt(X)= πt ([πs (0) − πs (X)]e1) 6= πt ([πs (0) − πs (X)]e1)= ρt(ρs(0) − ρs(X)).~~

More generally, the essential suprema over S ∈ St need not limit in the same direction for different times t and portfolios X. Since one is unable to merge the essential suprema at time t and s, the usual recursion ρt(X)= ρt(ρs(0) − ρs(X)) does not hold. T In fact, the recursive relation (4.10) can be viewed as (ρt)t=0 satisfying the dynamic S T T programming principle for (πt )t=0 for a fixed scalarization S. Recall now, that (ρt)t=0 T itself is a fixed scalarization of (Rt)t=0 (in the direction of the cash asset). In [34] we show that scalarized risk measures are in general only recursive with itself if the scalarizations change over time in a certain way (see also [54, 57]). Since here the cash T asset and thus the scalarization is fixed, (ρt)t=0 is not recursive, but by Corollary 4.21 satisfies the weaker recursive property (4.10) with respect to a fixed scalarization of S T (πt )t=0. T S The above recursive form for (ρt)t=0 simplifies when πs (0) = 0 for every S ∈ St, T e.g., if (ρt)t=0 is a coherent risk measure. The following proposition provides further results under the same setting as Corollary 4.21 above. In particular, we find a relation ∗ between the sets St over time, and specifically we can deduce that S = S0. Proposition 4.22. Consider the setting of Lemma 4.20 for closed and normalized T ∗ acceptance sets (At)t=0. Then St ⊆ Ss for any times t

e 28 ˜π π the closure of the set of random variables as with At for At defined in Proposition 4.5. The existence of these random sets is guaranteed by [63, Theorem 2.1.6]. Finally, by ∗ construction in Definition 4.13, S = t≥0 St = S0 by the monotonicity of these sets of frictionless price processes. T We will conclude our discussion of time consistency by relating the usual definition T of time consistency of the scalar mappings (ρt)t=0 to multiportfolio time consistency T of the set-valued risk measures (Rt)t=0 under the same single eligible asset space. T Lemma 4.23. Consider the setting of Theorem 3.2. (Rt)t=0 is multiportfolio time T consistent if and only if (ρt)t=0 is ρ-time consistent (i.e., time consistent as defined in (4.1)).

~~∞ Proof. First, by Theorem 3.2, we know that Rt(X) = (ρt(X)+ Lt (R+)) e1 for all times t and portfolios X ∈ L∞(Rd). T Let (Rt)t=0 be multiportfolio time consistent. For any t~~

~~ρs(X) ≤ ρs(Y ) ⇒ Rs(X) ⊇ Rs(Y ) ⇒ Rt(X) ⊇ Rt(Y ) ⇒ ρt(X) ≤ ρt(Y ).~~

~~T ∞ d Let (ρt)t=0 be time consistent. Additionally let t~~

~~Rs(X) ⊆ Rs(Y ) ⇒ (u ≥ ρs(X) ⇒∃Y ∈Y : u ≥ ρs(Y )) Y [∈Y ⇒∃Y ∈Y : ρs(X) ≥ ρs(Y ) ⇒∃Y ∈Y : ρt(X) ≥ ρt(Y )~~

~~⇒∃Y ∈Y : Rt(X) ⊆ Rt(Y ) ⇒ Rt(X) ⊆ Rt(Y ) Y [∈Y~~

~~Naturally, ρ-time consistency is a stronger property than π-time consistency.~~

~~T Corollary 4.24. If (ρt)t=0 is ρ-time consistent, then it is also π-time consistent. Proof. This follows directly from Lemmas 4.20 and 4.23.~~

~~5 Examples~~

Due to Lemma 4.20, we immediately know that every multiportfolio time consistent T set-valued risk measure generates a multivariate scalar risk measure (ρt)t=0 which is π- time consistent. Thus we refer to [29, 31, 32] for some examples. In this section we will focus on results on the superhedging risk measure with a single eligible asset, followed by an example for composed risk measures like the composed average value at risk. For these examples we will demonstrate the concept of π-time consistency and show that the prior, strong, time consistency property (ρs(X) ≤ ρs(Y ) ⇒ ρt(X) ≤ ρt(Y )) is too strong of a concept.

29 5.1 Superhedging under proportional transaction costs Let us first consider a market with proportional transaction costs only. As introduced in Section 3.1 and discussed in more details in [52, 72, 53], the market model will be defined T by a sequence of solvency cones (Kt)t=0. We will assume that the market satisfies the robust no-arbitrage property and Assumption 3.3. Such a superhedging risk measure was studied at time 0 in the discrete time setting in, e.g., [50, 62]. By the definition of the superhedging risk measure, with trading at discrete times t ∈ {0, 1, ..., T }, we can define the dual representation via

SHP Q T ρt (X)= esssup E −ST X Ft (Q,S)∈QSHP t h i where SHP e ¯ 1 + Qt := (Q,S) ∈ Qt | ξt,s(Q)Ss ∈ Ls(Ks ) ∀s ≥ t . S In particular, for the construction of the decomposition πt we care about

~~e SHP EMM(S) := {Q ∈ M | S is a Q-mtg} if S ∈ S Qt (S) := . (∅ else~~

In fact, in this case St = S for any time t ∈ [0, T ]. In particular, this implies that ∗ S S = S as well. This implies that the dual variables for the construction of πt are provided by the set of equivalent martingale measures EMM(S) for the price process S S. With this construction we can deﬁne the acceptance set for φt (the univariate scalar S risk measure corresponding with πt as deﬁned in (4.6)) as

~~φ ∞ Q SHP At := Z ∈ L (R) | E [Z| Ft] ≥ 0 ∀Q ∈ Qt (S)~~

n ∞ Q o = Z ∈ L (R) | E [Z| Ft] ≥ 0 ∀Q ∈ EMM(S) . n o φ Immediately this implies that At is the acceptance set of the scalar superhedging risk S T S measure, and therefore we recover that φt t=0 is time consistent (and thus so is π by Proposition 4.14) as is expected since the set-valued superhedging risk measure is multiportfolio time consistent in the case with full eligible space M˜ = Rd (see, SHP T e.g., [29]). Thus, ρt t=0 is π-time consistent by its definition (Definition 4.13), respectively by Lemma 4.20. Remark 5.1. SHP T Corollary 4.21 provides a recursive relation for ρt t=0, which we would like to note is similar to, but distinct from, that utilized in [62, Corollary 6.2]. SHP T We conclude by demonstrating that the superhedging risk measure ρt t=0, even though it is π-time consistent, is itself not ρ-time consistent. This follows by T Lemma 4.23. The underlying set-valued risk measure (Rt)t=0 is a closed and condi- T ∞ tionally coherent risk measure with acceptance sets At = s=t Ls (Ks) for all times t and M = R×{0}d−1. It is clear from the definition of the acceptance sets and the space P of eligible portfolios that, in general, At 6= At ∩ Ms + As. Thus, by [29, Theorem 3.4], T we can immediately conclude that (Rt)t=0 is not multiportfolio time consistent with SHP T this choice of eligible assets, and by Lemma 4.23 ρt t=0 is not ρ-time consistent as defined in (4.1). 30 5.2 Superhedging under convex transaction costs Continuing with the superhedging example, let us now consider a market with convex transaction costs, e.g., one that also includes price impact. As discussed in [64, 31, 32], T we will model such a market with convex solvency regions (Ct)t=0. We will assume that this market model satisfies the no-scalable robust no-arbitrage condition and has recession cones satisfying Assumption 3.3. As in the case with proportional transaction costs above, we can introduce the dual SHP representation, in the form of those provided in [50], for the superhedging price ρt via T SHP s dQ Q T ρt (X) = esssup σt (E Fs Ss) − E ST X Ft , Q e dP ( ,S)∈Qt s=t ! X h i where the penalty function is defined as the support function of the solvency regions

~~σs Y E Y TZ . t ( ) := ess∞ inf Ft Z∈Ls (Cs) h i~~

~~dQ 1 + In particular, the penalty function is ﬁnite only if E dP Fs Ss ∈ Ls(recc (Cs) ) for all times s ∈ {t, ..., T }. h i~~

T Remark 5.2. If we consider a conical market model (Kt)t=0, then the penalty function is determined by 0 on {Y ∈ K+} σs Y s . t ( )= + (−∞ on {Y 6∈ Ks } Thus we immediately recover the superhedging price presented in Section 5.1 as a special case. Therefore we recover the formulation from Jouini, Kallal [50] as seen in (1.1) in the static case as well. As with the case under proportional transaction costs only, it immediately follows from Lemma 4.20 and the multiportfolio time consistency of the set-valued superhedg- ˜ Rd SHP T ing risk measure with full eligible space M = (see, e.g., [29, 31]) that ρt t=0 is π-time consistent. As in the case with proportional transaction costs only, though we ﬁnd that the superhedging price is π-time consistent, it is not ρ-time consistent (with respect to deﬁnition (4.1)). This follows by Lemma 4.23 as the superhedging risk measure is not multiportfolio time consistent when the space of eligible assets M is only the cash asset.

5.3 Composed risk measures Composing the set-valued risk measures backwards in time automatically guarantees T −1 multiportfolio time consistency. That is, consider one-step risk measures (Rt,t+1)t=0 and a terminal risk measure RT to deﬁne the composed risk measure by:

R˜t(X) := Rt,t+1(R˜t+1(X)) = Rt,t+1(−Z) ∀t ∈ {0, 1, ..., T − 1} ˜ Z∈R[t+1(X) R˜T (X)= RT (X). This was studied for, e.g., the set-valued average value at risk in [29, 31]. As with the superhedging example above, often we consider the full space of eligible assets M˜ = Rd

31 as it is natural when considering trading in a market model [32] (as is studied in this paper) or with systemic risk measures [35]. In that setting, by Lemmas 4.20 and 4.23, ∞ the scalarization of the composed risk measureρ ˜t(X) := ess inf{m ∈ Lt (R) | me1 ∈ ∞ d R˜t(X)} for X ∈ L (R ) is π-time consistent but not ρ-time consistent. For illustrative purposes, let us brieﬂy consider the composed average value at risk with full space of eligible assets M˜ = Rd. We refer to [31, Section 6.1] for details on the set-valued composed average value at risk. In particular, consider the stepped average t ∞ d value at risk with levels λ ∈ Lt ([ǫ, 1] ) for the stepped risk measure from time t to ∞ d ∞ t + 1 and some lower threshold ǫ> 0. The scalarizationρ ˜t : L (R ) → Lt (R) of the composed average value at risk at time t, deﬁned by (3.1), can be given by the dual representation:

~~Q T ρ˜t(X) = esssup ess sup −E ST X Ft S∈St Q ˜λ S ∈Q ( ) h i ˜λ e s ¯ Q (S)= Q ∈ M | Ss,i ≥ λi ξs,s+1( Q)Ss+1,i ∀s = 0, 1, ..., T − 1, i = 1, 2, ..., d .~~

λ Note that any probability measure Q ∈ Q˜ (S) for some S ∈ St (and any ﬁxed time t) will also be a dual variable for the univariate composed average value at risk with t T −1 sequence of levels (λ1)t=0 as described in [19]. As discussed above, this multivariate risk measure is π-time consistent but not ρ-time consistent.

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