arXiv:2103.00905v2 [q-fin.RM] 7 Jul 2021 yNtoa aua cec onaino hn N.119011 (No. China 2020JJ5025). of (No. Foundation Science Natural National by esrsvateotoa lrto sdmntae n e.g in, equiva demonstrated as be filtration to optional proven the be via can measures processes to [ applied Schied and measures Föllmer risk 4], [3, al. et Artzner dis e.g., of appear of, measures measures risk risk the these for Though allow framework. measures axiomatic risk por These a of time). risk in the backwards (i.e., principle ov Bellman’s consistent to equivalent are be preferences risk introd (i.e., were consistency models time multi-period for measures risk Coherent Introduction 1 ∗ † e-audDnmcRs esrsfrPoessadVectors and Processes for Measures Risk Dynamic Set-Valued tvn nttt fTcnlg,Sho fBsns,Hobok Business, of School Technology, of Institute Stevens ua nvriy olg fFnneadSaitc,Chang Statistics, and Finance of College University, Hunan etddfiiinfrmliotoi iecnitnyo se proposed. of is consistency prov time multiportfolio are for vectors definition and mented processes for measures risk mu set-valued of equivalence the Finally, represent framework. dual set-valued new the a in deduce to result the this utilize of We augmentation cesses. an requires pro equivalence are this formulations measures, function penalty measure their risk of of equivalence equivalence The investigated. is filtration h eainhpbtenstvle ikmaue o proc for measures risk set-valued between relationship The ahn Chen Yanhong uy8 2021 8, July ∗ Abstract 1 4 n aua cec onaino ua Province Hunan of Foundation Science Natural and 84) ahr Feinstein Zachary n J000 USA. 07030, NJ en, h,Cia408.Rsac sprilysupported partially is Research 410082. China sha, rtm)o ikmaue a hw to shown was measures risk of time) er 3 n rtel n oaz inn[24], Gianin Rosazza and Frittelli and 23] floi qiaett h trtdrisks iterated the to equivalent is tfolio ,Acaoe l [1]. al. et Acciaio ., tprflotm ossec between consistency time ltiportfolio -audrs esrsfrprocesses for measures risk t-valued o rcse n etr n the and vectors and processes for s ie.I otatwt clrrisk scalar with contrast In vided. to o ikmaue o processes for measures risk for ation dd oacmls hs naug- an this, accomplish to ided; cdb rze ta.[,6,where 6], [5, al. et Artzner by uced e-audrs esrsfrpro- for measures risk set-valued ic rmtemr rdtoa risk traditional more the from tinct se n etr nteoptional the on vectors and esses ett hs oetaiinlrisk traditional more these to lent ahflw ob unie nan in quantified be to flows cash [email protected] † Within this work, we are concerned with dynamic risk measures. That is, those risk measures which allow for updates to the minimal capital requirement over time due to revealed information as encoded in a filtration. First, for dynamic risk measures for random variables which describe
financial positions, for instance: Bion-Nadal [8] studied the dual or robust representation; Bion-
Nadal [9, 10] gave the equivalent characterization of time consistency of dynamic risk measures for random variables by a cocycle condition on the minimal penalty function (i.e., such that the penalty functions can be decomposed as a summation over time); Barrieu and El Karoui [31] studied some applications of static and dynamic risk measures in the aspect of pricing, hedging and designing derivative; and Delbaen et al. [17] studied the representation of the penalty function of dynamic risk measures for random variables induced by backward stochastic differential equations. There also exist many studies about dynamic risk measures applied to cash flows, for example: Riedel [32] studied the dual representation for time consistent dynamic coherent risk measures for processes with real values; Cheridito et al. [13, 14] studied dynamic risk measures on the space of all bounded and unbounded càdlàg processes, respectively, with real values or generalized real values; Frittelli and Scandolo [25] proposed dynamic risk measures for processes from a perspective of acceptance sets; Cheridito et al. [15] studied dynamic risk measures for bounded discrete-time processes with random variables as values, and provided a dual representation and an equivalent characterization of time consistency in terms of the additivity property of the corresponding acceptance sets; and
Acciaio et al. [1] provided supermartingale characterisation (i.e., that the sum of a risk measure and its minimal penalty function form a supermartingale w.r.t. the dual probability measure) of time consistency of dynamic risk measures for processes. As highlighted in [1], as with the static risk measures, these dynamic risk measures for processes can be found to be equivalent to risk measures over random variables through the use of the optional filtration. Utilizing the optional σ-algebra, the equivalence of time consistency for traditional dynamic risk measures and those for processes are also found to be equivalent.
Set-valued risk measures were introduced in Jouini et al. [29], Hamel [26], Hamel et al. [28] and extended to the dynamic framework in Feinstein and Rudloff [18, 19]. All of those works present risk measures for random vectors. Set-valued risk measures were introduced so as to consider risks in markets with frictions. Such risk measures have, more recently, been utilized for quantifying systemic risk in Feinstein et al. [22] and Ararat and Rudloff [2]. In Chen and Hu [11, 12], set-
2 valued risk measures for processes were introduced. However, as demonstrated in those works, the equivalence of risk measures for processes and risk measures for random vectors no longer holds except under certain strong assumptions. This limits the application of known results for, e.g., multiportfolio time consistency (i.e., a version of Bellman’s principle for set-valued risk measures) as proven in Feinstein and Rudloff [20, 21] to set-valued risk measures for processes without rigorous direct proof.
In this work we prove a new relation between set-valued risk measures for processes and those for random vectors (on the optional filtration). Notably, the formulation for equivalence indicates that risk measures for processes at time t need to be augmented to capture the risks prior to time t due to the non-existence of a unique “0” element in the set-valued framework; such an augmentation is not necessary in the scalar case as in, e.g., [1]. By utilizing this new equivalence relation, we are able to derive a novel dual representation for set-valued risk measures for processes akin to that done in [1]. Additionally, we extend these results to present the equivalent formulation for multiportfolio time consistency; since the equivalence between risk measure settings requires an augmentation, we present a new definition for time consistency for risk measures for processes.
The organization of this paper is as follows. In Section 2, we present preliminary and back- ground information on the set-valued risk measures of interest within this work, i.e., as functions of stochastic processes and random vectors. Section 3 demonstrates the equivalence of these risk measures both in primal and dual representations. With the focus on dynamic risk measures, the equivalence of multiportfolio time consistency for processes and vectors is presented in Section 4.
Section 5 concludes.
2 Set-valued risk measures
In this section we will summarize the definitions for set-valued risk measures. First, in Section 2.1, we will provide details of the filtrations and the spaces of claims of interest within this work.
Section 2.2 will provide the necessary background on risk measures for processes as defined in, e.g., [11, 12]. Section 2.3 will provide the necessary background on risk measures for random vectors as defined in, e.g., [18, 19].
3 2.1 Background and notation
Fix a finite time horizon T ∈ N := {1, 2,... } and denote T := {0, 1,...,T } and Tt := {s ∈
T : s ≥ t} for t ∈ T. Fix a filtered probability space (Ω, F, (Ft)t∈T, P) with F0 the complete and trivial σ-algebra. Without loss of generality, take F = FT . Let d ≥ 1 be the number of assets under consideration and |·| be an arbitrary but fixed norm on Rd. Define the space of p-integrable random
p d p d p d p d 0 d d-vectors for p ∈ {1, ∞} by Lt (R ) := L (Ω, Ft, P; R ) (with L (R ) := LT (R )). Similarly, Lt (R ) denotes the linear space of the equivalence classes of Ft-measurable vector-valued functions, where
p d we specify random vectors P-a.s. In this way, Lt (R ) denotes the linear space of the equivalence d classes of Ft-measurable functions X :Ω → R with bounded p norm where these norms are given by
EP[|X|]= |X(ω)|dP if p = 1 kXkp = Ω R ess supω∈Ω |X(ω)| if p = ∞. In this paper, we denote the expectation EP[X] by E[X], and denote the conditional expectation
P ∞ d E [X|Ft] by Et[X]. Throughout this work, we consider the weak* topology on Lt (R ) such that ∞ d 1 d the dual space of Lt (R ) is Lt (R ). Fix m ∈ {1,...,d} of the assets to be eligible for covering the risk of a portfolio. Denote by
M := Rm × {0}d−m the subspace of eligible assets (those assets which can be used to satisfy capital
p p d requirements, e.g., US dollars and Euros). Write Lt (B) := {X ∈ Lt (R ) : X ∈ B, P-a.s.} for those random vectors that take P-a.s. values in B ⊆ Rd for p ∈ {0, 1, ∞}. In particular, we denote
d the closed convex cone of uniformly bounded R -valued Ft-measurable random vectors with P-a.s.
∞ d ∞ d non-negative components by Lt (R+), and let Lt (R++) be the Ft-measurable random vectors ∞ with P-a.s. strictly positive components. Additionally, we denote Mt := Lt (M) to be the space of time t measurable eligible portfolios. Throughout we consider the non-negative eligible portfolios
∞ d ∗ 1 d ⊤ Mt,+ := Mt ∩ Lt (R+), the positive dual cone Mt,+ := {u ∈ Lt (R ) : E[u m] ≥ 0 for any m ∈ ⊥ 1 d ⊤ Mt,+} and the perpendicular space Mt,+ := {u ∈ Lt (R ) : E[u m] = 0 for any m ∈ Mt,+}. ∗ 1 d ⊤ Generally, we will define C := {u ∈ Lt (R ) : E[u m] ≥ 0 for any m ∈ C} for any convex ∞ d cone C ⊆ Lt (R ). (In)equalities between random vectors, stochastic processes and between sets
4 are always understood componentwise in the P-a.s. sense. The multiplication between a random
∞ ∞ d variable λ ∈ Lt (R) and a set of random vectors Bt ⊆ Lt (R ) is understood as in the elementwise sense, i.e., λBt := {λX : X ∈ Bt} with (λX)(ω) = λ(ω)X(ω). Given a set A ⊆ Mt, cl(A) means the closure of A w.r.t. the subspace topology on Mt and co(A) means the convex hull of A. Denote the spaces of upper and closed-convex upper sets respectively by
U(Mt; Mt,+) := {D ⊆ Mt : D = D + Mt,+}
G(Mt; Mt,+) := {D ⊆ Mt : D = cl co(D + Mt,+)},
1 d where the + sign denotes the usual Minkowski addition. We will denote Γt(w) with w ∈ Lt (R )\{0} ∞ d ⊤ by Γt(w) = {u ∈ Lt (R ) : w u ≥ 0 P-a.s.}. As will be utilized in the dual representations presented below, the Minkowski subtraction for sets A, B ⊆ Mt is defined as
• A − B = {m ∈ Mt : B + {m}⊆ A}.
The indicator function for some D ∈ F is denoted by ID :Ω → {0, 1} and defined as
1 if ω ∈ D ID(ω)= 0 if ω∈ / D.
∞,d Define R the space of all d-dimensional adapted stochastic processes X := (Xt)t∈T on
∞ d 1,d (Ω, F, (Ft)t∈T, P) whose coordinates are in L (R ) uniformly over t. A denotes the space of all d- d dimensional adapted stochastic processes a on (Ω, F, (Ft)t∈T, P) with i=1 E[ t∈T |∆at,i|] < +∞, ∞,d where a−1,i := 0 and ∆at,i := at,i − at−1,i. Denote by ITt the elementP of PR with the j-th component equal to 0 for j = 0,...,t − 1 and 1 for j ∈ Tt and denote by I{s} the element of R∞,d where only the s-th component is equal to 1 and all other are equal to 0. As is stan- dard in the literature (see, e.g., [15]), for any time 0 ≤ r ≤ s ≤ T , we define the projec- tion πr,s(X)t := ITr Xt∧s,t ∈ T for any d-dimensional adapted stochastic process X. Denote
∞,d ∞,d 1,d 1,d Rt := πt,T (R ) and At := πt,T (A ). The space of non-negative processes is written as ∞,d ∞,d ∞,d R+ := {X ∈ R : Xt,i ≥ 0 for any t ∈ T, i = 1,...,d}. As in [1, 15], a process X ∈ Rt describes the evolution of a financial value on the time Tt. Throughout this work we will consider
5 ∞,d 1,d 1,d ∞,d ∞,d the coarsest topologies (denoted by σ(Rt , At ), σ(At , Rt ) respectively) such that Rt and 1,d At form a dual pair (and vice versa), i.e., the space of all continuous linear functionals on the ∞,d ∞,d 1,d 1,d 1,d ∞,d 1,d ∞,d topological space (Rt ,σ(Rt , At )) (resp. (At , σ(At , Rt )) is At (resp. Rt ). Denote by M(P) the set of probability measures Q on (Ω, F) which are absolutely continuous with respect to P. Denote those probability measures equal to P on Ft by Mt(P) := {Q ∈ M(P) :
Q = P on Ft}. For any given t ≤ s ∈ T, define
∞ Dt,s := {ξ ∈ Ls (R+) : Et[ξ] = 1} .
ξ ξ Then every ξ ∈ Dt,s defines a probability measure Q in Mt(P) with density dQ /dP = ξ. Con- versely, every Q ∈ M(P) induces a collection of non-negative random variables ξt,s(Q) ∈ Dt,s for t ≤ s ∈ T where ξt,s(Q) is defined as
dQ Es[ dP ] dQ dQ on {Et[ ] > 0} Et[ ] dP ξt,s(Q) := dP 1 otherwise. As in, e.g., Cheridito and Kupper [16], we will use a P-almost sure version of the Q-conditional expectation of X ∈ L∞(Rd) given by
Q Q Et [X] := E [X|Ft] := Et[ξt,T (Q)X]
d ⊤ for any time t ∈ T. For Q ∈ M(P) , we will utilize the vector representation ξt,s(Q) := (ξt,s(Q1),...,ξt,s(Qd))
s d 1 d 0 d for any time t ≤ s ∈ T. We further define the function wt : M(P) × Lt (R ) → Ls(R ) for any t,s ∈ T with t
s wt (Q, w) := diag(w)ξt,s(Q),
d 1 d for any Q ∈ M(P) and w ∈ Lt (R ), where diag(w) denotes the diagonal matrix with the compo- t d nents of w on the main diagonal. By convention, for any t ∈ T, wt(Q, w)= w for any Q ∈ M(P) 1 d and w ∈ Lt (R ). As we seek to relate risk measures for random vectors to those of processes, as in [1], we
6 consider the optional filtration, i.e., (Ω¯, F¯, (F¯t)t∈T, P¯) with sample space Ω=Ω¯ × T, σ-algebra
F¯ = σ({At × {t} : At ∈ Ft,t ∈ T}), filtration F¯t = σ({Dr × {r}, Dt × Tt : Dr ∈ Fr,r < t, Dt ∈ Ft}), t ∈ T and probability measure P¯ = P ⊗ µ where µ = (µt)t∈T is some adapted reference process such that t∈T µt = 1 and P[mint∈T µt > ǫ] = 1 for some ǫ > 0. Under this measure ¯ ¯P¯ on the optional σ-algebra,P the expectation is E[X] := E [X] := E[ t∈T Xtµt] for any measurable function X on (Ω¯, F¯); we emphasize that this expectation is takenP over the optional σ-algebra by utilizing the E¯ notation throughout this work. A random vector X = (Xt)t∈T on (Ω¯, F¯, P¯) is F¯t-measurable if and only if Xr is Fr-measurable for all r = 0,...,t and Xr = Xt for any ¯p d p ¯ ¯ ¯ d ¯p d ¯p d r>t. Denote Lt (R ) := L (Ω, Ft, P; R ) (with L (R ) := LT (R )) for p ∈ {0, 1, ∞}; similarly ¯ d ¯p ¯p d denote the set of Ft-measurable random vectors taking value in B ⊆ R by Lt (B) ⊆ Lt (R ) d for p ∈ {0, 1, ∞}. Often we consider the non-negative orthant B = R+ or the positive orthant d ¯ ¯∞ ¯∞ d ¯ B = R++. Denote Mt = Lt (M) := {X ∈ Lt (R ) : X ∈ M P-a.s.}. Throughout this work, we ¯∞ d ¯∞ d ¯1 d consider the weak* topology on Lt (R ) such that the dual space of Lt (R ) is Lt (R ). As on the original probability space, denote by M¯ (P¯) the set of probability measures Q¯ on (Ω¯, F¯) which are absolutely continuous with respect to P¯. Denote by E¯ Q¯ [X] as the expectation of X ∈ L¯∞(Rd) with respect to the probability measure Q¯ . The following Theorem 2.1, adapted from [1], provides a representation for these probability measures on the optional σ-algebra.
Theorem 2.1. For any probability measure Q¯ ∈ M¯ (P¯) on the optional σ-algebra, there exist a ∞,1 probability measure Q ∈ M(P) and an optional random measure ψ ∈ Ψ(P) := {ψ ∈ R+ :
t∈T ψt = 1 P-a.s.} such that Q¯ Q P E¯ [X]= E ψtXt (1) "t T # X∈ for any X ∈ L¯∞(R).
Conversely, any Q ∈ M(P) and ψ ∈ Ψ(P) define a probability measure Q¯ := Q ⊗ ψ ∈ M¯ (P¯) such that (1) holds.
Proof. This follows directly from Theorem 3.4 and Remark B.1 of [1].
As we often are interested in those probability measures that are equivalent to P¯ on the filtration, we present the following corollary.
7 Corollary 2.2. Let Q¯ ∈ M¯ (P¯) with decomposition Q¯ = Q ⊗ ψ (as defined in Theorem 2.1). Then, for any time t ∈ T, it follows that
Q = P on F ¯ ¯ ¯ t Q = P on Ft ⇔ ψs = µs P-a.s. ∀s Proof. This follows directly from [1, Lemma B.5]. We note that, from Corollary 2.2, it follows that M¯ t(P¯) := {Q¯ ∈ M¯ (P¯) : Q¯ = P¯ on F¯t} = {Q ⊗ ψ : Q ∈ Mt(P), ψ ∈ Ψ(P), ψs = µs P-a.s. ∀s As for the original probability space, we will use a P¯-almost sure version of the Q¯ -conditional ¯∞ d ¯Q¯ ¯Q¯ ¯ expectation of X ∈ L (R ). Denote by Et [X] := E [X|Ft] as the conditional expectation of X ∈ L¯∞(Rd). This P¯-almost sure version can be provided by ξ¯(Q¯ ) construction as for the original probability space; in the following Corollary 2.3 we provide an equivalent representation. Corollary 2.3. For Q¯ ∈ M¯ (P¯) with decomposition Q¯ = Q ⊗ ψ, the conditional expectation of ∞ X ∈ L¯ (R) given F¯t takes the form t−1 Q ψs I t−1 E −1 X ¯ {P ψr<1} t s∈Tt t s ¯ Q r=0 1−Pr=0 ψr Et [X]= XsI{s} + ITt . h µs i s=0 +I t−1 E P −1 X {P ψr=1} t s∈Tt t s X r=0 1−Pr=0 µr hP i Proof. First, by definition of the conditional expectation and Theorem 2.1, t−1 Q Q ¯ Q¯ ¯ Q¯ E ψsXs = E ψsEt [X]s + ψs Et [X]t "s T # "s=0 s T ! # X∈ X X∈ t for any X ∈ L¯∞(R). By matching terms, ¯ Q¯ Et [X]s = Xs Q-a.s. on {ψs > 0} for s Q¯ -conditional expectation. s ¯ ¯ d ¯1 d ¯0 d We further define the functions w¯t : M(P) × Lt (R ) → Ls(R ) for any t,s ∈ T with t t−1 s−1 s ¯ ¯ ¯ ¯ ¯ w¯t (Q, w¯) := w¯rI{r} + diag(w ¯t) ξt,s(Q)rI{r} + ξt,s(Q)sITs r=1 r=t ! X X d where ξ¯t,s(Q¯ ) is defined comparably to ξt,s(Q) for Q ∈ M(P) . By convention, for any t ∈ T, t ¯ ¯ ¯ ¯ d ¯1 d ¯ w¯t(Q, w¯) =w ¯ for any Q ∈ M(P) and w¯ ∈ Lt (R ). We note that, through the decomposition of Q s ¯ as in Theorem 2.1, an equivalent formulation for w¯t (Q, w¯) can be given in the style of Corollary 2.3. We conclude this section with a brief result relating the topologies of L¯∞(Rd) and R∞,d. This will be utilized later in this work. n ¯∞ d ¯1 d n ∞ d 1 d Lemma 2.4. (1) Let X → X in σ(L (R ), L (R )). Then Xt → Xt in σ(Lt (R ),Lt (R )) and n ∞,d 1,d X ITt → XITt in σ(Rt , At ) for every time t ∈ T. n ∞ d 1 d n (2) Fix t ∈ T. Let Xs → Xs in σ(Ls (R ),Ls(R )) for every s Proof. (1) Let Xn → X in σ(L¯∞(Rd), L¯1(Rd)). And fix t ∈ T. 1 d −1 ¯1 d (a) Let Zt ∈ Lt (R ) and define Y := µt ZtI{t} ∈ L (R ). ⊤ n ⊤ n ¯ ⊤ n ¯ ⊤ ⊤ E[Zt Xt ]= E[ µsYs Xs ]= E[Y X ] → E[Y X]= E[Zt Xt]. s T X∈ 1,d −1 I ¯1 Rd (b) Let a ∈At . Define Y := s∈Tt µs ∆as {s} ∈ L ( ). P ⊤ n ⊤ n ¯ ⊤ n ¯ ⊤ ⊤ E[ ∆as Xs ]= E[ µsYs Xs ]= E[Y X ] → E[Y X]= E[ ∆as Xs]. s T s T s T X∈ t X∈ X∈ t n ∞ d 1 d n (2) Fix t ∈ T. Let Xs → Xs in σ(Ls (R ),Ls(R )) for every s 9 r 1,d s ∈ Tt (with ∆as := 0 for s 2.2 Risk measures for processes In this section we provide a quick overview of the definition of set-valued risk measures for processes as defined in [11, 12]. Herein we present an axiomatic framework for such functions in Definition 2.5. We then summarize prior results on the primal representation w.r.t. an acceptance set. We conclude this section with considerations for a novel dual representation for these conditional risk measures. ∞,d Definition 2.5. A function ρt : Rt →U(Mt; Mt,+) for t ∈ T is called a set-valued conditional ∞,d risk measure for processes if it satisfies the following properties for all X,Y ∈ Rt , (1) Cash invariant: for any m ∈ Mt, ρt(X + mITt )= ρt(X) − m; (2) Monotone: ρt(X) ⊆ ρt(Y ) if X ≤ Y component-wise; (3) Finite at zero: ρt(0) 6= ∅ is closed (w.r.t. the subspace topology on Mt) and Ft-decomposable (i.e., IAρt(0) + IAc ρt(0) ⊆ ρt(0) for any A ∈ Ft; see, e.g., Chapter 2 of Molchanov [30]), and P[˜ρt(0) = M] = 0 where ρ˜t(0) is a Ft-measurable random set (i.e., graphρ ˜t(0) := {(ω,x) ∈ d d d d Ω × R : x ∈ ρt(0; ω)} is Ft ⊗ B(R )-measurable for Borel σ-algebra B(R ) of R ; see, e.g., Tahar ∞ and Lépinette [7]) such that ρt(0) = Lt (˜ρt(0)). A conditional risk measure for processes at time t ∈ T is said to be: ∞,d • Normalized if ρt(X)= ρt(X)+ ρt(0) for every X ∈ Rt ; 10 ∞,d ∞ • Conditionally convex if for all X,Y ∈ Rt and λ ∈ Lt ([0, 1]) ρt(λX + (1 − λ)Y ) ⊇ λρt(X) + (1 − λ)ρt(Y ); ∞,d ∞ • Conditionally positive homogeneous if for all X ∈ Rt and λ ∈ Lt (R++) ρt(λX)= λρt(X); • Conditionally coherent if it is conditionally convex and conditionally positive homogeneous; • Closed if the graph of ρt ∞,d graph ρt := {(X, u) ∈ Rt × Mt : u ∈ ρt(X)} is closed in the product topology; • Conditionally convex upper continuous (c.u.c.) if − ∞,d ρt (D) := {X ∈ Rt : ρt(X) ∩ D 6= ∅} is closed for any Ft-conditionally convex set D ∈ G(Mt; −Mt,+). A dynamic risk measure for processes is a sequence of conditional risk measures for processes (ρt)t∈T. And a dynamic risk measure for processes is said to have one of the above properties if ρt has the corresponding property for any t ∈ T. For the interpretation of these risk measures, recall that X ∈ R∞,d denotes the evolution of a financial value over time. As such, for instance, cash invariance implies that an addition of m ∞,d eligible assets at time t to X ∈ Rt which is held until T will transform the value into X + mITt and the risk is decreased by m at time t. We now consider the primal representation of a set-valued conditional risk measure for processes. That is, for any risk measure, let the acceptance set be defined as those claims that do not require 11 the addition of any eligible asset in order to be “acceptable” to the risk manager: ∞,d At := {X ∈ Rt : 0 ∈ ρt(X)}. As shown in [11, 12], these acceptance sets uniquely define the risk measure via the relation ρt(X)= {m ∈ Mt : X + mITt ∈ At} ∞,d for any X ∈ Rt . With this construction of risk measures for processes, we give a consideration of the dual representation. The following dual representation – Corollary 2.6 – is novel in the literature for set-valued risk measures for processes, we will leave the proof until Section 3.2. Though this result can be proven using the duality theory of Hamel et al. [27] and following the same logic as the proof of [20, Corollary 2.4], we will take a different approach by utilizing the prior results on risk measures for vectors and the yet-to-be-proven equivalence between these formulations. For this result, we first need to consider the set of dual variables s 1 d Wt := (Q, w) ∈ Dt : wt (Qs, ws) ∈ Ls(R+) ∀s ∈ Tt , T −t+1 n d×(T −t+1) ∗ ⊥ o Dt := Mt(P) × Mt,+ \ Mt . s 1 d Notably (Q, w) ∈Wt implies ws ≥ 0 Qs-a.s. since wt (Qs, ws) ∈ Ls(R+). ∞,d Corollary 2.6. A function ρt : Rt → G(Mt; Mt,+) is a closed conditionally convex risk measure if and only if Qs • ρt(X)= Et [−Xs]+Γt(ws) ∩ Mt − αt(Q, w) , (2) (Q,w)∈Wt s∈Tt \ X where αt is the minimal conditional penalty function given by Qs αt(Q, w)= Et [−Ys]+Γt(ws) ∩ Mt . Y ∈At s∈Tt \ X 12 ρt is additionally conditionally coherent if and only if Qs ρt(X)= Et [−Xs]+Γt(ws) ∩ Mt , max (Q,w)∈W s∈Tt \ t X for max ⊤ Qs Wt = (Q, w) ∈Wt : ws Et [Zs] ≥ 0 for any Z ∈ At . ( s T ) X∈ t We conclude this section with a brief introduction to a time consistency notion, called multi- portfolio time consistency, for these set-valued risk measures for processes. This notion for risk measures for processes was introduced in [11]. We will revisit this definition in Section 4. Definition 2.7. A dynamic risk measure for processes (ρt)t∈T is multiportfolio time consistent if for any times t ρs(X) ⊆ ρs(Y ) ⇒ ρt(ZI{[t,s)} + XITs ) ⊆ ρt(ZI{[t,s)} + Y ITs ) Y B Y B [∈ [∈ ∞,d ∞,d ∞,d for any X ∈ Rs , Z ∈ Rt , and B ⊆ Rs . Conceptually, a risk measure is multiportfolio time consistent if a cash flow X is guaranteed (almost surely) to be more risky than some collection of cash flows B at a time point in the future s and if all of those cash flows are identical up to time s, then X is also more risky at all earlier times. This can most easily be understood when the collection B is a singleton, i.e., when comparing two cash flows X and Y directly. 2.3 Risk measures for vectors In this section we provide a quick overview of the definition of set-valued risk measures for random vectors as defined in [18, 19]. Herein we present an axiomatic framework for such functions in Definition 2.8 with emphasis on the special case on the optional filtration. We then summarize prior results on the primal representation w.r.t. an acceptance set. We conclude this section with considerations for a dual representation for these conditional risk measures. ∞ d Definition 2.8. A function R¯t : L¯ (R ) → U(M¯ t; M¯ t,+) for t ∈ T is called a set-valued con- ditional risk measure for vectors on L¯∞(Rd) if it satisfies the following properties for all 13 X,Y ∈ L¯∞(Rd), (1) Cash invariant: for any m ∈ M¯ t, R¯t(X + m)= R¯t(X) − m; (2) Monotone: R¯t(X) ⊆ R¯t(Y ) if X ≤ Y component-wise; ˜ (3) Finite at zero: R¯t(0) 6= ∅ is closed and F¯t-decomposable, and such that P¯[R¯t(0) = M] = 0 ¯˜ ¯ ¯ ¯∞ ¯˜ where Rt(0) is a Ft-measurable random set such that Rt(0) = Lt (Rt(0)). A conditional risk measure for vectors at time t ∈ T is said to be: ∞ d • Normalized if R¯t(X)= R¯t(X)+ R¯t(0) for every X ∈ L¯ (R ); • Time decomposable if t−1 ¯ ¯ ¯ Rt(X)= Rt(XI{s})I{s} + Rt(XITt )ITt s=0 X for any X ∈ L¯∞(Rd); ¯∞ d ¯∞ • Conditionally convex if for all X,Y ∈ L (R ) and λ ∈ Lt ([0, 1]) R¯t(λX + (1 − λ)Y ) ⊇ λR¯t(X) + (1 − λ)R¯t(Y ); ¯∞ d ¯∞ • Conditionally positive homogeneous if for all X ∈ L (R ) and λ ∈ Lt (R++) R¯t(λX)= λR¯t(X); • Conditionally coherent if it is conditionally convex and conditionally positive homogeneous; • Closed if the graph of R¯t ∞ d graph R¯t := {(X, u) ∈ L¯ (R ) × M¯ t : u ∈ R¯t(X)} is closed in the product topology; 14 • Conditionally convex upper continuous (c.u.c.) if ¯− ¯∞ d ¯ Rt (D) := {X ∈ L (R ) : Rt(X) ∩ D 6= ∅} is closed for any F¯t-conditionally convex set D ∈ G(M¯ t; −M¯ t,+). A dynamic risk measure for vectors is a sequence of conditional risk measures for vectors (R¯t)t∈T. A dynamic risk measure for vectors is said to have one of the above properties if R¯t has the corresponding property for any t ∈ T. Remark 2.9. We note that the time decomposable property presented above is, as far as the authors are aware, novel to this work. Time decomposability conceptually means that the time s We now consider the primal representation of a set-valued conditional risk measure for vectors. That is, for any risk measure, let the acceptance set be defined as those claims that do not require the addition of any eligible asset in order to be “acceptable” to the risk manager: ∞ d A¯t := {X ∈ L¯ (R ) : 0 ∈ R¯t(X)}. As shown in [18], these acceptance sets uniquely define the risk measure via the relation R¯t(X)= {m ∈ M¯ t : X + m ∈ A¯t} for any X ∈ L¯∞(Rd). We now provide a dual representation for risk measures for vectors. For comparison to the dual representation presented above for risk measures for processes, we present the dual representation provided first in [20]. For this result, we first need to consider the set of dual variables ¯ ¯ ¯ ¯ d ¯ ∗ ¯ ⊥ T ¯ ¯1 d Wt := (Q, w¯) ∈ Mt(P) × (Mt,+ \ Mt ) :w ¯t (Q, w¯) ∈ L (R+) . n o 15 As above, (Q¯ , w¯) ∈ W¯ t implies w¯ ≥ 0 Q¯ -a.s. ∞ d Corollary 2.10. [20, Corollary 2.4] A function R¯t : L¯ (R ) → G(M¯t; M¯ t,+) is a closed condi- tionally convex risk measure if and only if ¯ ¯ ¯Q ¯ ¯ • ¯ Rt(X)= Et [−X]+ Γt(w ¯) ∩ Mt − α¯t(Q, w¯) , (Q¯ ,w¯)∈W¯ \ t where α¯t is the minimal conditional penalty function given by ¯ ¯ Q¯ ¯ ¯ α¯t(Q, w¯)= Et [−Y ]+ Γt(w ¯) ∩ Mt. ¯ Y\∈At R¯t is additionally conditionally coherent if and only if ¯ ¯Q¯ ¯ ¯ Rt(X)= Et [−X]+ Γt(w ¯) ∩ Mt, Q¯ ¯ max ( ,w¯)\∈Wt for ¯ max ¯ ¯ T ¯ ¯∗ Wt = (Q, w¯) ∈ Wt :w ¯t (Q, w¯) ∈ At . We conclude this introduction to risk measures for vectors with a consideration of multiportfolio time consistency. This notion for risk measures for vectors is a modification of that introduced in [18, 19]; we demonstrate in Proposition 2.12 that this coincides with the typical definition for time decomposable risk measures. Definition 2.11. A dynamic risk measure for vectors (R¯t)t∈T is multiportfolio time consistent if for any times t R¯s(X) ⊆ R¯s(Y ) ⇒ R¯t(X) ⊆ R¯t(Y ) Y B Y B [∈ [∈ ∞ for any X ∈ L¯ and B := {(b0, b1, . . . , bs−1, bs) : br ∈ Br, r < s, bs ∈ Bs} for any arbitrary ∞ d ∞,d sequence of sets Br ⊆ Lr (R ) for r As for risk measures for processes, conceptually, multiportfolio time consistency allows for the comparison of a single cash flow with a collection of portfolios over time in which a guaranteed 16 ordering must hold backwards in time. Proposition 2.12. Let (R¯t)t∈T be a normalized, time decomposable risk measure. (R¯t)t∈T is mul- tiportfolio time consistent if and only if for any times t R¯s(X) ⊆ R¯s(Y ) ⇒ R¯t(X) ⊆ R¯t(Y ) Y B Y B [∈ [∈ for any X ∈ L¯∞(Rd) and B ⊆ L¯∞(Rd). ¯ ¯ Proof. This result follows trivially via the application of the recursive relation Rt(X)= Z∈R¯s(X) Rt(−Z) for any times t Remark 2.13. Throughout much of the remainder of this work, the restriction of set-valued ∞ d risk measures for vectors to a sub-σ-algebra is utilized. Herein we will define Rt : Lt (R ) → U(Mt; Mt,+) to be the restriction of a set-valued risk measure (with filtered probability space ∞ d (Ω, F, (Fs)s∈T, P)) to Lt (R ). The primal and dual representation of such risk measures will be utilized extensively below defined in analogous ways to those provided above; for closed and convex risk measures – as are utilized for the dual representation – the codomain of Rt is G(Mt; Mt,+) ⊆ U(Mt; Mt,+). Notably, the penalty function αRt for these restricted risk measures only depends on ∗ a single dual variable w ∈ Mt,+, i.e., αRt (w) does not depend on a vector of probability measures d Q ∈ Mt(P) . 3 Equivalence of risk measures for processes and vectors In this section we will focus on the relation between set-valued risk measures for processes and vec- tors. This was studied under restrictive conditions in [11] which match many of the same properties of the scalar setting (i.e., with a full space of eligible assets M = Rd and with the strong normaliza- ∞ d tion property ρt(0) = Lt (R+)). Herein we drop these restrictions and determine the equivalence for both the primal and dual representations. Notably, this equivalence requires the introduction of an augmentation of the risk measure for processes with a series of risk measures for vectors on 17 the original filtered probability space; when the aforementioned strong assumptions are imposed these augmenting risk measures can be dropped without incident as detailed in Corollary 3.2. 3.1 Relation for primal representations First, we compare the relation between set-valued risk measures for processes and for vectors through their axiomatic definitions. This is akin to the results presented in [1] for scalar risk measures. Notably, in comparison to that work, the set-valued risk measures for vectors in the optional filtration are a larger class than those for processes rather than the equivalence as found for scalar risk measures. In particular, we find risk measures for vectors on the optional filtration are equivalent to the risk measures for processes augmented with the restricted set-valued risk measures for vectors which are discussed in Remark 2.13. Intuitively, this augmentation is necessary to account for the possible risk prior to the current time that cannot trivially be accounted for by eligible assets; that is, the augmentation is required for the M 6= Rd setting. In the full eligible asset setting M = Rd, the equivalence of risk measures for processes and vectors no longer requires an augmentation by restricted set-valued risk measures as is detailed in Corollary 3.2 below. Theorem 3.1. Fix time t ∈ T. ∞,d (1) Any conditional risk measure for processes ρt : Rt →U(Mt; Mt,+) and series of conditional ∞ d risk measures for vectors Rs : Ls (R ) → U(Ms; Ms,+) (restricted to Fs-measurable random vec- tors), s = 0, 1,...,t − 1, define a time decomposable conditional risk measure for random vectors ∞ d on the optional filtration R¯t : L¯ (R ) →U(M¯ t; M¯ t,+) via t−1 ¯ Rt(X) := Rs(Xs)I{s} + ρt(πt,T (X))ITt . (3) s=0 X t−1 Additionally, if (Rs)s=0, ρt are normalized, conditionally convex, or conditionally coherent, then the corresponding risk measure R¯t has the same property. ∞ d (2) Any conditional risk measure for random vectors on the optional filtration R¯t : L¯ (R ) → ¯ ¯ ∞,d U(Mt; Mt,+) defines a conditional risk measure for processes ρt : Rt →U(Mt; Mt,+) and a series ∞ d of conditional risk measures Rs : Ls (R ) → U(Ms; Ms,+) restricted to Fs-measurable random 18 vectors, s = 0, 1,...,t − 1, via ¯ ρt(X) := Rt(XITt )t (4) ¯ Rs(Z) := Rt(ZI{s})s, s = 0, 1,...,t − 1. (5) Additionally, if R¯t is normalized, conditionally convex, or conditionally coherent, then the corre- t−1 sponding risk measures (Rs)s=0, ρt have the same property. ¯ t−1 (3) If Rt is time decomposable then it can be decomposed into form (3) such that ρt and (Rs)s=0 are defined as in (4) and (5) respectively. Additionally, if R¯t is conditionally convex and either t−1 closed or conditionally convex upper continuous then ρt and (Rs)s=0 have the same property and vice versa. Proof. (1) It is easy to check that R¯t defined in (3) is a time decomposable conditional risk mea- sure on the optional filtration as defined in Definition 2.8. This is done directly by considering the equivalent definitions for the risk measure for processes and the sequence of conditional risk measures. Due to simplicity, we omit the direct constructions herein. (2) It is easy to check that ρt defined in (4) is a risk measure for processes and Rs, for s ∞ d in (5) is a conditional risk measure for vectors on Ls (R ). Notably, for ρt, the result follows directly from [11, Proposition 3.4] with only verification required as we no longer require the full eligible space, i.e., M ⊆ Rd; due to simplicity of those results, we omit those proofs. However, due to the updated definition for finiteness at zero, that result needs to be proven for ρt separately from the approach of [11] but still follows trivially by construction (recalling that P[mint∈T µt > ǫ] = 1 for some ǫ> 0) and thus we omit the details. ∞ d (3) If R¯t is time decomposable and X ∈ L¯ (R ) then: t−1 ¯ ¯ ¯ Rt(X)= Rt(XI{s})I{s} + Rt(XITt )ITt s X=0 t−1 = Rs(Xs)I{s} + ρt(πt,T (X))ITt . s=0 X Finally, we want to study the closedness properties for conditionally convex risk measures. 19 (a) Closed: • Assume R¯t is closed. First, we will show that Rs is closed for any s ∞ d graph Rs = {(Xs,ms) ∈ Ls (R ) × Ms : ms ∈ Rs(Xs)} ∞ d ¯ = {(Xs,ms) ∈ Ls (R ) × Ms : ms ∈ Rt(XsI{s})s} ∞ d = {(X,m) ∈ L¯ (R ) × M¯ t : m ∈ R¯t(X)}s = (graph R¯t)s. n n ∞ d 1 d Let (Xs ,ms )n∈N ⊆ graph Rs → (Xs,ms) in σ(Ls (R ),Ls(R )) be a convergent net indexed n n 0 by N. By Lemma 2.4 and assumption, it must follow that (Xs I{s},ms I{s} + m I{s}c )n∈N → 0 ¯ 0 ¯ (XsI{s},msI{s} + m I{s}c ) ∈ graph Rt for any arbitrary m ∈ Rt(0). As (Xs,ms) ∈ graph Rs, closedness is proven. Consider now the closedness of ρt. By the same logic as above, ¯ graph ρt = {(XITt ,mt) : (X,m) ∈ graph Rt}. 0 Therefore, as above by concatenation with (0,m ) ∈ graph R¯t, graph ρt is closed. t−1 • Assume ρt and (Rs)s=0 are all closed risk measures. ∞ d graph R¯t = {(X,m) ∈ L¯ (R ) × M¯ t : m ∈ R¯t(X)} m ∈ R (X ) ∀s n n Let (X ,m ) ⊆ graph R¯t → (X,m) in the product topology. By the above construction and Lemma 2.4, it must follow that (Xs,ms) ∈ graph Rs for every time s Therefore closedness is proven as (X,m) ∈ graph R¯t. (b) Conditionally convex upper continuous: • Assume R¯t is conditionally convex upper continuous. First we will show that Rs is conditionally 20 convex upper continuous. Let Ds ∈ G(Ms; −Ms,+) be closed and conditionally convex. − ∞ d Rs (Ds)= {Xs ∈ Ls (R ) : Rs(Xs) ∩ Ds 6= ∅} ∞ d ¯ = {Xs ∈ Ls (R ) : Rt(XsI{s})s ∩ Ds 6= ∅} ∞ d = {X ∈ L¯ (R ) : R¯t(X)s ∩ Ds 6= ∅}s ¯− ¯ = Rt (DsI{s} + MtI{s}c )s. n − ∞ d 1 d Let Xs ⊆ Rs (Ds) → Xs in σ(Ls (R ),Ls(R )). By Lemma 2.4 and assumption, it must follow n ¯− ¯ that Xs I{s} → XsI{s} ∈ Rt (DsI{s} + MtI{s}c ). Therefore conditional convex upper continuity − is proven as Xs ∈ Rs (Ds). Consider now the conditional convex upper continuity of ρt. Let Dt ∈ G(Mt; −Mt,+) be closed and conditionally convex. By the same logic as above, t−1 − ¯− ρt (Dt)= Rt ( MsI{s} + DtITt )ITt . s=0 X − Therefore, as above, ρt (Dt) must be closed and the result follows. t−1 ¯ ¯ • Assume ρt and (Rs)s=0 are all conditionally convex upper continuous. Let D ∈ G(Mt; −Mt,+) be t−1 closed and conditionally convex. By decomposability in the optional filtration, D = s=0 DsI{s} + DtITt for Ds ∈ G(Ms; −Ms,+) closed and conditionally convex for every time s ≤ t. P ¯− ¯∞ d ¯ Rt (D)= {X ∈ L (R ) : Rt(X) ∩ D 6= ∅} ∞ d = {X ∈ L¯ (R ) : R¯t(X)s ∩ Ds 6= ∅∀s ≤ t} ∞ d = {X ∈ L¯ (R ) : Rs(Xs) ∩ Ds 6= ∅∀s < t, ρt(πt,T (X)) ∩ Dt 6= ∅} t−1 − − = Rs (Ds)I{s} + ρt (Dt)ITt . s=0 X n ¯− ¯∞ d ¯1 d n − Let X ⊆ Rt (D) → X in σ(L (R ), L (R )). By Lemma 2.4 and assumption, Xs → Xs ∈ Rs (Ds) n − for every s Given these results on the equivalence of risk measures for processes and vectors, we consider 21 the special case in which every asset is eligible to cover risk, i.e., M = Rd. This setting provides a simpler equivalent relation insofar as the restricted risk measures Rs utilized in Theorem 3.1 above can be fully defined by its value at 0. This setting more clearly demonstrates the relationship with ∞ d scalar risk measures, as Rs(0) = Ls (R+) in the d = 1 asset setting for any choice of normalized risk measure with closed values. Corollary 3.2. Let M = Rd. The equivalences in Theorem 3.1 can be simplified insofar as the conditional risk measures Rs can be replaced with (conditionally convex) upper sets Cs, i.e., Cs := R¯t(0)s t−1 ¯ Rt(X)= (−Xs + Cs)I{s} + ρt(πt,T (X))ITt . s X=0 Proof. This follows from Theorem 3.1 by noting that Rs(XI{s}) = Rs(0) − Xs by cash invariance for this choice of eligible assets. Therefore the full risk measure Rs is no longer required, but only the risk measure at 0 which we denote Cs. We conclude this section on the equivalence of primal representations by providing the relation between acceptance sets for processes and vectors on the optional filtration. Corollary 3.3. Fix time t ∈ T. ∞,d (1) Consider a conditional risk measure for processes ρt : Rt → U(Mt; Mt,+) and series of ∞ d ¯ conditional risk measures Rs : Ls (R ) →U(Ms; Ms,+) for s = 0, 1,...,t − 1, and Rt is defined as ¯ t−1 (3), then At = s=0 ARs I{s} + AtITt . P ∞ d (2) Given a conditional risk measure for random vectors on the optional filtration R¯t : L¯ (R ) → ¯ ¯ ∞,d ¯ U(Mt; Mt,+), define ρt and Rs via (4) and (5) respectively, then At = {X ∈ Rt : XITt ∈ At} ¯ and ARs = (At)s for every s = 0, 1,...,t − 1. Proof. This result follows directly by construction of the equivalences in Theorem 3.1. 3.2 Relation for dual representations Before presenting the following results, we want the reader to recall the dual representations for risk measures for processes in Corollary 2.6 and vectors in Corollary 2.10. We also highlight the 22 ∞ d form for the penalty functions for risk measures Rs : Ls (R ) → G(Ms; Ms,+) as provided in Remark 2.13. Finally, we remind the reader that any conditionally convex risk measure for vectors is time decomposable by construction as given in Remark 2.9. Lemma 3.4. Fix time t ∈ T. ∞,d (1) Let ρt : Rt → G(Mt; Mt,+) be a closed conditionally convex risk measure for processes and ∞ d let Rs : Ls (R ) → G(Ms; Ms,+), s G(M¯t; M¯ t,+) defined in (3) has dual representation w.r.t. the penalty function t−1 ¯ α¯t(Q, w¯) =α ¯t(Q ⊗ ψ, w¯)= αRs (w ¯s)I{s} + αt(Wt(Q ⊗ ψ, w¯))ITt s=0 X where Wt(Q ⊗ ψ, w¯) := (Qˆ , w) ∈Wt is defined as dQˆ s,i ψs,i µs := Q ξt,s(Qi)I Qi + I Qi , P i {Et [ψs,i]>0} E {Et [ψs,i]=0} d Et [ψs,i] t[µs] Qi Et [ψs,i] ws,i := t−1 w¯t,i 1 − r=0 µr P for indices i = 1, 2,...,d and times s ∈ Tt. (2) Consider a closed conditionally convex risk measure for random vectors on the optional fil- ∞ d tration R¯t : L¯ (R ) → G(M¯ t; M¯t,+). The associated closed conditionally convex risk measure ∞,d for processes ρt : Rt → G(Mt; Mt,+) and series of closed conditionally convex risk measures ∞ d Rs : Ls (R ) → G(Ms; Ms,+) defined in (4) and (5), respectively, have dual representations w.r.t. the penalty functions ¯ αRs (ws) =α ¯t(P, wsI{s})s, αt(Q, w) =α ¯t(W¯ t(Q, w))t, where W¯ t(Q, w) := (Q¯ , w¯) ∈ W¯ t is defined as w¯ := diag I{ws>0} wsITt , s∈Tt X 23 ¯ t−1 dQi 1 − r=0 µr ws,i := I + I + I ξt,s(Qs,i)IT dP¯ {s∈[0,t)} µ w¯ {w¯t,i>0} {w¯t,i=0} t s Ps t,i for indices i = 1, 2,...,d and times s ∈ Tt. Proof. (1) Recall the definition of the minimal penalty function α¯t from Corollary 2.10, i.e., ¯ ¯ Q¯ ¯ ¯ α¯t(Q, w¯)= Et [−Z]+ Γt(w ¯) ∩ Mt ¯ Z\∈At ¯ ⊤ ⊤ ¯ Q¯ ¯ = u ∈ Mt :w ¯ u ≥ ess sup w¯ Et [−Z] P-a.s. . ( Z∈A¯t ) Consider now the decomposition of the acceptance set A¯t as given in Corollary 3.3. Additionally, assume Q¯ i = Qi ⊗ ψi be the decomposition as provided in Corollary 2.2 for each component i of d the vector of measures Q¯ ∈ M¯ t(P¯) . Recall from Corollary 2.2 that ψs,i = µs for every asset i and t−1 time s α¯t(Q¯ , w¯) ¯ w¯⊤u ≥ t−1 ess sup w¯⊤E¯Q[−Z I ] s=0 Zs∈ARs t s {s} = u ∈ M¯ t : ¯ +esssupP w¯⊤E¯ Q[−ZI ] P¯-a.s. Z∈At t Tt t−1 t−1 ¯ ⊤ ⊤ ⊤ = u ∈ Mt : w¯s usI{s} +w ¯t utITt ≥ ess sup w¯s (−Zs)I{s} ( s=0 s=0 Z∈ARs X X 1 ⊤EQ I P¯ +esssup w¯t t [ t−1 diag(ψs)(−Zs)] Tt -a.s. Z∈At s T 1 − r=0 µr ) X∈ t t−1 ⊤ P ⊤ = us ∈ Ms :w ¯s us ≥ ess sup w¯s (−Zs) P-a.s. I{s} s=0 ( Z∈ARs ) X diag(ψ ) ⊤ ⊤EQ s P I + ut ∈ Mt :w ¯t ut ≥ ess sup w¯t t [ t−1 (−Zs)] -a.s. Tt . ( Z∈At s T 1 − r=0 µr ) X∈ t P By construction of (Qˆ , w)= Wt(Q¯ , w¯), ˆ 1 ⊤EQs ⊤EQ ws t [Ys] =w ¯t t [ t−1 diag(ψs)Ys] s T s T 1 − r=0 µr X∈ t X∈ t P 24 ∞,d ¯ ˆ for any Y ∈ Rt . The result follows if Wt(Q, w¯) = (Q, w) ∈Wt. In fact, this holds by the relation s ˆ t−1 T ¯ 1 d wt (Qs, ws)= µs/(1 − r=0 µr)w ¯t (Q, w¯)s ∈ Ls(R+). P (2) First, recall the definition of the minimal penalty function αRs for the risk measure Rs : ∞ d Ls (R ) → G(Ms; Ms,+) as provided in Remark 2.13, i.e., αRs (w)= {(−Z +Γs(w)) ∩ Ms} Z A ∈\Rs ⊤ ⊤ = u ∈ Ms : w u ≥ ess sup w (−Z) P-a.s. ( Z∈ARs ) 1 d ⊥ for any w ∈ Ls(R+)\Ms . Using the construction of ARs as given in Corollary 3.3, the construction ¯ ¯ of αRs in this lemma is trivial since (P, wI{s}) ∈ Wt by inspection. Second, recall the definition of the minimal penalty function αt for the risk measure ρt for processes as given in Corollary 2.6, i.e., Qs αt(Q, w)= Et [−Zs]+Γt(ws) ∩ Mt Z A s T \∈ t X∈ t ⊤ ⊤ Qs = u ∈ Mt : ws u ≥ ess sup ws Et [−Zs] P-a.s. ( s T Z∈At s T ) X∈ t X∈ t ⊤ I s∈Tt ws diag( {ws>0})u ≥ = u ∈ Mt : Pess sup w⊤ diag(I )EQs [−Z ] P-a.s. Z∈At s∈Tt s {ws>0} t s P for any (Q, w) ∈ Wt. The terminal equality above follows from the construction of the eligible assets and ws ≥ 0 Qs-a.s.. By construction of (Q¯ , w¯)= W¯ t(Q, w), ¯ ⊤ ¯Q ⊤ Qs w¯ Et [Y ] = ws Et [Ys] r T sX∈ t ∞ d for any Y ∈ L¯ (R ) and r ∈ Tt. Therefore, using the construction of At as given in Corollary 3.3, the definition of αt as provided in this lemma holds so long as W¯ t(Q, w) ∈ W¯ t. In fact, W¯ t(Q, w) ∈ ¯ T ¯ Q t−1 −1 s Q I ¯1 Rd Wt holds by the relation w¯t (Wt( , w)) = (1 − r=0 µr) s∈Tt µs wt ( s, ws) {s} ∈ L ( +). P P Remark 3.5. We highlight that, in the full eligible space setting M = Rd with the notation given in 25 ∗ Corollary 3.2, the penalty function representation simplifies insofar as αRs (w ¯s)=Γs(w ¯s) if w¯s ∈ Cs and ∅ otherwise. We, additionally, consider the relation between the dual representations for conditionally co- herent risk measures. This is provided in the following corollary utilizing the maximal set of dual variables. Corollary 3.6. Fix time t ∈ T. ∞,d (1) Let ρt : Rt → G(Mt; Mt,+) be a closed conditionally coherent risk measure for processes and ∞ d let Rs : Ls (R ) → G(Ms; Ms,+), s G(M¯t; M¯ t,+) defined in (3) has dual representation w.r.t. the set of dual variables ¯ max ¯ ¯ max ¯ max Wt = {(Q, w¯) ∈ Wt :w ¯s ∈WRs ∀s = 0,...,t − 1, Wt(Q, w¯) ∈Wt }. (2) Consider a closed conditionally coherent risk measure for vectors on the optional filtration R¯t : ∞ d L¯ (R ) → G(M¯t; M¯ t,+). The associated closed conditionally coherent risk measure for processes ∞,d ∞ d ρt : Rt → G(Mt; Mt,+) and series of closed conditionally coherent risk measures Rs : Ls (R ) → G(Ms; Ms,+) defined in (4) and (5), respectively, have dual representations w.r.t. the set of dual variables max ¯ ¯ max WRs = {w¯s : (Q, w¯) ∈ Wt }, ¯ ¯ ¯ d Q∈M[t(P) max ¯ ¯ max Wt = {(Q, w) ∈Wt : Wt(Q, w) ∈ Wt }. Proof. This is an immediate consequence of Lemma 3.4 utilizing the indicator notion of the penalty functions, i.e., Q max s∈Tt Γt(ws) ∩ Mt if ( , w) ∈Wt αt(Q, w)= P ∅ else, 26 max Γs(ws) ∩ Ms if ws ∈WRs αRs (ws)= ∅ else, ¯ ¯ ¯ ¯ max Γt(w ¯) ∩ Mt if (Q, w¯) ∈ Wt α¯t(Q¯ , w¯)= ∅ else. We now use the results of this section in order to prove the dual representation for risk measures for processes as presented in Corollary 2.6. Specifically, we take advantage of the known dual representation for set-valued risk measures for vectors (Corollary 2.10) and the equivalence of forms for the set-valued risk measures for processes and vectors presented above in order to prove this new dual representation for risk measures for processes. ∞,d Proof of Corollary 2.6. Let ρt : Rt → G(Mt; Mt,+) be some closed conditionally convex risk measure for processes. Additionally, fix a sequence of closed and conditionally convex risk measures ∞ d Rs : Ls (R ) → G(Ms; Ms,+) for s ¯ ¯ Q ¯ ¯ • ¯ ρt(X)= Et [−XITt ]+ Γt(w ¯) ∩ Mt − α¯t(Q, w¯) t (Q¯ ,w¯)∈W¯ t \ Q ψs E −1 t − s∈Tt t Xs +Γt(w ¯t) ∩ Mt 1−Pr=0 ψr = •h i (Q⊗ψ,w¯)∈W¯ t − α¯t(PQ ⊗ ψ, w¯)t \ ˆ Qs • ˆ = Et [−Xs]+Γt(ws) ∩ Mt − αt(Q, w) (Q⊗ψ,w¯)∈W¯ t s∈Tt \ X where (Qˆ , w)= Wt(Q¯ , w¯) is defined as in Lemma 3.4(1). It remains to show that s s ¯ ¯ ¯ {(wt (Qs, ws))s∈Tt : (Q, w) ∈Wt} = {(wt (Wt(Q, w¯)s))s∈Tt : (Q, w¯) ∈ Wt}. 27 First, by construction, {Wt(Q¯ , w¯) : (Q¯ , w¯) ∈ W¯ t}⊆Wt which implies ⊇ in the desired equality. Second, for any (Q, w) ∈Wt there exists W¯ t(Q, w) ∈ W¯ t (as constructed in Lemma 3.4(2)) so that s s ¯ wt (Qs, ws)= wt (Wt(Wt(Q, w))s) for any s ∈ Tt which implies ⊆ in the desired equality. Thus the proof in the convex case is complete. For the conditionally coherent case. Define ⊤EQs P Γt(ws) ∩ Mt if ess inf s∈Tt ws t [Zs] = 0 -a.s. s∈T Z∈At βt(Q, w)= t P P ∅ else. Then from the positive homogeneity of ρ, we have that βt(Q, w) ⊆ αt(Q, w), which yields that Qs • Et [−Xs]+Γt(ws) ∩ Mt − αt(Q, w) s∈Tt X Qs • ⊆ Et [−Xs]+Γt(ws) ∩ Mt − βt(Q, w). s∈Tt X Note that Qs • Et [−Xs]+Γt(ws) ∩ Mt − ( Γt(ws)) ∩ Mt s T s T X∈ t X∈ t Qs = Et [−Xs]+Γt(ws) ∩ Mt, s T X∈ t Qs • Et [−Xs]+Γt(ws) ∩ Mt − ∅ = Mt, T sX∈ t we obtain that Qs ρt(X) ⊆ Et [−Xs]+Γt(ws) ∩ Mt. Q max s∈T ( ,w)\∈Wt Xt Finally, from (2), it follows that ut ∈ ρt(X) if and only if ⊤ ⊤ Qs ⊤ Qs ess inf ws ut + esssup ws Et [−Zs]+ ws Et [Xs] ≥ 0. (Q,w)∈W t ( T Z∈At T T ) sX∈ t sX∈ t sX∈ t 28 Qs Assume u ∈ max (E [−Xs]+Γt(ws)) ∩ Mt. Then, immediately, (Q,w)∈Wt s∈Tt t T P ⊤ ⊤ Qs ⊤ Qs ess inf ws u + esssup ws Et [−Zs]+ ws Et [Xs] ≥ 0 P-a.s. (Q,w)∈Wmax t T Z∈At T T sX∈ t sX∈ t sX∈ t Additionally and clearly, P-a.s. ⊤ ⊤EQs ⊤EQs ess inf max ws u + esssup ws t [−Zs]+ ws t [Xs] ≥ 0. (Q,w)∈Wt\W t (s T Z∈At s T s T ) X∈ t X∈ t X∈ t Hence, the essential infimum taken over the full set of dual variables Wt must also be bounded from below by 0 P-a.s. which implies that u ∈ ρt(X) and the equivalence is shown. 4 Equivalence of multiportfolio time consistency Though multiportfolio time consistency of set-valued risk measures for processes was presented in [11], in this work we find that a slight variation of that definition is useful for this work. In particular, motivated by Theorem 3.1, we consider a joint definition of both the risk measure for processes ρt and the series of restricted conditional risk measures for vectors Rs. t t−1 Definition 4.1. The pair (ρt, (Rs)s=0)t∈T is jointly multiportfolio time consistent if: (1) ρ is multiportfolio time consistent as in Definition 2.7; ∞ d ∞ d ∞,d (2) for any times t s s Rr(Xr) ⊆ Rr(Yr) ∀r ∈ [t,s) Y B r[∈ r s−1 s−1 ⇒ ρt( XrI{r} + ZITs ) ⊆ · · · ρt( YrI{r} + ZITs ); r=t Y B Y −1 B −1 r=t X t[∈ t s [∈ s X ∞ d ∞ d (3) for any times r s s t t Rr(Xr) ⊆ Rr(Yr) ⇒ Rr(Xr) ⊆ Rr(Yr). Y B Y B r[∈ r r[∈ r Conceptually, (ρ, R) is jointly multiportfolio time consistent if, in addition to multiportfolio 29 time consistency it satisfies two additional consistency properties. (2) If two claims are identical after time s ∈ T and the restricted risk measures (associated with ρs) on the claims at r ∈ [t,s) are ordered as multiportfolio time consistency, then the claims as measured by ρt at time t must also satisfy the ordering of risks; i.e., the restricted risk measures are consistent in time with the risk measures for processes. And (3) the ordering of portfolios induced by the restricted risk measure at time r t s of the risk measure for processes with which it is associated, i.e., Rr = Rr for r We now turn to the final main result of this work – the equivalence of multiportfolio time consistency for set-valued risk measures for processes and vectors. This is, again, akin to the results presented for scalar risk measures in [1]. Fundamentally, as the following proof is demonstrated for the definition of multiportfolio time consistency, these results can be used to construct the various equivalent formulations as well; in particular, we highlight the recursive formulation presented in [18, 11] and the cocycle condition for penalty functions in [19]. Theorem 4.2. (1) Consider a jointly multiportfolio time consistent pair of risk measures (ρ, R). ∞ d The associated time decomposable risk measure on the optional filtration R¯t : L¯ (R ) →U(M¯ t; M¯ t,+) defined in (3) is multiportfolio time consistent. (2) Consider a multiportfolio time consistent, time decomposable risk measure for random vectors ¯ ∞,d on the optional filtration R. The associated risk measure for processes ρt : Rt → U(Mt; Mt,+) t ∞ d and series of risk measures Rs : Ls (R ) →U(Ms; Ms,+), s = 0,...,t − 1 for t = 0,...,T , defined in (4) and (5) are jointly multiportfolio time consistent. Proof. Throughout we will fix times t (1) Let B := {(b0, b1, . . . , bs−1, bs) : br ∈ Br, r < s, bs ∈ Bs} for any arbitrary sequence of sets ∞ d ∞,d ¯∞ d Br ⊆ Lr (R ) for r R¯s(X) ⊆ R¯s(Y ). Y B [∈ s s I I By construction, this implies Rr(Xr) ⊆ Yr∈Br Rr(Yr) for every r ρt(XITt ) ⊆ · · · ρt( YrI{r} + XITs ) Y B Y −1 B −1 r=t t[∈ t s [∈ s X s−1 ⊆ · · · ρt( YrI{r} + YsITs )= ρt(Y ITt ), Y B Y −1 B −1 Y B r=t Y B t[∈ t s [∈ s s[∈ s X [∈ where the first inclusion follows from the second condition of joint multiportfolio time consistency and the second inclusion from the first condition of joint multiportfolio time consistency. Therefore, ¯ ¯ ¯ by construction of Rt it immediately follows that Rt(X) ⊆ Y ∈B Rt(Y ). S (2) Note, first, since (R¯t)t∈T is time decomposable, from Theorem 3.1, we have that t−1 ¯ t Rt(X)= Rr(Xr)I{r} + ρt(πt,T (X))ITt r X=0 for any time t ∈ T. For this proof, we will only directly demonstrate the first condition of joint multiportfolio time consistency; the latter two properties follow from identical arguments. Fix X ∈ R∞,d and B ⊆ R∞,d such that ρs(πs,T (X)) ⊆ ρs(πs,T (Y )). Y B [∈ This implies, by the decomposition of R¯s, that ¯ ¯ Rs(ZI{[t,s)} + XITs ) ⊆ Rs(ZI{[t,s)} + Y ITs ) Y B [∈ for any Z ∈ R∞,d. Utilizing multiportfolio time consistency of R¯, we recover ¯ ¯ Rt(ZI{[t,s)} + XITs ) ⊆ Rt(ZI{[t,s)} + Y ITs ) Y B [∈ 31 ∞,d for any Z ∈ R . By the decomposition of R¯t, we immediately conclude ρt(ZI{[t,s)} + XITs ) ⊆ ρt(ZI{[t,s)} + Y ITs ) Y B [∈ for any Z ∈ R∞,d, i.e., ρ is multiportfolio time consistent. From Theorems 3.1 and 4.2 and using results in [19, 20], one can steadily deduce some equivalent characterizations of multiportfolio time consistency for set-valued dynamic risk measures for pro- cesses, such as the cocycle condition on the sum of minimal penalty functions and supermartingale relation. Remark 4.3. Consider the setting in which all assets are eligible, i.e., assume that M = Rd. Let ρ be a multiportfolio time consistent risk measure for processes (see Definition 2.7) and define ∞ d ∞ d ∞ d Rs : Ls (R ) →U(Ls (R ); Ls (R+)) by ∞ d ∞ d Rs(Xs) := −Xs + Ls (R+), Xs ∈ Ls (R ). By monotonicity, (ρ, R) is jointly multiportfolio time consistent. Therefore, by Theorem 4.2, ρ is multiportfolio time consistent if and only if the associated risk measure for vectors R¯ is multiport- folio time consistent. We conclude this discussion by remarking that, in this discrete time setting, it is sufficient to consider multiportfolio time consistency defined for single time steps only through a sequential application of the recursive definition (i.e., setting s = t + 1). This is discussed in, e.g., [19] as well. 5 Conclusion In this work we demonstrated the equivalence between set-valued risk measures for processes with those for random vectors on the optional filtration. Such considerations allow for the application of results from one stream of literature to the other. In particular, we highlight that the set-valued risk measures for vectors is a more mature field with more results on dual representations and equivalent 32 forms for multiportfolio time consistency. Herein, we used this equivalence to prove a new dual representation for risk measures for processes. 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