Math Finan Econ https://doi.org/10.1007/s11579-017-0205-0

Time consistency for set-valued dynamic risk measures for bounded discrete-time processes Yanhong Chen1 · Yijun Hu1

Received: 20 January 2017 / Accepted: 18 November 2017 © Springer-Verlag GmbH Germany, part of Springer Nature 2017

Abstract In this paper, we introduce two kinds of time consistent properties for set-valued dynamic risk measures for discrete-time processes that are adapted to a given filtration, named time consistency and multi-portfolio time consistency. Equivalent characterizations of multi-portfolio time consistency are deduced for normalized dynamic risk measures. In the normalized case, multi-portfolio time consistency is equivalent to the recursive form for risk measures as well as a decomposition property for the acceptance sets. The relations between time consistency and multi-portfolio time consistency are addressed. We also provide a way to construct multi-portfolio time consistent versions of any dynamic risk measure. Finally, we investigate the relationship about time consistency and multi-portfolio time consistency between risk measures for processes and risk measures for random vectors on some product space. Keywords Dynamic risk measures · Set-valued risk measures · Bounded discrete-time processes · Time consistency · Multi-portfolio time consistency Mathematics Subject Classification 91B30 · 91B32 · 91B70 JEL Classification G32 · C61 · D81 · G15

Supported by the National Natural Science Foundation of China (Nos. 11371284, 11771343).

B

Yanhong Chen [email protected] Yijun Hu [email protected]

1

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, Hubei, People’s Republic of China

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1 Introduction In their seminal paper, Artzner et al. [1,2] first introduced the concept of coherent risk measure, by proposing four basic properties to be satisfied by every sound financial risk measure. Further, Föllmer and Schied [19] and, independently, Frittelli and Rosazza Gianin [17] introduced the broader class, named convex risk measure, by dropping one of the coherency axioms. To consider risk evaluations of random vectors motivated by market models with transaction costs, Jouini et al. [27] introduced set-valued coherent risk measures in a static setting (also called one-period setting). Later, Hamel [22] developed the approach of Jouini et al. [27], and studied set-valued convex risk measures. For more details on set-valued risk measures in a static setting, see Hamel and Heyde [23], Hamel et al. [24,25], Labuschagne and Offwood-Le Roux [31], Farkas et al. [11], Molchanov and Cascos [33], and the references therein. Recently, Feinstein and Rudloff [12] addressed set-valued dynamic convex and coherent risk measures and introduced a set-valued version of time consistency, called multi-portfolio time consistency. Multi-portfolio time consistency has been further studied by Feinstein and Rudloff [13,15,16]. From a viewpoint different from that of Feinstein and Rudloff [12,13,15,16], Tahar and Lépinette [39] also studied set-valued dynamic coherent risk measures. Feinstein and Rudloff [14] gave an overview of results for set-valued dynamic risk measures. In all the above-mentioned works, a dynamic financial position is described by a random vector. However, as pointed out by Jobert and Rogers [26], risk assessment in the dynamic setting should also account for uncertainty about the time value of money and it is not sufficient to consider only the total amount of cash accumulated by some arbitrary time in the future, as the experience of Long Term Capital Management demonstrated. This requires to consider entire cash flow processes rather than total amounts at terminal dates as risky objects. In other words, a dynamic financial position should be described by a stochastic process rather than a random vector. Based on this consideration, Chen and Hu [5] studied the set-valued dynamic risk measures for stochastic processes, where a dynamic financial position is described by a stochastic process rather than a random vector. In the present paper, we will study time consistency of set-valued dynamic risk measures for discrete-time processes introduced by Chen and Hu [5]. Namely, we will introduce two kinds of time consistent properties, named time consistency and multi-portfolio time consistency, and will discuss the relations between them. Equivalent characterizations of multi-portfolio time consistency will be given in terms of a decomposition property on acceptance sets and the recursive form for risk measures. We will also provide a way to construct multi-portfolio time consistent versions of any dynamic risk measure. Finally, we will also investigate the relationship about time consistency and multi-portfolio time consistency between risk measures for processes and risk measures for random vectors on some product space. It is also worth mentioning that scalar dynamic risk measures for stochastic processes have also been extensively studied in the last two decades. For example, see Ridel [36], Cheridito et al. [6–8], Frittelli and Scandolo [18], Artzner et al. [3], Jobert and Rogers [26], Cheridito and Kupper [9], Acciaio et al. [4], Penner and Réveillac [34], and the references therein. The rest of the paper is organized as follows. In Sect. 2, we will review some notations and properties of set-valued dynamic risk measures for discrete-time processes from Chen and

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Hu [5], and introduce the definitions of time consistencies. The main results will be stated in Sect. 3, and their proofs will be postponed to Sect. 4.

2 Preliminaries In this section, we will introduce some notations and review some definitions and results for set-valued dynamic risk measures for bounded discrete-time processes from Chen and Hu [5]. Throughout this paper, the notations of τ , s and t are all understood as deterministic times (rather than stopping times). We denote N := {1, 2, . . . }, N0 := {0} ∪ N. Fix a finite time horizon T ∈ N. We denote by T the set of time points, i.e. T := {0, 1, . . . , T } and Tt := {s ∈ T : s ≥ t} for t ∈ T. Let (, F , (Ft )t∈T , P) be a filtered probability space with F0 = {, ∅} and FT = F . 0,d Let d ∈ N be the number of assets under consideration. the space  By R 1we denote  of all d-dimensional adapted stochastic processes X := X t := (X t , . . . , X td )tr ; t ∈ T on (, F , (Ft )t∈T , P), where (X t1 , . . . , X td )tr means the transpose of (X t1 , . . . , X td ). Denote by X i the adapted stochastic process {X ti ; t ∈ T} for each 1 ≤ i ≤ d, by X t the random vector (X t1 , . . . , X td )tr for each t ∈ T. In this paper, all equalities and inequalities between random vectors or stochastic processes are c-wise in the P-a.s. sense. Define     R∞,d := X ∈ R0,d : X R∞,d < ∞ and A1,d := a ∈ R0,d : aA1,d < ∞ with the corresponding norms X R∞,d := where

max X i R∞

and

i∈{1,...,d}

aA1,d :=

d 

a i A1 ,

i=1

     X i R∞ := inf r ∈ R : sup X ti  ≤ r , t∈T

     i i := E := 0, ati := ati − at−1 ,  ati  , a−1

a i A1

t∈T

and E



 

 i   i t∈T  at  means the integral of t∈T  at  with respect to the probability P.

For every time 0 ≤ τ ≤ s ≤ T , we define the projection πτ,s : R0,d → R0,d by πτ,s (X )t := 1{τ ≤t} X t∧s , t ∈ T, ∞,d and A1,d ⊆ A1,d by and the spaces R∞,d τ,s ⊆ R τ,s     ∞,d 1,d and A1,d , R∞,d τ,s := πτ,s R τ,s := πτ,s A

and use the notions 



R∞,d := πτ,T R∞,d = {0}τ × τ

T 

L∞ d ( Ft )



and



1,d . A1,d τ := πτ,T A

t=τ

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Moreover, denote  

  i := a ∈ A1,d τ : at ≥ 0 for all t ∈ T, i = 1, . . . , d , +     ∞,d Rτ := X ∈ R∞,d : X ti ≥ 0 for all t ∈ T, i = 1, . . . , d . τ A1,d τ

+

A stochastic process X ∈ R∞,d is meant to describe the evolution of a financial value on τ the discrete time interval [τ, T ] ∩ N0 . As in Kabanov [28] and discussed by Schachermayer [38], Kabanov and Safarian [29] and Feinstein and Rudloff [12,13], the value processes in this paper are in ‘physical units’ of an asset rather than the value in a fixed numéraire. That is, for a value process X ∈ R∞,d , the values of X ti (for 1 ≤ i ≤ d, t ∈ [τ, T ] ∩ N0 ) are the τ number of units of asset i in the value process at time t. Moreover, the bilinear form ·, ·d on R∞,d × A1,d τ τ is given by

T   tr X, ad := E X t at . t=τ ∞,d such that for all a ∈ A1,d , X → , A1,d σ (R∞,d τ τ ) denotes the coarsest topology on Rτ τ ∞,d ∞,d ) denotes the coarsest X, ad is a continuous linear functional on Rτ . σ (A1,d τ , Rτ ∞,d , a → X, a is a continuous linear functional topology on A1,d d τ such that for all X ∈ Rτ 1,d on Aτ . ∞ We denote by L ∞ d (Fτ ) := L d (, Fτ , P) the linear space of Fτ -measurable functions X :  → Rd such that X ∞ = ess supω∈ |X (ω)| < +∞, where | · | is an arbitrary norm in Rd , and by L 1d (Fτ ) := L 1d (, Fτ , P) the linear space of Fτ -measurable functions X :  → Rd such that X 1 =  |X (ω)|d P < +∞. In this paper, we will consider the   1 norm topology on L 1d (Fτ ) and the weak∗ -topology (i.e. σ L ∞ d (Fτ ), L d (Fτ ) -topology) on L ∞ d (Fτ ). For the purpose of presentation, we adopt the notations as in Feinstein and p p d , P-a.s.} Rudloff [12–14]. For p ∈ {1, ∞}, write L d (Fτ )+ := {X ∈ L d (Fτ ) : X ∈ R+ d for the closed convex cone of R -valued Fτ -measurable random vectors with P-a.s. nonp p d , P-a.s.} for negative components, and write L d (Fτ )++ := {X ∈ L d (Fτ ) : X ∈ R++ d the closed convex cone of R -valued Fτ -measurable random vectors with P-a.s. positive ∞ components. We denote by L ∞ d (Fτ ; Bτ ) those random vectors in L d (Fτ ) that take P-a.s. ∞ values in Bτ . The multiplication between a random variable λ ∈ L (Fτ ) and a set of random vectors B ⊆ L ∞ d (Fτ ) is understood as in the elementwise sense, i.e. λB := {λX : X ∈ B} ∞ ⊆ L d (Fτ ) with (λX )(ω) = λ(ω)X (ω). Let Mτ be an Fτ -measurable set such that Mτ (ω) ∗ is a linear subspace of Rd for almost every ω ∈ . Then Mτ := L ∞ d (Fτ ; Mτ ) is aweak ∞ 1 closed linear subspace of L d (Fτ ). We endow Mτ with the topology σ Mτ , L d (Fτ ) , which   is the coarsest topology on Mτ such that for all v ∈ L 1d (Fτ ), u → u, v := E u tr v is a   continuous linear functional on Mτ . Note that the topology σ Mτ , L 1d (Fτ ) is equivalent    1 ∗ to the subspace topology D ∩ Mτ : D ∈ σ L ∞ d (Fτ ), L d (Fτ ) of the weak -topology for ∞ ∞ ∞ L d (Fτ ). Denote (Mτ )+ := Mτ ∩ L d (Fτ )+ and (Mτ )++ := Mτ ∩ L d (Fτ )++ . We can analogously define (Mτ )− and (Mτ )−− . We will assume that (Mτ )+ is non-trivial, i.e. (Mτ )+  = {0}. Denote ((Mτ )+ )+ := {v ∈ L 1d (Fτ ) : E[u tr v] ≥ 0 for any u ∈ (Mτ )+ } and Mτ⊥ := {v ∈ L 1d (Fτ ) : E[u tr v] = 0 for any u ∈ Mτ }. Let 2Mτ denote the power set of Mτ including the empty set. Given a set A ⊆ Mτ , co (A) means the convex hull  of A, and cl (A) means the closure of A with respect to the topology σ Mτ , L 1d (Fτ ) . Denote P (Mτ ; (Mτ )+) := {D ⊆ Mτ : D = D + (Mτ )+ }, G (Mτ ; (Mτ )+ ) := {D ⊆ Mτ : D = cl co (D + (Mτ )+ )}.

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Note that for X ∈ R∞,d and γ ∈ L ∞ d (Fτ ) the processes Y = X 1[τ,s) and Z = γ 1[τ,s) are given by  X ti (ω), t = τ, τ + 1, . . . , s − 1, i Yt (ω) = 0, otherwise,  γ i (ω), t = τ, τ + 1, . . . , s − 1, i Z t (ω) = 0, otherwise for t ∈ T, ω ∈ , i ∈ {1, 2, . . . , d}. We denote by Md (P) the set of d-dimensional probability measures which are absolutely continuous with respect to P. Let diag(w) be the diagonal matrix with the elements of a vector w as the main diagonal. We will consider a P-a.s. version of the Q-conditional expectation of X ∈ L ∞ d (FT ). For any Q := (Q 1 , . . . , Q d ) ∈ Md (P), any τ, t ∈ {0, 1, . . . , T } with τ < t, and X ∈ L ∞ d (Ft ), let   E Q [X | Fτ ] := E diag (ξτ +1 (Q)) · · · diag (ξt (Q)) X | Fτ , where ξs (Q) := (ξ s (Q 1 ), . . . , ξ s (Q d ))tr for any time 1 ≤ s ≤ T with  ⎧ dQ  i ⎪ ⎨ E d P |F s (ω)  , if E ddQPi | Fs−1 (ω) > 0, d Qi ξ s (Q i )[ω] := E d P |F s−1 (ω) ⎪ ⎩ 1, otherwise for any ω ∈ . In this way the conditional expectation E Q [X | Fτ ] is defined P-almost surely. See e.g. Cheridito and Kupper [9], Feinstein and Rudloff [12–14]. We further define the function w τt : Md (P) × L 1d (Fτ ) → L 0d (Ft ) for any t, τ ∈ {0, 1, . . . , T } with τ < t by w τt (Q, w) := diag(w)diag (ξτ +1 (Q)) · · · diag (ξt−1 (Q)) ξt (Q) ττ (Q, w) := w for any for any Q ∈ Md (P) and w ∈ L 1d (Fτ ). Additionally, we define w τ ∈ {0, 1, . . . , T }, Q ∈ Md (P) and w ∈ L 1d (Fτ ). Finally, in order to study the relationship between conditional risk measures for processes and conditional risk measures for random vectors, similar to Acciaio et al. [4], we also  defined by , P) , F consider the product space (  =  × T, 

 = σ ({At × {t} : At ∈ Ft , t ∈ T}) , F

where μ = (μt )t∈T is some adapted reference process such that for any t ∈ T, and where 

 P⊗μ E X t μt [X ] := E

 = P ⊗ μ, P t∈T μt

= 1 and μt > 0

t∈T

). , F for any bounded measurable function X on (  coincides with the optional σ -field generated by all adapted processes. Every Note that F  and in particular we , P), , F adapted process can be identified with a random variable on ( have     ∞     R∞,d =  L∞ d F := L d , F , P . ) given by τ )τ ∈T on ( , F We introduce the optional filtration (F   τ = σ D j × { j}, Dτ × Tτ : D j ∈ F j , j < τ, Dτ ∈ Fτ , τ ∈ T. F

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 is F , P) τ -measurable if and only if X t is Ft , F A random variable X = (X t )t∈T on ( measurable for all t = 0, . . . , τ and X t = X τ for any t > τ . In particular,     ∞,d ∞     R0,τ =  L∞ d Fτ := L d , Fτ , P . ∗  τ :=   L∞ Denote Mτ∗ := M0 × · · · × Mτ −1 × Mτ × Mτ × · · · × Mτ . M d (Fτ ; Mτ ). Then  τ if and only if m t ∈ Mt for all  belongs to M , P) , F a random vector m = (m t )t∈T on ( t = 0, . . . , τ , and m t = m τ for any t > τ . On a general level, a set-valued conditional risk measure on R∞,d is a mapping τ

→ P (Mτ ; (Mτ )+ ) ρτ : R∞,d τ a subset of the power set of the space Mτ of eligible assigning a value process X ∈ R∞,d τ portfolios at time τ that can depend on the information available at time τ . The set ρτ (X ) contains those portfolios that can be used to cover the risk of X . The definition of set-valued conditional risk measures for processes was given in Chen and Hu [5]. Here, we will make some modification, and the original definition in Chen and Hu [5] and the differences between these two definitions will be addressed in Remark 2.1 below. Definition 2.1 A function ρτ : R∞,d → P (Mτ ; (Mτ )+ ) for τ ∈ T is called a set-valued τ conditional risk measure (for processes) if it satisfies the following properties for all X, Y ∈ R∞,d , τ , (A1) Mτ -translative: for any m ∈ Mτ and X ∈ R∞,d τ   ρτ X + m1Tτ = ρτ (X ) − m; (A2) Monotone: ρτ (X ) ⊆ ρτ (Y ) if X ≤ Y component-wisely; (A3) Finite at zero: ∅  = ρτ (0)  = Mτ . Conceptually, Mτ -translativity implies that the infusion of an amount of money m at time τ transforms a value process X ∈ R∞,d into X + m1Tτ and its risk decreases by m. τ Monotonicity also has a clear interpretation: if a value process Y ∈ R∞,d dominates another τ value process X ∈ R∞,d , then there should be more possibilities to compensate the risk of τ Y than for X . Finiteness at zero means that there is an eligible portfolio at time τ that covers the risk of the zero payoff, but not all portfolios compensate for it. In this paper, we will use the notation ρτ (X ) := ρτ ◦ πτ,T (X ) for any X ∈ R∞,d . Definition 2.2 A set-valued conditional risk measure ρτ : R∞,d → P (Mτ ; (Mτ )+ ) is τ called (0) normalized if ρτ (X ) = ρτ (X ) + ρτ (0) for every X ∈ R∞,d ; τ (1) convex if, for all X, Y ∈ R∞,d and all λ ∈ [0, 1], τ ρτ (λX + (1 − λ)Y ) ⊇ λρτ (X ) + (1 − λ)ρτ (Y ); (2) conditionally convex if, for all X, Y ∈ R∞,d and all λ ∈ L ∞ (Fτ ) with 0 ≤ λ ≤ 1, τ ρτ (λX + (1 − λ)Y ) ⊇ λρτ (X ) + (1 − λ)ρτ (Y ); (3) positively homogeneous if, for all X ∈ R∞,d and all λ > 0, τ ρτ (λX ) = λρτ (X );

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(4) conditionally positively homogeneous if, for all X ∈ R∞,d and all λ ∈ L ∞ (Fτ ) with τ λ > 0, ρτ (λX ) = λρτ (X ); (5) subadditive if, for all X , Y ∈ R∞,d , ρτ (X + Y ) ⊇ ρτ (X ) + ρτ (Y ); τ (6) coherent if it is convex and positively homogeneous, or subadditive and positively homogeneous; (7) conditionally coherent if it is conditionally convex and conditionally positively homogeneous, or subadditive and conditionally positively homogeneous; (8) closed if the graph of ρτ defined by   graph ρτ = (X, u) ∈ R∞,d × M : u ∈ ρ (X ) τ τ τ      ∞,d 1 , A1,d is closed in the product topology on Rτ , σ (R∞,d τ τ ) × Mτ , σ Mτ , L d (Fτ ) . → P (Mτ ; (Mτ )+ ) for τ ∈ T is Remark 2.1 In Chen and Hu [5], a function ρτ : R∞,d τ called a set-valued conditional risk measure (for processes) if it satisfies (A1), (A2) and  3): (Mτ )+ ⊆ ρτ (0) and ρτ (0) (Mτ )−− = ∅, (A 3) was also named the finiteness at zero. In fact, (A 3) is a property of normalization where (A in a ‘weaker’ sense, as was done by Hamel and Rudloff [21] and Hamel and Heyde [23]. 3) implies (A3) in Definition 2.1, the property of finiteness at zero. Similar Meanwhile, (A to Feinstein and Rudloff [12,13], in this paper, we will use the property of normalization given in Definition 2.2 for several reasons. First, it turns out to be the appropriate property when discussing time consistency of dynamic risk measures. Also, in the closed coherent set-valued case, both notions coincide (see Property 3.1 of Jouini et al. [27]). Second, a 3). Third, any conditional risk normalized closed conditional risk measure ρτ satisfies (A measure (that is finite at zero) can be normalized in the following way. The normalized version ρˆτ of a conditional risk measure ρτ is defined by ρˆτ (X ) := ρτ (X ) −· ρτ (0) := {u ∈ Mτ : u + ρτ (0) ⊆ ρτ (X )} , for any X ∈ R∞,d . The operation A −· B for sets A and B is sometimes called the Minkowski τ difference [20] or geometric difference [35]. This different notation trivially shows how the normalization procedure introduced above relates to the normalized version for scalar risk 3). measures given by ρτ (X ) − ρτ (0). The normalized version ρˆτ will also satisfy (A A sequence (ρτ )τ ∈T is called a set-valued dynamic risk measure (for processes) if for each τ , ρτ : R∞,d → P (Mτ ; (Mτ )+ ) is a set-valued conditional risk measure (for processes). It τ is said to have one of the properties given in Definition 2.2 if ρτ has this property for every τ ∈ T. Expanding upon the definition given by Feinstein and Rudloff [12] of acceptance sets for set-valued dynamic risk measures for random vectors, we now define acceptance sets for set-valued dynamic risk measures (for processes). Definition 2.3 Aτ ⊆ R∞,d is called an acceptance set at time τ if it satisfies the following: τ (1) Mτ 1Tτ ∩ Aτ  = ∅; (2) Mτ 1Tτ ∩ (R∞,d \Aτ )  = ∅; τ (3) Aτ + (R∞,d ) ⊆ Aτ ; + τ   where Mτ 1Tτ := m1Tτ : m ∈ Mτ .

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The following proposition, which is similar to Remark 2 of Feinstein and Rudloff [12], Proposition 2.5 of Hamel et al. [24], Proposition 2.1 of Artzner et al. [2] and other references, shows that there is a one-to-one relationship between acceptance sets and risk measures in our framework, and its proof will be given in Sect. 4. Proposition 2.1 There is a one-to-one relationship between acceptance sets and risk measures in our framework. (i) Suppose that ρτ is a conditional risk measure for processes, define   : 0 ∈ ρτ (X ) , Aρτ := X ∈ R∞,d τ

(2.1)

then Aρτ is an acceptance set. (ii) Let Aτ ⊆ R∞,d be an acceptance set, and define τ ρτAτ (X ) := {u ∈ Mτ : X + u1Tτ ∈ Aτ },

(2.2)

then ρτAτ is a conditional risk measure for processes. The set ρτAτ (X ) contains the eligible portfolios u that, when added to X at time τ and carried forward, make X +u1Tτ acceptable. (iii) Further, suppose that Aτ ⊆ R∞,d is an acceptance set, and that ρτ is a conditional τ A risk measure for processes, then it can be shown that Aτ = Aρ Aτ and ρτ (·) = ρτ ρτ (·). τ

The following proposition, which is similar to Proposition 6.5 of Hamel, Heyde and Rudloff [24] and Proposition 2.11 of Feinstein and Rudloff [12], is a list of corresponding properties between classes of conditional risk measures and classes of acceptance sets. If a risk measure ρτ has the (risk measure) property, then Aρτ has the corresponding (acceptance set) property. Vice versa, if an acceptance set Aτ satisfies the (acceptance set) property, then ρτAτ satisfies the corresponding (risk measure) property. All these properties can be verified by the same arguments as in Proposition 6.5 of Hamel, Heyde and Rudloff [24]. Hence, we will omit their proofs here. Proposition 2.2 The following properties are in a one-to-one relationship for a conditional risk measure ρτ : R∞,d → P (Mτ ; (Mτ )+ ) and an acceptance set Aτ ⊆ R∞,d : τ τ with Y − X ∈ B, we have ρτ (X ) ⊆ (a) ρτ is B-monotone (that is, if for any X, Y ∈ R∞,d τ ρτ (Y )), and Aτ + B ⊆ Aτ , where B ⊆ R∞,d ; τ (b) ρτ maps into the set P (Mτ ; C) := {D ⊆ Mτ : D = D + C},

(c) (d) (e) (f) (g) (h)

  and Aτ + C1Tτ = Aτ , where 0 ∈ C ⊆ Mτ , and C1Tτ := m1Tτ : m ∈ C ; ρτ is (conditionally) convex, and Aτ is (conditionally) convex; ρτ is (conditionally) positively homogeneous, and Aτ is a (conditional) cone; ρτ is subadditive, and Aτ + Aτ ⊆ Aτ ; ρτ has a closed graph, and Aτ is closed; ρτ (X )  = ∅ for all X ∈ R∞,d , and R∞,d = Aτ + Mτ 1Tτ ; τ τ ρτ (X )  = Mτ for all X ∈ R∞,d , and R∞,d = R∞,d \Aτ + Mτ 1Tτ . τ τ τ

For the remainder of this paper the convention that Aτ = Aρτ for a conditional risk measure ρτ will be used. The stepped acceptance set from time τ to s > τ is defined as below.

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Definition 2.4 The set Aτ,s ⊆ R∞,d τ,s , defined by   , : 0 ∈ ρ (X ) and X ∈ M Aτ,s = X ∈ R∞,d τ s s τ,s is called the (s − τ )-stepped acceptance set at time τ . Expanding upon the definition of time consistency for scalar dynamic risk measures for bounded discrete-time processes, such as dynamical consistency introduced by Riedel [36, Axiom 2], time-consistency introduced by Cheridito et al. [8, Proposition 4.4] and (strongly) time consistency introduced by Acciaio et al. [4, Definition 4.1], we will define time consistency for set-valued dynamic risk measures for bounded discrete-time processes as follows. Definition 2.5 A dynamic risk measure (ρτ )τT=0 is called time consistent if, for all times 0 ≤ τ < s ≤ T and any X , Y ∈ R∞,d with ρs (X ) ⊆ ρs (Y ), then for any X ∗ , Y ∗ ∈ R∞,d with X ∗ 1[τ,s) = Y ∗ 1[τ,s) , X ∗ 1[s,T ] = X 1[s,T ] and Y ∗ 1[s,T ] = Y 1[s,T ] , we have ρτ (X ∗ ) ⊆ ρτ (Y ∗ ). The meaning of time consistency can be interpreted as follows. If a value process X coincides with another value process Y between times τ and s − 1, and at time s, X is more risky than Y in every possible state of the world, then at time τ , X should also be more risky than Y . A value process X is considered more risky than value process Y if any portfolio that compensates for the risk of X would also cover the risk of Y . Next, we will introduce a stronger property than time consistency, named multi-portfolio time consistency. An example will be given in Example 3.1 below. The formulation of multiportfolio time consistency is motivated by the multi-portfolio time consistency introduced by Feinstein and Rudloff [12]. Hence the multi-portfolio time consistency here can be considered as an analogue of the one of Feinstein and Rudloff [12]. Furthermore, Proposition 3.5 below will show that, under certain conditions, multi-portfolio time consistency for conditional risk measures for processes and that for random vectors on some product space are equivalent. Definition 2.6 A dynamic risk measure (ρτ )τT=0 is called multi-portfolio time consistent if, for all times 0 ≤ τ < s ≤ T , all sets A, B ⊆ R∞,d and any  X ∈ R∞,d , the implication         X 1[τ,s) + X 1[s,T ] ⊆ X 1[τ,s) + Y 1[s,T ] ρs (X ) ⊆ ρs (Y ) ⇒ ρτ  ρτ  X ∈A

Y ∈B

X ∈A

Y ∈B

is satisfied. Remark 2.2 Note that a dynamic risk measure (ρτ )τT=0 is multi-portfolio time consistent if and only if the following condition holds: For all times 0 ≤ τ < T , all subsets A, B ⊆ R∞,d and any  X ∈ R∞,d ,      ρτ +1 (X ) ⊆ ρτ +1 (Y ) ⇒ ρτ  X 1{τ } + X 1[τ +1,T ] X ∈A

Y ∈B

⊆



ρτ



X ∈A

  X 1{τ } + Y 1[τ +1,T ] .

Y ∈B

The multi-portfolio time consistency means that if a market sector A, i.e. a collection of value processes, is more risky than another sector B at time s, and they are of the same risk between times τ and s − 1, then at time τ , sector A should also be more risky than B. A market sector A is considered more risky than sector B if any portfolio that compensates for the risk of a value process in A would also cover the risk of a value process in B.

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Finally, we will provide the duality result for set-valued closed convex conditional risk measures which will be used in Theorem 3.2. For the special case of coherency and conditional convexity and coherency, similar to Corollaries 3.1 and 3.2 of Chen and Hu [5], we can also obtain the corresponding duality results. Considering the emphasis of this paper is the discussion about time consistency, we will not restate them here. We will define the set of dual variables to be Wτ := {(Q, w) : Q = (Q τ , Q τ +1 , . . . , Q T ) ∈ Md (P) × · · · × Md (P), w = (wτ , wτ +1 , . . . , wT ) ∈ L 1d (Fτ )+ × · · · × L 1d (Fτ )+ , T 

 + wt ∈ (Mτ )+ \Mτ⊥ ,

t=τ

w τt (Q t , wt ) ∈ L 1d (Ft )+ for any t ∈ {τ, τ + 1, . . . , T }



for any 0 ≤ τ ≤ T . Definition 2.7 A function −ατ : Wτ → G (Mτ ; (Mτ )+ ) is a penalty function at time τ ∈ T if it satisfies  !T " #

tr (P1) −ατ (Q, w) = cl −ατ (Q, w) + u ∈ Mτ : E wt u ≥ 0 for all (Q, w) ∈ t=τ

(P2)

Wτ , 

(Q,w)∈Wτ

−ατ (Q, w)  = ∅ and −ατ (Q, w)  = Mτ for at least one (Q, w) ∈ Wτ .

Similar to Theorem 3.1 of Chen and Hu [5], we can obtain the following Proposition 2.3, see Remark 2.3 below. Proposition 2.3 A function ρτ : R∞,d → P (Mτ ; (Mτ )+ ) is a set-valued closed convex τ conditional risk measure if and only if there is a penalty function −ατ at time τ such that  $ τ −ατ (Q, w) + F (2.3) ρτ (X ) = (Q,w) (−X ) , (Q,w)∈Wτ

where



T  %    τ tr Qt  wt u − E (X t | Fτ ) ≥ 0 . F(Q,w) (X ) := u ∈ Mτ : E t=τ

In particular, for ρτ with aforementioned properties, (2.3) is satisfied with the minimal penalty function −ατmin defined by  τ (2.4) − ατmin (Q, w) := cl F (Q,w) (X ), X ∈Aτ

where

  Aτ := X ∈ R∞,d : 0 ∈ ρτ (X ) . τ

The penalty function −ατmin has the property that, for any penalty function −ατ satisfying (2.3), it holds that −ατmin (Q, w) ⊆ −ατ (Q, w) for all (Q, w) ∈ Wτ . Remark 2.3 In Chen and Hu [5], (P2) in Definition 2.7 was replaced by

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 (P2):

(Mτ )+ ⊆ −ατ (Q, w) for all (Q, w) ∈ Wτ , ⎞ ⎛ $ $ −ατ (Q, w)⎠ (Mτ )− = ∅. and ⎝ (Q,w)∈Wτ

 implies (P2) in Definition 2.7. The modification of the definition of set-valued Note that (P2) conditional risk measures for processes from Chen and Hu [5] prevents us from making directly use of Theorem 3.1 of Chen and Hu [5]. Nevertheless, under the corresponding  from Chen and Hu [5], by the same arguments as modification (P2) in Definition 2.7 of (P2) in the proof of Theorem 3.1 of Chen and Hu [5], we can steadily show Proposition 2.3. In the following proposition, similar to Theorem 2.3 of Feinstein and Rudloff [13], we extend the above duality result by showing that it is sufficient to only consider the set of probability measures which are equal to the physical probability measure P up to time τ , that is, to use the smaller set   τ := (Q, w) ∈ Wτ : Q t = P|F τ for any t = τ, . . . , T W in the dual representation as opposed to Wτ as in Proposition 2.3. This result is an extension of the scalar dual representation given in Detlefsen and Scandolo [10], Ruszczynski and Shapiro [37] and Klöppel and Schweizer [30], and of the set-valued dual representation given in Feinstein and Rudloff [13]. We will say Q = P|F τ for vector probability measures Q and some time τ ∈ T if for every D ∈ Fτ , it follows that Q i (D) = P(D) for all i = 1, . . . , d. The proof of Proposition 2.4 below will be given in Sect. 4. → P (Mτ ; (Mτ )+ ) is a set-valued closed convex Proposition 2.4 A function ρτ : R∞,d τ conditional risk measure if and only if  $ τ ρτ (X ) = −ατmin (Q, w) + F (2.5) (Q,w) (−X ) . τ (Q,w)∈W

min Based on the above duality results, we define the minimal stepped penalty function −ατ,s (stepped from τ to s > τ ) as  min τ (Q, w) := cl (2.6) − ατ,s F (Q,w) (X ) X ∈Aτ,s

τ,s , where for any (Q, w) ∈ W  τ,s := (Q, w) : Q = (Q τ , Q τ +1 , . . . , Q T ) ∈ Md (P) × · · · × Md (P), W Q t = P|F τ for any t = τ, . . . , T, w = (wτ , wτ +1 , . . . , wT ) ∈ L 1d (Fτ )+ × · · · × L 1d (Fτ )+ , T 

wt ∈ ((Mτ )+ )+ \Mτ⊥ ,

t=τ w τt (Q t , wt )

∈ L 1d (Ft )+ for any t ∈ {τ, τ + 1, . . . , s − 1}  t for any t ∈ {s, s + 1, . . . , T } , w τt (Q t , wt ) ∈ B((M + s )+ )

and

  t 1 tr B((M + := v ∈ L d (Ft ) : E[v u] ≥ 0for anyu ∈ (Ms )+ . s )+ )

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Math Finan Econ

τ ⊆ W τ,s for any 0 ≤ τ < s ≤ T . Obviously, for any choice of Ms , it follows that W

3 Main results In this section, we will state the equivalent characterizations of multi-portfolio time consistency, and will discuss the relations between time consistency and multi-portfolio time consistency. We will also provide a way to construct multi-portfolio time consistent versions of any dynamic risk measure. Finally, we will also investigate the relationship about time consistency and multi-portfolio time consistency between risk measures for processes and risk measures for random vectors on some product space. We need first to give a lemma, whose proof will be given in the next section. This lemma can be considered as a discrete-time processes version of Lemma 3.6 of Feinstein and Rudloff [12]. Lemma 3.1 Suppose that (ρτ )τT=0 is a normalized dynamic risk measure. Let 0 ≤ τ < s ≤ T and X ∈ R∞,d . It holds that   * (1) X 1[τ,T ] ∈ As + Aτ,s ⇔ Y ∈ρs (X ) X 1[τ,s) − Y 1[s,T ] Aτ,s  = ∅;   * (2) (a) ρτ (X ) ⊆ Y ∈ρs (X ) ρτ X 1[τ,s) − Y 1[s,T ] ⇒ Aτ ⊆ As + Aτ,s ; * (b) if, additionally, Mτ ⊆ Ms , then Aτ ⊆ As + Aτ,s ⇒ ρτ (X ) ⊆ Y ∈ρs (X ) ρτ   X 1[τ,s) −*Y 1[s,T ] ;   (3) (a) ρτ (X ) ⊇ Y ∈ρs (X ) ρτ X 1[τ,s) − Y 1[s,T ] ⇒ Aτ ⊇ As + Aτ,s ; * (b) if, additionally, Mτ ⊆ Ms , then Aτ ⊇ As + Aτ,s ⇒ ρτ (X ) ⊇ Y ∈ρs (X ) ρτ   X 1[τ,s) − Y 1[s,T ] . The following Theorem 3.1, which is similar to Theorem 3.4 of Feinstein and Rudloff [12], characterizes multi-portfolio time consistency in terms of a decomposition property of the acceptance sets and in terms of the recursive form for a normalized dynamic risk measure. The recursive form (3.1) below can be considered as a set-valued version of that in Cheridito et al. [8]. Theorem 3.1 For a normalized dynamic risk measure (ρτ )τT=0 , the following are equivalent: (i) (ρτ )τT=0 is multi-portfolio time consistent; (ii) for every time 0 ≤ τ < s ≤ T , any A, B ⊆ R∞,d and any  X ∈ R∞,d ,         ρs (X ) = ρs (Y ) ⇒ ρτ  ρτ  X 1[τ,s) + X 1[s,T ] = X 1[τ,s) + Y 1[s,T ] ; X ∈A

Y ∈B

X ∈A

Y ∈B

(iii) (ρτ )τT=0 is recursive, that is for all times 0 ≤ τ < s ≤ T and every X ∈ R∞,d ,      ρτ X 1[τ,s) − Y 1[s,T ] . ρτ (X ) = ρτ X 1[τ,s) − ρs (X )1[s,T ] :=

(3.1)

Y ∈ρs (X )

If, additionally, Mτ ⊆ Mτ +1 for every time τ ∈ {0, 1, . . . , T − 1}, then all of the above is also equivalent to (iv) for every time 0 ≤ τ < s ≤ T , Aτ = As + Aτ,s .

(3.2)

Using Proposition 2.4, we further have that multi-portfolio time consistency implies a property of the penalty function when the risk measure is convex, which is similar to Theorem 3.2 of Feinstein and Rudloff [13].

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Math Finan Econ

Theorem 3.2 Assume that Mτ ⊆ Mτ +1 for every time τ ∈ {0, 1, . . . , T − 1}. Suppose that (ρτ )τT=0 is a normalized closed convex multi-portfolio time consistent dynamic risk measure, τ , then for any 0 ≤ τ < s ≤ T and any (Q, w) ∈ W ⎛ ⎞  min τ ⎠ −ατmin (Q, w) = cl ⎝−ατ,s F (Q, w) + cl (Q,w) (X ) . X ∈As

Example 3.1 below, which is a modification of Example 3.7 in Feinstein and Rudloff [12], shows that time consistency is a weaker property than multi-portfolio time consistency. However, Proposition 3.1 below, which is similar to Lemma 3.8 of Feinstein and Rudloff [12], shows a sufficient condition for time consistency and multi-portfolio time consistency to be equivalent. Example 3.1 Consider a one-period model with τ ∈ {0, T }. Let   A0 := X ∈ R∞,d : X 01 ∈ L ∞ (F0 )+ , X T1 ∈ L ∞ (FT )+ and

  : X T1 ∈ L ∞ (FT )++ . A T := X ∈ R∞,d T   i Let Mτ := X ∈ L ∞ d (Fτ ) : for any i ∈ {2, 3, . . . , d}, X = 0 for τ ∈ {0, T }. Then A0 and A T are acceptance sets. Denote the corresponding risk measure ρτ (X ) := {u ∈ Mτ : X + u1Tτ ∈ Aτ } for τ ∈ {0, T }. Then   ρ0 (X ) = (u 1 , 0, 0, . . . , 0) ∈ Rd : X 01 + u 1 ≥ 0, X T1 + u 1 ≥ 0 ,   1 1 ρT (X ) = u ∈ L ∞ d (FT ) : X T + u > 0 .

Thus

  A0,T := X ∈ R∞,d 0,T : 0 ∈ ρ0 (X ), X T ∈ MT   1 1 1 ∞ i : X ≥ 0, X ≥ 0, X ∈ L ( F ), X = 0, i = 2, 3, . . . , d . = X ∈ R∞,d T 0 T T T 0,T   1 1 Hence A T + A0,T = X ∈ R∞,d 0,T : X 0 ≥ 0, X T > 0 , which yields A T + A0,T  A0 . Therefore, (ρτ )τT=0 is not multi-portfolio time consistent. However, ρT (X ) ⊆ ρT (Y ) if and only if YT1 ≥ X T1 . Consequently, for any  X ∈ R∞,d 0,T −1 , we have     X 1{0} + X = u ∈ M0 :  X 01 + u 1 ≥ 0, X T1 + u 1 ≥ 0 ρ0    ⊆ u ∈ M0 :  X 01 + u 1 ≥ 0, YT1 + u 1 ≥ 0   = ρ0  X 1{0} + Y , which implies that (ρτ )τT=0 is time consistent. Proposition 3.1 Let (ρτ )τT=0 be a normalized time consistent dynamic risk measure for processes such that, for all times 0 ≤ τ < s ≤ T and every X ∈ R∞,d , there exists a  Z ∈ ρs (X ) such that ρs (−  Z 1Ts ) ⊇ ρs (X ). Then (ρτ )τT=0 is multi-portfolio time consistent. The next proposition shows a way to construct multi-portfolio time consistent versions of any set-valued dynamic risk measure on R∞,d . For multi-portfolio time consistent versions of any scalar dynamic risk measure or set-valued dynamic risk measure on L ∞ d (FT ), we refer to Cheridito and Kupper [9] and Feinstein and Rudloff [12], respectively.

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Proposition 3.2 Let (ρτ )τT=0 be a dynamic risk measure on R∞,d and let Mτ ⊆ Mτ +1 for every time τ ∈ {0, 1, . . . , T − 1}, then ( ρτ )τT=0 defined for all X ∈ R∞,d by ρ T (X ) := ρT (X ),    for any τ ∈ {0, 1, . . . , T − 1}, ρ τ (X ) := ρτ X 1{τ } − Z 1[τ +1,T ]

(3.3) (3.4)

Z ∈ ρτ +1 (X )

is multi-portfolio time consistent. Furthermore, ( ρτ )τT=0 has properties (A1) and (A2) in T Definition 2.1. Additionally, if (ρτ )τ =0 is (conditionally) convex (coherent), then ( ρτ )τT=0 is (conditionally) convex (coherent). Although ( ρτ )τT=0 defined by (3.3) and (3.4) may not be normalized or finite at zero, the following Proposition 3.3, which is similar to Proposition 3.12 of Feinstein and Rudloff [12], will give us some sufficient conditions to ensure ( ρτ )τT=0 is normalized and finite at zero. Meanwhile, it shows that time consistency may be a useful property for the creation of multi-portfolio time consistent risk measures. However, we need first to give a lemma, which is similar to Lemma 3.13 of Feinstein and Rudloff [12], and its proof will be given in the next section. Lemma 3.2 If (ρτ )τT=0 is a time consistent dynamic risk measure, and ρs is normalized ρτ is ρs (0)1Ts -monotone for any time τ ≤ s, where ρs (0)1Ts := for some time s, then  h1Ts : h ∈ ρs (0) . That is, let X, Y ∈ R∞,d , if there exists a Z ∈ ρs (0) such that Y − X = Z 1Ts , then ρτ (X ) ⊆ ρτ (Y ). ρτ )τT=0 be defined Proposition 3.3 Let (ρτ )τT=0 be a dynamic risk measure on R∞,d , and let ( T T as in (3.3) and (3.4). Then ( ρτ )τ =0 is normalized and finite at zero if (ρτ )τ =0 is normalized and either of the following is true: (i) ρτ (0) = (Mτ )+ for every time τ , or (ii) (ρτ )τT=0 is a time consistent risk measure with 0 ∈ ρτ (0) for all times τ . If Mτ ⊆ Mτ +1 for every time τ ∈ {0, 1, . . . , T −1}, and (ρτ )τT=0 is a normalized coherent risk measure with 0 ∈ ρτ (0) for all times τ , then ( ρτ )τT=0 is normalized, coherent and for τ ∈ {0, 1, . . . , T } either finite at zero or ρ τ (X ) ∈ {∅, Mτ } for every X ∈ R∞,d . Next, we investigate the relationship between conditional risk measures for processes and conditional risk measures for random vectors. Similar to Proposition 3.6 of Acciaio, Föllmer and Penner [4], under certain conditions, there is a one-to-one relationship between conditional risk measures for processes and conditional risk measures for random vectors on the optional σ -field. Proposition 3.4 Suppose that M0 = · · · = MT = Rd . (i) Any conditional risk measure for processes ρτ : R∞,d → P (Mτ ; (Mτ )+ ), t ∈ T, defines τ    a conditional risk measure for random vectors ρ τ :  L∞ d (F ) → P (Mτ ; (Mτ )+ ) via  ρ τ (X ) := [−X 0 , +∞) 1{0} + · · · + −X τ −1 , +∞) 1{τ −1}   + h1Tτ : h ∈ ρτ (X ) , X ∈ R∞,d , (3.5) where

  [X t , +∞) 1{t} := h ∈ L ∞ d (Ft ) : h ≥ X t , P−a.s. 1{t}

for any t = 0, . . . , T . Additionally, if ρτ is normalized, (conditionally) convex, or (conditionally) coherent, then the corresponding risk measure ρ τ has the same property.

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Math Finan Econ

   (ii) Conversely, any ρ τ :  L ∞ d (F ) → P (Mτ ; (Mτ )+ ) which is a conditional risk measure for random vectors with ρ τ (0) τ  = Mτ , defines a conditional risk measure for processes ρτ : R∞,d → P (Mτ ; (Mτ )+ ) via τ    ρτ (X ) := ρ τ X 1Tτ τ , X ∈ R∞,d . (3.6) τ If ρ τ is normalized, (conditionally) convex, or (conditionally) coherent, then the corresponding risk measure ρτ has the same property.  τ )+ , then Additionally, if ρ τ is conditionally convex and satisfies ρ τ (0) = (M  ρ τ (X ) = [−X 0 , +∞) 1{0} + · · · + −X τ −1 , +∞) 1{τ −1}   + h1Tτ : h ∈ ρτ (X ) , X ∈ R∞,d , (3.7) where ρτ is defined by (3.6). Remark 3.1 (a) Note that the representations (3.5) and (3.6) require the strong property  τ )+ for equivalence. However, if proportional transaction costs are included ρ τ (0) = (M  τ )+ . In this situation, we can not link conditional (via solvency cones), then ρ τ (0)  = (M risk measures for processes and conditional risk measures for random vectors on the optional σ -field by representations (3.5) and (3.6) in order to make them equivalent. (b) Suppose that M0 = · · · = MT = Rd . (b1) Let ρτ : R∞,d → P (Mτ ; (Mτ )+ ) be a conditional risk measure for processes, then τ  one can construct a conditional risk measure for random vectors ρ τ :  L∞ d (F ) →   P (Mτ ; (Mτ )+ ) defined by (3.5). Furthermore, using ρ τ , one can construct another conditional risk measure for processes Rτ : R∞,d → P (Mτ ; (Mτ )+ ) defined via τ (3.6). Obviously, it holds that ρτ (·) = Rτ (·).    (b2) Let ρτ :  L∞ d (F ) → P (Mτ ; (Mτ )+ ) be a conditional risk measure for random vectors with ρ τ (0) τ  = Mτ , then one can construct a conditional risk measure for processes ρτ : R∞,d → P (Mτ ; (Mτ )+ ) through (3.6). Furthermore, one can construct another τ    conditional risk measure for random vectors R τ :  L∞ d (F ) → P (Mτ ; (Mτ )+ ) via (3.5). From Proposition 3.4, we know that if ρ τ is conditionally convex and satisfies  τ )+ , then it holds that ρ τ (·) = R τ (·). ρ τ (0) = (M Proposition 3.5 below shows that, under certain conditions, a dynamic risk measure for processes is time consistent (multi-portfolio time consistent, respectively) if and only if the corresponding dynamic risk measure for random vectors, as defined by (3.5), is time consistent (multi-portfolio time consistent, respectively). Proposition 3.5 Suppose that M0 = · · · = MT = Rd . (i) Assume that a dynamic risk measure for processes (ρτ )τT=0 is time consistent, then the corresponding dynamic risk measure for random vectors (ρ τ )τT=0 defined by (3.5) is time consistent. (ii) Conversely, assume that (ρ τ )τT=0 is a conditionally convex dynamic risk measure on T    L∞ d (F ) with ρ τ (0) = (Mτ )+ for any τ = 0, 1, . . . , T . If (ρ)τ =0 is time consistent, then T the corresponding dynamic risk measure for processes (ρτ )τ =0 , as defined by (3.6), is also time consistent. (iii) Let (ρτ )τT=0 be a normalized dynamic risk measure for processes, if (ρτ )τT=0 is multiportfolio time consistent, then the corresponding dynamic risk measure for random vectors (ρ τ )τT=0 , as defined by (3.5), is also multi-portfolio time consistent.

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(iv) Conversely, assume that (ρ τ )τT=0 is a conditionally convex dynamic risk measure on T    L∞ d (F ) with ρ τ (0) = (Mτ )+ for any τ = 0, 1, . . . , T . If (ρ τ )τ =0 is multi-portfolio time consistent, then the corresponding dynamic risk measure for processes (ρτ )τT=0 , as defined by (3.6), is also multi-portfolio time consistent. We will end this section with an example. We will construct a multi-portfolio time consistent dynamic risk measure. Namely, we will first consider specific acceptance sets consisting of non-negative value processes, because a value process represents the gains of a financial position. Then we will deduce from the acceptance sets a multi-portfolio time consistent dynamic risk measure. Example 3.2 Let Aτ := (R∞,d )+ , M := Rm × {0}d−m , where 1 ≤ m ≤ d, Mτ := τ L∞ ( F ; M) for any τ ∈ {0, 1, . . . , T }. Then Aτ is an acceptance set. The corresponding τ d T risk measure (ρτ )τ =0 is multi-portfolio time consistent. In fact, ρτ (X ) := {u ∈ Mτ : X + u1Tτ ∈ Aτ } = {u ∈ Mτ : u ≥ −X t , for any t = τ, . . . , T }. Obviously, ρτ is normalized. For any 0 ≤ τ < s ≤ T ,   Aτ,s := X ∈ R∞,d τ,s : 0 ∈ ρτ (X ) and X s ∈ Ms  = X ∈ R∞,d τ,s : X t ≥ 0 for any t = τ, . . . , s, X t = X s for anyt = s + 1, . . . , T,  i and X s = 0 for any i = m + 1, . . . , d . For any X ∈ Aτ , let Y := X 1[τ,s) , Z := X 1[s,T ] , then Y ∈ Aτ,s , Z ∈ As and X = Y + Z , which implies Aτ ⊆ As + Aτ,s . Obviously, Aτ,s + As ⊆ Aτ . Hence Aτ = As + Aτ,s . By Theorem 3.1, it follows that (ρτ )τT=0 is multi-portfolio time consistent. In particular, if m = d, then, from Propositions 3.4 and 3.5, we know that we can define a conditional risk measure for random vectors ρ τ via (3.5) for any τ = 0, . . . , T , and (ρ τ )τT=0 is multi-portfolio time consistent.

4 Proofs of main results In this section, we will provide the proofs of Propositions 2.1 and 2.4 and all the proofs of the results stated in Sect. 3. Proof of Proposition 2.1 (i) Suppose that ρτ is a conditional risk measure for processes, it is sufficient for us to show that Aρτ defined by (2.1) satisfies three properties in Definition 2.3. In fact, since ρτ (0)  = ∅, there exists an m ∈ Mτ such that m ∈ ρτ (0). By the Mτ -translative property of ρτ , we know that ρτ (m1Tτ ) = ρτ (0) − m  0, which yields m1Tτ ∈ Aρτ . Hence Mτ 1Tτ ∩ Aρτ  = ∅. Since ρτ (0)  = Mτ , there exists an m ∈ Mτ \ρτ (0). Thus by the Mτ -translative property of  ρτ , we have 0 ∈ / ρτ (m1Tτ ), which implies m1Tτ ∈ / Aρτ . Hence Mτ 1Tτ (R∞,d \A ρτ )  = ∅. τ Finally, let X ∈ Aρτ , Y ∈ (R∞,d ) , then X + Y ≥ X . Using the monotone property of + τ ρτ , we have ρτ (X ) ⊆ ρτ (X + Y ). Hence 0 ∈ ρτ (X + Y ), X + Y ∈ Aρτ , which implies Aρτ + R∞,d ⊆ A ρτ . τ + is an acceptance set. In order to prove the proposition, we (ii) Suppose that Aτ ⊆ R∞,d τ only need to show that ρτAτ defined by (2.2) satisfies (A1)–(A3).

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To this end, for any m ∈ Mτ , we know that     ρτAτ X + m1Tτ = u ∈ Mτ : X + (m + u)1Tτ ∈ Aτ   = m + u ∈ Mτ : X + (m + u)1Tτ ∈ Aτ − m = ρτAτ (X ) − m, which yields ρτAτ satisfies (A1). For any X, Y ∈ R∞,d with X ≤ Y , we have τ   ρτAτ (X ) = u ∈ Mτ : X + u1Tτ ∈ Aτ   = u ∈ Mτ : Y + u1Tτ ∈ Aτ + (Y − X )   ⊆ u ∈ Mτ : Y + u1Tτ ∈ Aτ = ρτAτ (Y ), which implies ρτAτ satisfies (A2). Finally, since Mτ 1Tτ ∩ Aτ  = ∅, there exists an m ∈ Mτ such that 0 ∈ ρτAτ (m1Tτ ). From the Mτ -translative property of ρτAτ , we know that ρτAτ (m1Tτ ) = ρτAτ (0) − m, thus  \Aτ )  = ∅, we m ∈ ρτAτ (0), which yields that ρτAτ (0)  = ∅. Similarly, from Mτ 1Tτ (R∞,d τ can check that ρτAτ (0)  = Mτ . (iii) It is easily checked. Proposition 2.1 is proved.   Proof of Proposition 2.4 By Proposition 2.3, we have these duality results with respect to the full dual variables Wτ . It remains to show that we only need to consider the probability measures that are equal to P on Fτ . Since the dual representation only involves ( wτT (Q τ , wτ ), . . . , w τT (Q T , wT )), which can be seen from τ − ατmin (Q, w) + F (Q,w) (−X )

T %  

T   wttr E Q t (Z t − X t | Fτ ) ≤ E wttr u = u ∈ Mτ : inf E 

Z ∈Aτ

= u ∈ Mτ : inf E Z ∈Aτ

t=τ

T 

 w τT (Q t , wt )tr

t=τ

(Z t − X t ) ≤ E

t=τ

T 

% w τT

(Q t , wt ) u tr

t=τ

for any X ∈ R∞,d , we need to show τ    w τT (Q τ , wτ ), . . . , w τT (Q T , wT ) : (Q, w) ∈ Wτ    τ τT (Q T , wT ) : (Q, w) ∈ W = w τT (Q τ , wτ ), . . . , w to prove the result. τ . For the other direction, let (Q, w) ∈ Wτ , and define Trivially, ⊇ holds by Wτ ⊇ W d Ri

Rt := (Rt1 , . . . , Rtd ) ∈ Md (P) by d Pt = ξ τ +1 (Q it ) . . . ξ T (Q it ) for any i = 1, . . . , d and τ , and by construction w τT (Q t , wt ) = w τT (Rt , wt ) t = τ, . . . , T . It follows that (R, w) ∈ W for any t = τ, . . . , T . Proposition 2.4 is proved.   Proof of Lemma 3.1 (1) “Only if” part. Take X 1[τ,T ] ∈ As + Aτ,s , then there exists a Z ∈ As and an H ∈ Aτ,s such that X 1[τ,T ] = Z + H and X 1[τ,s) = H 1[τ,s) . From the definition of

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Math Finan Econ

Aτ,s , we know that Hs ∈ Ms . Thus from the Ms -translative property of ρs , it follows that ρs (X ) = ρs (Z + Hs 1[s,T ] ) = −Hs + ρs (Z ), which implies (4.1) − Hs ∈ ρs (X ).  On the other hand, ρτ X 1[τ,s) + H *s 1[s,T ] = ρτ (H )  0, thus (X 1[τ,s) + Hs 1[s,T ] ) ∈ Aτ,s , which, together with (4.1), yields Y ∈ρs (X ) (X 1[τ,s) − Y 1[s,T ] ) ∩ Aτ,s  = ∅. “If” part. Let Z ∈ −ρs (X ) such that (X 1[τ,s) + Z 1[s,T ] ) ∈ Aτ,s , and trivially X 1[τ,T ] = X 1[s,T ] − Z 1[s,T ] + X 1[τ,s) + Z 1[s,T ] . By the Ms -translative property of ρs , we have   ρs X 1[s,T ] − Z 1[s,T ] = ρs (X ) + Z  0, 

which implies X 1[s,T ] − Z 1[s,T ] ∈ As . Hence, * X 1[τ,T ] ∈ As + Aτ,s . (2) (a) Let X ∈ Aτ , then 0 ∈ ρτ (X ) ⊆ Y ∈ρs (X ) ρτ (X 1[τ,s) − Y 1[s,T ] ), which yields that there exists a Y ∈ ρs (X ) such that 0 ∈ ρτ (X 1[τ,s) − Y 1[s,T ] ). Thus, X 1[τ,s) − Y 1[s,T ] ∈ Aτ,s .

(4.2)

On the other hand, by the Ms -translativity of ρs , we have ρs (X 1[s,T ] +Y 1[s,T ] ) = ρs (X )−Y  0, which implies X 1[s,T ] + Y 1[s,T ] ∈ As .

(4.3)

Trivially, X = X 1[s,T ] + Y 1[s,T ] + X 1[τ,s) − Y 1[s,T ] , which, together with (4.2) and (4.3), yields that X ∈ As + Aτ,s . (b) Let X ∈ R∞,d and Y ∈ ρτ (X ), then X 1[τ,T ] + Y 1[τ,T ] ∈ Aτ , which, together with Aτ ⊆ As + Aτ,s , implies X1[τ,T ] + Y 1[τ,T ] ∈ As + Aτ,s . By (1), there exists a Z ∈ ρs (X 1[τ,T ] +Y 1[τ,T ] ) such that X 1[τ,T ] + Y 1[τ,T ] 1[τ,s) −Z 1[s,T ] ∈ Aτ,s , i.e. X 1[τ,s) + Y 1[τ,s) − Z 1[s,T ] ∈ Aτ,s . Since Mτ ⊆ Ms and Y ∈ ρτ (X ), we have that Y ∈ Ms . From the Ms -translative property of ρs , it follows that Z ∈ ρs (X +Y 1[τ,T ] ) = ρs (X )−Y , which implies there exists a Z ∈ ρs (X ) such that Z = Z − Y . Hence, X 1[τ,s) + Y 1[τ,T ] − Z 1[s,T ] ∈ Aτ,s .     0 ∈ ρτ X 1[τ,s) + Y 1[τ,T ] − Z 1[s,T ] = ρτ X 1[τ,s) − Z 1[s,T ] − Y.   Thus, Y ∈ ρτ X 1[τ,s) − Z 1[s,T ] . Furthermore, we have     ρτ (X ) = Y ⊆ ρτ X 1[τ,s) − Z 1[s,T ] . Y ∈ρτ (X )

Z ∈ρs (X )

and X 1[τ,s) = H 1[τ,s) . (3) (a) Take Z ∈ As and H ∈ Aτ,s , then X := Z + H ∈ R∞,d τ From the definition of Aτ,s , we have Hs ∈ Ms . Thus, from the Ms -translative property of ρs , it follows that ρs (X ) = ρs (Z + Hs 1[s,T ] ) = −Hs + ρs (Z ), which implies −Hs ∈ ρs (X ). Furthermore,    ρτ X 1[τ,s) − Y 1[s,T ] ρτ (X ) ⊇ Y ∈ρs (X )

  ⊇ ρτ X 1[τ,s) + Hs 1[s,T ] = ρτ (H )  0, which yields X ∈ Aτ .   (b) Take X ∈ R∞,d , Y ∈ ρs (X ) and Z ∈ ρτ X 1[τ,s) − Y 1[s,T ] , we only need to show that Z ∈ ρτ (X ). In fact, by the Mτ -translative property of ρτ , we have that     ρτ X 1[τ,s) − Y 1[s,T ] + Z 1[τ,T ] = ρτ X 1[τ,s) − Y 1[s,T ] − Z  0,

123

Math Finan Econ

which, together with Mτ ⊆ Ms , yields X 1[τ,s) − Y 1[s,T ] + Z 1[τ,T ] ∈ Aτ,s . By the Ms translative property of ρs , it follows that   ρs X 1[s,T ] + Y 1[s,T ] = ρs (X ) − Y  0, which implies X 1[s,T ] + Y 1[s,T ] ∈ As . Hence X 1[τ,T ] + Z 1[τ,T ] = X 1[s,T ] + Y 1[s,T ] + X 1[τ,s) − Y 1[s,T ] + Z 1[τ,T ] ∈ As + Aτ,s . Furthermore, X 1[τ,T ] + Z 1[τ,T ] ∈ Aτ , which yields 0 ∈ ρτ (X + Z 1[τ,T ] ) = ρτ (X ) − Z , thus Z ∈ ρτ (X ). The proof of Lemma 3.1 is completed.   Proof of Theorem 3.1 It can trivially be seen that property (i) yields property (ii). (ii) ⇒ (iii): Let X ∈ R∞,d and Y ∈ ρs (X ), then it holds that   ρs (−Y 1[s,T ] ) = (ρs (0) + Y ) Y ∈ρs (X )

Y ∈ρs (X )



=

Y + ρs (0) = ρs (X ) + ρs (0) = ρs (X ),

Y ∈ρs (X )

i.e.



ρs (X ) =

ρs (Z ).

Z ∈{−Y 1[s,T ] :Y ∈ρs (X )}

  X := Thus by property (ii) and setting A := X 1[s,T ] , B := −Y 1[s,T ] : Y ∈ ρs (X ) and  X 1[τ,s) , we have ρτ (X ) = ρτ (X 1[τ,s) + X 1[s,T ] )    = ρτ X 1[τ,s) + Z 1[s,T ] Z ∈{−Y 1[s,T ] :Y ∈ρs (X )}    ρτ X 1[τ,s) − Y 1[s,T ] . = Y ∈ρs (X )

(iii)⇒ (ii): Let  X ∈ R∞,d , A, B ⊆ R∞,d with   ρs (X ) ⊆ ρs (Y ), X ∈A

(4.4)

Y ∈B

and let (ρτ )τT=0 be recursive (as defined in (3.1)), then        X 1[τ,s) + X 1[s,T ] = X 1[τ,s) − Z 1[s,T ] ρτ  ρτ  X ∈A

(4.5)

X ∈A Z ∈ρs (X )



= Z∈

ρs (X )



⊆ Z∈

=

* X ∈A

* Y ∈B



  X 1[τ,s) − Z 1[s,T ] ρτ 

  ρτ  X 1[τ,s) − Z 1[s,T ]

(4.6)

ρs (Y )

  X 1[τ,s) + Y 1[s,T ] , ρτ 

Y ∈B

where (4.5) follows from the recursive property of (ρτ )τT=0 , i.e. (3.1). (4.6) is implied by (4.4).

123

Math Finan Econ

(iii) ⇒ (i): If A, B ⊆ R∞,d with property (4.4) and let (ρτ )τT=0 be recursive, then for any  X ∈ R∞,d ,  X ∈A

=

  ρτ  X 1[τ,s) + X 1[s,T ] 





ρτ

   X 1[τ,s) + X 1[s,T ] 1[τ,s) − Z 1[s,T ]

(4.7)

X ∈A Z ∈ρs (  X [τ,s)+X 1[s,T ] )

=





  ρτ  X 1[τ,s) − Z 1[s,T ]

X ∈A Z ∈ρs (X 1[s,T ] )



= Z∈

⊆ Z∈

=

* X ∈A

* Y ∈B



  X 1[τ,s) − Z 1[s,T ] ρτ 

ρs (X 1[s,T ] )



  ρτ  X 1[τ,s) − Z 1[s,T ]

(4.8)

ρs (X 1[s,T ] )

  ρτ  X 1[τ,s) + Y 1[s,T ] ,

Y ∈B

where (4.7) follows from the recursive property of (ρτ )τT=0 , i.e. (3.1). (4.8) is implied by (4.4). Finally, if Mτ ⊆ Mτ +1 for every time τ ∈ {0, 1, . . . , T − 1}, then by Lemma 3.1 property (iv) is equivalent to property (iii). The proof of Theorem 3.1 is completed.   Proof of Theorem 3.2 Assume that Mτ ⊆ Mτ +1 for every time τ ∈ {0, 1, . . . , T − 1}, and (ρτ )τT=0 is a normalized closed convex multi-portfolio time consistent dynamic risk measure. Then by Theorem 3.1, we know that (3.2) holds. Furthermore, from Proposition 2.4, it follows τ , that for any (Q, w) ∈ W − ατmin (Q, w)





= cl

u ∈ Mτ : E

X ∈Aτ

 wttr E Q t

t=τ





= cl

T 

u ∈ Mτ : E

Y ∈Aτ,s , Z ∈As

+E ⎛ = cl ⎝

T  t=τ



wttr E Q t

u ∈ Mτ : E

Y ∈Aτ,s

+



⎛

= cl ⎝cl

u ∈ Mτ : E

t=τ

T 

% wttr u

wttr E Q t

(4.9) 

(Yt | Fτ ) ≤ E

T 



T 

wttr E Q t

(Z t | Fτ ) ≤ E

u ∈ Mτ : E

T 

%⎞

wttr u

⎠

t=τ





% wttr u

t=τ



t=τ

Y ∈Aτ,s

123

wttr E Q t (Yt | Fτ )

t=τ T 

% wttr u

t=τ

t=τ



Z ∈As



(Z t | Fτ ) ≤ E





T 

(X t | Fτ ) ≤ E

T 

T  t=τ

 wttr E Q t

(Yt | Fτ ) ≤ E

T  t=τ

% wttr u

Math Finan Econ



+cl

 u ∈ Mτ : E

T 

Z ∈As

= cl



 wttr E Q t (Z t | Fτ ) ≤ E

T 

t=τ

%⎞ wttr u ⎠

t=τ

(4.10) min −ατ,s (Q, w)



+cl ⎛





u ∈ Mτ : E

X ∈As

T 

 wttr E Q t (X t | Fτ ) ≤ E

t=τ

min = cl ⎝−ατ,s (Q, w) + cl

⎞



T 

%⎞ wttr u ⎠

t=τ

τ ⎠ F (Q,w) (X ) ,

X ∈As

where (4.9) follows from (3.2), and (4.10) follows from Proposition 1.23 in Löhne [32]. The proof of Theorem 3.2 is completed.   Proof of Proposition 3.1 Let X ∈ R∞,d , 0 ≤ τ < s ≤ T , and Z ∈ ρs (X ) arbitrarily chosen. Then, Ms -translativity and normalization of ρs imply that ρs (−Z 1Ts ) = ρs (0)+ Z ⊆ ρs (X ). Furthermore by time consistency, for any  X ∈ R∞,d , it follows that ρτ (  X 1[τ,s) − Z 1[s,T ] ) ⊆ ρτ (  X 1[τ,s) + X 1[s,T ] ), which yields      X 1[τ,s) − Z 1[s,T ] ⊆ ρτ  X 1[τ,s) + X 1[s,T ] . ρτ  (4.11) Z ∈ρs (X )

On the other hand, by assumption there exists a  Z ∈ ρs (X ) such that ρs (−  Z 1[s,T ] ) ⊇ ρs (X ). Hence by time consistency, we further have     Z 1[s,T ] ⊇ ρτ  ρτ  X 1[τ,s) −  X 1[τ,s) + X 1[s,T ] , from which and (4.11) it follows that      ρτ  X 1[τ,s) − Z 1[s,T ] = ρτ  X 1[τ,s) + X 1[s,T ] .

(4.12)

Z ∈ρs (X )

By taking  X := X 1[τ,s) in (4.12), we have    ρτ (X ) = ρτ X 1[τ,s) − Z 1[s,T ] , Z ∈ρs (X )

which implies (ρτ )τT=0 is multi-portfolio time consistent by Theorem 3.1. The proof of Proposition 3.1 is completed.   * τ +1 (X ) ⊆ Proof of Proposition 3.2 Let 0 ≤ τ < T , A, B ⊆ R∞,d such that X ∈A ρ * ∞,d  ρ  (Y ), then for any X ∈ R , we know that Y ∈B τ +1        ρ τ  ρτ  X 1{τ } + X 1[τ +1,T ] = X 1{τ } − Z 1[τ +1,T ] X ∈A

X ∈A Z ∈ ρτ +1 (X )



= Z∈

* X ∈A

  X 1{τ } − Z 1[τ +1,T ] ρτ 

ρ τ +1 (X )

123

Math Finan Econ



⊆ Z∈

=

* Y ∈B



  ρτ  X 1{τ } − Z 1[τ +1,T ]

ρ τ +1 (Y )

  ρ τ  X 1{τ } + Y 1[τ +1,T ] .

Y ∈B

Thus, ( ρτ )τT=0 is multi-portfolio time consistent. that ρ τ +1 is Mτ +1 -translative, we show that ρ τ is 

  ρ τ X + m τ 1[τ,T ] =

Obviously, ρ T is MT -translative. Assume Mτ -translative. In fact, for any m τ ∈ Mτ ,   ρτ X 1{τ } + m τ 1{τ } − Z 1[τ +1,T ] . (4.13)

Z ∈ ρτ +1 ( X +m τ 1[τ,T ] )

Since Mτ ⊆ Mτ +1 , we have m τ ∈ Mτ +1 . From the Mτ +1 -translativity of ρ τ +1 , it follows that   ρ τ +1 X + m τ 1[τ,T ] = ρ τ +1 (X ) − m τ . (4.14) From (4.13), (4.14) and the Mτ -translativity of ρτ , we can conclude that   τ (X ) − m τ . ρ τ X + m τ 1[τ,T ] = ρ The monotonicity follows from the corresponding property for ρτ . If (ρτ )τT=0 is (conditionally) convex (coherent), then by backwards induction ( ρτ )τT=0 is (conditionally) convex (coherent). The proof of Proposition 3.2 is completed.   Proof of Lemma 3.2 Let X, Y ∈ R∞,d , and there exists a Z ∈ ρs (0) such that Y − X = Z 1[s,T ] , we need to show that ρτ (X ) ⊆ ρτ (Y ). In fact,   ρs (Y ) = ρs X + Z 1[s,T ] = ρs (X ) − Z = ρs (X ) + ρs (0) − Z ⊇ ρs (X ), (4.15) where the second equality follows from the Ms -translative property of ρs , and the third equation follows from the normalized property of ρs . For any 0 ≤ τ ≤ s, from (4.15) and time consistency of (ρτ )τT=0 , it follows that     ρτ (X ) = ρτ X 1[τ,s) + X 1[s,T ] ⊆ ρτ X 1[τ,s) + Y 1[s,T ] = ρτ (Y ). The proof of Lemma 3.2 is completed.

 

Proof of Proposition 3.3 Obviously, ρ T (0) = ρT (0). Hence, using backward induction, we want to show that ρ τ (X ) = ρ τ (X ) + ρ τ (0). (i) Let ρ τ +1 (0) = ρτ +1 (0) = (Mτ +1 )+ . Then for any Z ∈ ρ τ +1 (0), by monotonicity we have ρτ (−Z 1[τ +1,T ] ) ⊆ ρτ (0), which implies    ρτ −Z 1[τ +1,T ] = ρτ (0) = (Mτ )+ . ρ τ (0) = Z ∈ ρτ +1 (0)

τ (X ) + ρ τ (0). Trivially, it then holds that ρ τ (X ) = ρ (ii) Let ρ  (0) = ρ (0). For any Z ∈ ρ  (0) = ρτ +1 (0), we have 0−(−Z 1[τ +1,T ] ) ∈ τ +1 τ +1 τ +1   h1[τ +1,T ] : h ∈ ρτ +1 (0) . By Lemma 3.2, we know that ρτ (−Z 1[τ +1,T ] ) ⊆ ρτ (0), which, together with 0 ∈ ρτ +1 (0) = ρ τ +1 (0), implies that    ρτ −Z 1[τ +1,T ] = ρτ (0). ρ τ (0) = Z ∈ ρτ +1 (0)

123

Math Finan Econ

Hence ( ρτ )τT=0 is finite at zero. ρ τ is normalized since by the normalized property of ρτ , we have that      ρ τ (X ) + ρ ρτ X 1{τ } − Z 1[τ +1,T ] + ρτ (0) τ (0) = Z ∈ ρτ +1 (X )

=



  ρτ X 1{τ } − Z 1[τ +1,T ]

Z ∈ ρτ +1 (X )

=ρ τ (X ). If (ρτ )τT=0 is a normalized closed coherent risk measure, and Mτ ⊆ Mτ +1 for every time τ ∈ {0, 1, . . . , T −1}, then ( ρτ )τT=0 is coherent by Proposition 3.2, and thus ρ τ (X ) ⊇ ρ τ (X )+ ρ τ (0) by subadditivity. To show the other direction*assume that 0 ∈ ρ τ +1 (0) (by 0 ∈ ρ T (0) and backwards induction). It follows that ρ τ (0) = Z ∈ ρ (−Z 1 ) ⊇ ρ (0) 0 τ [τ +1,T ] τ ρτ +1 (0) since 0 ∈ ρτ (0). This implies ρ τ (X ) + ρ τ (0) ⊇ ρ τ (X ). Therefore, ρ τ is a normalized risk measure and ρ τ (0)  = ∅. However, it still may be the case that ρ τ (0) = Mτ . By normalization, if ρ τ (0) = Mτ , then ρ τ (X ) ∈ {∅, Mτ } for any X ∈ R∞,d . The proof of Proposition 3.3 is completed.   Proof of Proposition 3.4 (i) It is easy to check that ρ τ defined via (3.5) is a conditional risk  τ -translativity: Let m ∈ M τ , measure in the sense of Feinstein and Rudloff [12]. In fact, (1) M i.e. m = (m 0 , . . . , m τ −1 , m τ , m τ , . . . , m τ ) with m t ∈ Mt for t = 0, . . . , τ . Then by the Mτ -translativity of ρτ , we have ρ τ (X + m)

 = [−X 0 − m 0 , +∞) 1{0} + · · · + −X τ −1 − m τ −1 , +∞) 1{τ −1}   + h1Tτ : h ∈ ρτ (X + m)  = [−X 0 − m 0 , +∞) 1{0} + · · · + −X τ −1 − m τ −1 , +∞) 1{τ −1}    + h1Tτ : h ∈ ρτ X + m τ 1Tτ  = [−X 0 , +∞) 1{0} − m 0 1{0} + · · · + −X τ −1 , +∞) 1{τ −1} − m τ −1 1{τ −1}   + h1Tτ : h ∈ ρτ (X ) − m τ    = [−X 0 , +∞) 1{0} + · · · + −X τ −1 , +∞) 1{τ −1} + h1Tτ : h ∈ ρτ (X ) − m = ρ τ (X ) − m.

 (2) Monotonicity: If Y − X ∈  L∞ d (F )+ , then Yt ≥ X t for any t ∈ T. From the monotonicity of ρτ , it follows that    ρ τ (X ) = [−X 0 , +∞) 1{0} + · · · + −X τ −1 , +∞) 1{τ −1} + h1Tτ : h ∈ ρτ (X )    ⊆ [−Y0 , +∞) 1{0} + · · · + −Yτ −1 , +∞) 1{τ −1} + h1Tτ : h ∈ ρτ (Y ) = ρ τ (Y ). (3) Finiteness at zero: Since ∅  = ρτ (0)  = Mτ and ρ τ (0) = [0, +∞) 1{0} + · · · + [0, +∞) 1{τ −1} + {h1Tτ : h ∈ ρτ (0)}, τ . we have ∅  = ρ τ (0)  = M If ρτ is normalized, (conditionally) convex, or (conditionally) coherent, then, obviously, ρ τ has the same property.    (ii) Let ρτ :  L∞ d (F ) → P (Mτ ; (Mτ )+ ) be a conditional risk measure for random vectors with ρ τ (0) τ  = Mτ . In order to show that the risk measure ρτ defined by (3.6) is a

123

Math Finan Econ

conditional risk measure for processes, we only need to check that it satisfies Mτ -translativity, monotonicity and finiteness at zero. In fact, (1) Mτ -translativity: For any m ∈ Mτ and  τ , hence from the M  τ -translativity of ρ τ , we have that X ∈ R∞,d , obviously, m1Tτ ∈ M τ      ρτ X + m1Tτ = ρ τ X 1Tτ + m1Tτ τ     = ρ τ X 1Tτ − m1Tτ τ    = ρ τ X 1Tτ τ − m = ρτ (X ) − m. (2) Monotonicity: For any X, Y ∈ R∞,d with X ≤ Y , by the monotonicity of ρ τ , we τ know that ρ τ (X 1Tτ ) ⊆ ρ τ (Y 1Tτ ), which implies ρτ (X ) ⊆ ρτ (Y ). (3) Finiteness at zero: ρτ (0) = (ρ τ (0))τ , hence from ρ τ (0) τ  = Mτ , we have ρτ (0)  = Mτ , which, together with the finiteness at zero of ρ τ , yields ∅  = ρτ (0)  = Mτ . If additionally, ρ τ is normalized, (conditionally) convex, or (conditionally) coherent, then, obviously, ρτ has the same property. Additionally, if ρ τ is conditionally convex and satisfies ρ τ (0) = (Mτ )+ , then by Remark 2.5 of Feinstein and Rudloff [13], we have that ρ τ is decomposable, i.e.   τ , X, Y ∈   ρ τ (X 1 D + Y 1 D c ) = ρ τ (X )1 D + ρ τ (Y )1 D c , for any D ∈ F L∞ d F . τ , and M0 = · · · = MT = Rd , the decomposable Since Bτ :=  × {0, . . . , τ − 1} ∈ F  property and Mτ -translativity of ρ τ , imply that   ρ τ (X ) = ρ τ X 1 Bτ 1 Bτ + X 1 Bτc 1 Bτc     = ρ τ X 1 Bτ 1 Bτ + ρ τ X 1 Bτc 1 Bτc   = −X 0 1{0} − · · · − X τ −1 1{τ −1} + ρ τ (0)1 Bτ + ρ τ X 1 Bτc 1 Bτc   = −X 0 1{0} − · · · − X τ −1 1{τ −1} + ρ τ (0)1[0,τ −1] + ρ τ X 1Tτ 1Tτ      = −X 0 1{0} − · · · − X τ −1 1{τ −1} + ρ τ (0)1[0,τ −1] + h1Tτ : h ∈ ρ τ X 1Tτ τ ,  τ )+ , implies which, together with ρ τ (0) = (M       ρ τ (X ) = [−X 0 , +∞) 1{0} + · · · + −X τ −1 , +∞) 1{τ −1} + h1Tτ : h ∈ ρ τ X 1Tτ τ . Hence,    ρ τ (X ) = [−X 0 , +∞) 1{0} + · · · + −X τ −1 , +∞) 1{τ −1} + h1Tτ : h ∈ ρτ (X ) . The proof of Proposition 3.4 is completed.

 

Proof of Proposition 3.5 (i) Let 0 ≤ τ < s ≤ T , and X, Y ∈ R∞,d with ρ s (X ) ⊆ ρ s (Y ).

(4.16)

Then from (4.16) and (3.5), it follows that  [−X 0 , +∞) 1{0} + · · · + −X s−1 , +∞) 1{s−1} + {h1Ts : h ∈ ρs (X )}  ⊆ [−Y0 , +∞) 1{0} + · · · + −Ys−1 , +∞) 1{s−1} + {h1Ts : h ∈ ρs (Y )}, which implies − X 0 ≥ −Y0 , . . . , −X s−1 ≥ −Ys−1 , ρs (X ) ⊆ ρs (Y ).

123

(4.17)

Math Finan Econ

Thus, from (4.17), time consistency of (ρτ )τT=0 and the monotonicity of ρτ , we have   ρτ (X ) =ρτ X 1[τ,s) + X 1[s,T ]   ⊆ ρτ Y 1[τ,s) + X 1[s,T ]   ⊆ ρτ Y 1[τ,s) + Y 1[s,T ] = ρτ (Y ). Hence,

 ρ τ (X ) = [−X 0 , +∞) 1{0} + · · · + −X τ −1 , +∞) 1{τ −1} + {h1Tτ : h ∈ ρτ (X )}  ⊆ [−Y0 , +∞) 1{0} + · · · + −Yτ −1 , +∞) 1{τ −1} + {h1Tτ : h ∈ ρτ (Y )} = ρ τ (Y ).

Therefore, (ρ τ )τT=0 is time consistent.  T  τ )+ for any τ = (ii) Since ρ τ τ =0 is conditionally convex and satisfies ρ τ (0) = (M 0, . . . , T , from Proposition 3.4, we have  ρ τ (X ) = [−X 0 , +∞) 1{0} + · · · + −X τ −1 , +∞) 1{τ −1}   + h1Tτ : h ∈ ρτ (X ) , X ∈ R∞,d . For all times 0 ≤ τ < s ≤ T and any X, Y,  X ∈ R∞,d with ρs (X ) ⊆ ρs (Y ),

(4.18)

we need to show that

    X 1[τ,s) + X 1[s,T ] ⊆ ρτ  X 1[τ,s) + Y 1[s,T ] . ρτ 

In fact, let X := 01{0} + · · · + 01{τ −1} +  X 1[τ,s) + X 1[s,T ] ,  Y := 01{0} + · · · + 01{τ −1} + X 1[τ,s) + Y 1[s,T ] , then

  ρ s (X ) = [0, +∞)1{0} + · · · + [0, +∞)1{τ −1} + −  X τ , +∞ 1{τ }   + · · · + − X s−1 , +∞ 1{s−1}   + h1Ts : h ∈ ρs (X ) ,   ρ s (Y ) = [0, +∞)1{0} + · · · + [0, +∞)1{τ −1} + −  X τ , +∞ 1{τ }   + · · · + − X s−1 , +∞ 1{s−1}   + h1Ts : h ∈ ρs (Y ) .

(4.19)

(4.20)

From (4.18)–(4.20), it follows that ρ s (X ) ⊆ ρ s (Y ), which, together with the time consistency of (ρ τ )τT=0 , yields ρ τ (X ) ⊆ ρ τ (Y ).

(4.21)

ρ τ (X ) = [0, +∞)1{0} + · · · + [0, +∞)1{τ −1}    + h1Tτ : h ∈ ρτ  X 1[τ,s) + X 1[s,T ] ,

(4.22)

Since

123

Math Finan Econ

ρ τ (Y ) = [0, +∞)1{0} + · · · + [0, +∞)1{τ −1}    + h1Tτ : h ∈ ρτ  X 1[τ,s) + Y 1[s,T ] .

(4.23)

From (4.21)–(4.23), we can conclude that     ρτ  X 1[τ,s) + X 1[s,T ] ⊆ ρτ  X 1[τ,s) + Y 1[s,T ] . Therefore, (ρτ )τT=0 is time consistent.  (iii) Denote by Aτ the acceptance set of ρ τ , i.e. Aτ := {X ∈  L∞ ∈ ρ (X )}, and d (F ) : 0  τ  s . From the by Aτ,s the stepped acceptance set (from time τ to s > τ ), i.e. Aτ,s = Aτ M definition of ρ τ , we have that      Aτ = X ∈  L∞ d F : 0 ∈ ρ τ (X )      : X 0 ≥ 0, . . . , X τ −1 ≥ 0, 0 ∈ ρτ (X ) = X ∈ L∞ F  d∞  = Aτ + L d (F0 )+ × · · · × L ∞ (4.24) d (Fτ −1 )+ × {0} × · · · × {0} . $ s Aτ,s = Aτ M    ∞ = Aτ + L ∞ d (F0 )+ × · · · × L d (Fτ −1 )+ × {0} × · · · × {0} $   ∞ ∞ L∞ d (F0 ) × · · · × L d (Fs−1 ) × h1Ts : h ∈ L d (Fs ) $   ∞ ∞ L∞ = Aτ d (F0 ) × · · · × L d (Fs−1 ) × h1Ts : h ∈ L d (Fs )   ∞ + L∞ d (F0 )+ × · · · × L d (Fτ −1 )+ × {0} × · · · × {0} $  ∞  ∞ = Aτ R∞,d τ,s + L d (F0 )+ × · · · × L d (Fτ −1 )+ × {0} × · · · × {0} . (4.25) Note that ∞ Aτ + L ∞ d (F0 )+ × · · · × L d (Fτ −1 )+ × {0} × · · · × {0}

∞ = Aτ + L ∞ d (F0 )+ × · · · × L d (Fs−1 )+ × {0} × · · · × {0}, ∞ ∞ L d (F0 )+ × · · · × L d (Fs−1 )+ × {0} × · · · × {0}   ∞ = L∞ d (F0 )+ × · · · × L d (Fτ −1 )+ × {0} × · · · × {0}   ∞ + L∞ d (F0 )+ × · · · × L d (Fs−1 )+ × {0} × · · · × {0} ,

(4.26)

(4.27)

and when M0 = · · · = MT = Rd ,   $ Aτ,s := X ∈ R∞,d = Aτ : 0 ∈ ρ (X ), X ∈ M R∞,d τ s s τ,s τ,s .

(4.28)

Thus,   Aτ = Aτ + L ∞ (F0 )+ × · · · × L ∞ d (Fτ −1 )+ × {0} × · · · × {0}  d∞  (4.29) = Aτ + L d (F0 )+ × · · · × L ∞ d (Fs−1 )+ × {0} × · · · × {0}  ∞  ∞ = Aτ,s + As + L d (F0 )+ × · · · × L d (Fs−1 )+ × {0} × · · · × {0} (4.30) $  ∞  ∞,d ∞ = Aτ Rτ,s + As + L d (F0 )+ × · · · × L d (Fs−1 )+ × {0} × · · · × {0} =

$

= Aτ,s + As .

123





∞ ∞ R∞,d Aτ τ,s + L d (F0 )+ × · · · × L d (Fτ −1 )+ × {0} × · · · × {0}  ∞  +As + L d (F0 )+ × · · · × L ∞ d (Fs−1 )+ × {0} × · · · × {0}

(4.31)

(4.32) (4.33)

Math Finan Econ

where (4.29) is implied by (4.26), (4.30) follows from Theorem 3.1. (4.31)–(4.33) are implied by (4.28), (4.27) and (4.25), respectively. By Proposition 3.4, we know that (ρ τ )τT=0 is normalized. Hence, from Theorem 3.4 of Feinstein and Rudloff [12], it follows that (ρ τ )τT=0 is multi-portfolio time consistent.  T  τ )+ for any τ = (iv) Since ρ τ τ =0 is conditionally convex and satisfies ρ τ (0) = (M 0, . . . , T , from Proposition 3.4, we have that  ρ τ (X ) = [−X 0 , +∞) 1{0} + · · · + −X τ −1 , +∞) 1{τ −1}   + h1Tτ : h ∈ ρτ (X ) , X ∈ R∞,d . For all times 0 ≤ τ < s ≤ T , any  X ∈ R∞,d and any A, B ⊆ R∞,d with   ρs (X ) ⊆ ρs (Y ), X ∈A

(4.34)

Y ∈B

we need to prove that       ρτ  ρτ  X 1[τ,s) + X 1[s,T ] ⊆ X 1[τ,s) + Y 1[s,T ] . X ∈A

Y ∈B

For any X ∈ A and Y ∈ B, let X := 01{0} + · · · + 01{τ −1} +  X 1[τ,s) + X 1[s,T ] , Y := 01{0} + · · · + 01{τ −1} +  X 1[τ,s) + Y 1[s,T ] , then

  ρ s (X ) = [0, +∞)1{0} + · · · + [0, +∞)1{τ −1} + −  X τ , +∞ 1{τ }   + · · · + − X s−1 , +∞ 1{s−1}   + h1Ts : h ∈ ρs (X ) ,   ρ s (Y ) = [0, +∞)1{0} + · · · + [0, +∞)1{τ −1} + −  X τ , +∞ 1{τ }   + · · · + − X s−1 , +∞ 1{s−1}   + h1Ts : h ∈ ρs (Y ) .

From (4.34)–(4.36), it follows that  X ∈A

i.e.



ρ s (X ) ⊆



(4.35)

(4.36)

ρ s (Y ),

Y ∈B

     ρs  ρs  X 1[τ,s) + X 1[s,T ] ⊆ X 1[τ,s) + Y 1[s,T ] ,

X ∈A

Y ∈B

which, together with the multi-portfolio time consistency of (ρ τ )τT=0 , implies       ρτ  ρτ  X 1[τ,s) + X 1[s,T ] ⊆ X 1[τ,s) + Y 1[s,T ] . X ∈A

Y ∈B

Since

  X 1[τ,s) + X 1[s,T ] ρτ 

   = [0, +∞)1{0} + · · · + [0, +∞)1{τ −1} + h1Tτ : h ∈ ρτ  X 1[τ,s) + X 1[s,T ] ,

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Math Finan Econ

  ρτ  X 1[τ,s) + Y 1[s,T ]

   = [0, +∞)1{0} + · · · + [0, +∞)1{τ −1} + h1Tτ : h ∈ ρτ  X 1[τ,s) + Y 1[s,T ] ,

we have  X ∈A

     X 1[τ,s) + X 1[s,T ] ⊆ X 1[τ,s) + Y 1[s,T ] . ρτ  ρτ  Y ∈B

Therefore, (ρτ )τT=0 is multi-portfolio time consistent. The proof of Proposition 3.5 is completed.   Acknowledgements The authors are very grateful to the Editor-in-Chief, Professor Zachary Feinstein and the anonymous referees for their constructive comments and suggestions which led to the present greatly improved version of the manuscript. The interpretations of time consistency and multi-portfolio time consistency are motivated by the Editor-in-Chief Professor Frank Riedel and the anonymous referees. Propositions 2.2, 2.4, 3.4 and 3.5 and Remark 3.1 are motivated by the suggestions of the anonymous referees.

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