Scalar multivariate risk measures with a single eligible asset
Zachary Feinstein Birgit Rudloff



























theoremTheorem
corollary[theorem]Corollary
rem1[theorem]Remark
lemma[theorem]Lemma
definition[theorem]Definition
proposition[theorem]Proposition
conclusion[theorem]Conclusion
ex1[theorem]Example
ass1[theorem]Assumption

remark
example
assumption

equationsection
theoremsection
































































































































Scalar multivariate risk measures with a single eligible asset
    Zachary Feinstein Stevens Institute of Technology, School of Business, Hoboken, NJ 07030, USA, zfeinste@stevens.edu. Birgit Rudloff Vienna University of Economics and Business, Institute for Statistics and Mathematics, Vienna A-1020, AUT, brudloff@wu.ac.at.
    December 30, 2023 (Original: July 27, 2018)
=====================================================================================================================================================================================================================================================================


In this paper we present results on scalar risk measures in markets with transaction costs.  Such risk measures are defined as the minimal capital requirements in the cash asset.  First, some results are provided on the dual representation of such risk measures, with particular emphasis given on the space of dual variables as (equivalent) martingale measures and prices consistent with the market model.
Then, these dual representations are used to obtain the main results of this paper on time consistency for scalar risk measures in markets with frictions. It is well known from the superhedging risk measure in markets with transaction costs, as in <cit.>, that the usual scalar concept of time consistency is too strong and not satisfied. 

We will show that a weaker notion of time consistency can be defined, which corresponds to the usual scalar time consistency but under any fixed consistent pricing process. We will prove the equivalence of this weaker notion of time consistency and a certain type of backward recursion with respect to the underlying risk measure with a fixed consistent pricing process. Several examples are given, with special emphasis on the superhedging risk measure.




§ INTRODUCTION



The dynamic programming principle and time consistency are vital concepts both for the computation of dynamic optimization problems and guaranteeing that decisions made will not be reversed by the same decision maker at the next point in time.  In the context of risk management, these principles are encoded by properties on dynamic risk measures.  Such functions provide the minimal capital requirements necessary to make an investment acceptable to a risk manager.  In this paper we investigate time consistency properties for multivariate risk measures to allow for the possibility of transaction and market impact costs. That is, the frictionless conversion between assets is not possible and therefore the reduction of a portfolio to a single numéraire is either non-unique or implicitly requires a systematic increase in riskiness.  As will be detailed later in this work, our approach reduces this problem to the time consistency of the dynamic risk measures under all possible no-arbitrage pricing processes.

The organization of this paper is as follows. In Section <ref>, we provide a literature review  and in Section <ref> an overview of the main results along with our motivation for their study.  In Section <ref> we present background material on risk measures and notation that will be used throughout this paper.  Much of this notation is comparable to that utilized in the set-valued risk measure literature.  In Section <ref> we present the definition of the dynamic scalar risk measures we will consider in this paper and basic results on boundedness properties.  In Section <ref> we present dual representations for the scalar risk measures with a single eligible asset.  Special emphasis is placed on a representation akin to (<ref>) below considered in <cit.>, but for general convex and coherent dynamic risk measures in markets with transaction costs, which relates those risk measures to the set of no-arbitrage pricing processes.  In Section <ref> we present results on relevance or sensitivity of the scalar risk measures which provide a condition for the dual representation to be with respect to equivalent probability measures only.  In Section <ref> we introduce a new notion of time consistency for scalar risk measures with a single eligible asset.  This notion coincides with time consistency of the corresponding univariate scalar risk measure under any market-consistent frictionless (and no-arbitrage) price process.  As such, unlike much of the prior literature, we can and do demonstrate that such a property is satisfied by the usual examples; in particular we give details on the superhedging risk measure and composed risk measures in Section <ref>.




 §.§ Literature review


In their seminal paper, Artzner, Delbaen, Eber, and Heath <cit.> introduced the concept of coherent risk measures in a static univariate setting which provides the minimal capital necessary to compensate for the risk of a contingent claim.  Coherent risk measures were further studied in <cit.>.  Such risk measures were generalized to the convex case in <cit.> while retaining the same financial interpretation and a notion of diversification.

When a filtration ℱ is introduced, it is natural to consider dynamic risk measures, i.e., the minimal capital necessary to compensate for the risk of a contingent claim conditionally on the information at time t.  In this time-dynamic setting, the manner in which the risk of a contingent claim propagates through time is of great importance, as it has significant implications on risk management.  One such condition, called (strong) time consistency, consists of the condition that if one portfolio is riskier than another in the future, that same ordering must hold at all prior times as well.  This property is studied in the univariate setting in <cit.> in discrete time and <cit.> in continuous time.  Other, weaker, forms of time consistency have been presented for dynamic risk measure in, e.g., <cit.>.

In this work, we will consider multivariate risk measures.  The multivariate setting arises from markets with frictions, as the liquidation of a portfolio into some numéraire does not allow for repurchasing the same asset.  This setting was studied in a set-valued static (one period) framework in <cit.>.  Set-valued risk measures, in the one period framework, have recently been applied to studying systemic risk in <cit.>.  In a set-valued time-dynamic framework a new notion of time consistency, called multiportfolio time consistency, was introduced in <cit.>.  This property was further studied in <cit.>.  Computation of such set-valued risk measures was studied in <cit.>.

The focus of this work is on dynamic scalar multivariate risk measures with a single eligible asset.  Some results on dynamic scalar multivariate risk measures with multiple eligible assets have been discussed previously in <cit.>.  
Scalar risk measures in frictionless markets with either a single eligible asset <cit.> or multiple eligible assets <cit.> can be considered as scalar multivariate risk measures (see <cit.> for such a comparison).
Additionally, such functions have been considered in the context of systemic risk in, e.g., <cit.>.
Our focus in this paper is of the dynamic version of the multivariate risk measures with a single eligible asset as presented in <cit.> (see <cit.> for a brief discussion).

Of course time (in-)consistency has not just been studied in the context of risk measurement as considered here. There has been substantial development and investigation of time (in-)consistency for stochastic control and stopping problems coming, e.g., from behavioral economics where one considers the (in-)consistency of an agent's control (i.e., the choice of strategies or stopping times) over time.  Time (in-)consistency was surveyed for utility maximization problems by <cit.>. We wish to highlight the recent works of <cit.> on time consistency for stochastic control and <cit.> on time consistency for optimal stopping problems; these works utilize a standard equilibrium or consistent planning approach proposed in <cit.>.  We also wish to highlight <cit.> which very succinctly defines such problems of time (in-)consistency for control problems; that work proposes and uses several different approaches distinct from the equilibrium approach of the prior references.  Such problems in time (in-)consistency have drawn increased attention since the seminal work of Ekeland and Lazrak <cit.>.
In fact, such time (in-)consistency questions are intimately related with the same notions for risk measures.  This control viewpoint can be seen through the dual representation, which is a maximization problem over probability measures.  This is made explicit in the coherent case via stability <cit.>, i.e., the pasting of these probability measures through time.  Indeed, we can view time consistency of risk measures as the recursion of the value function from the control literature.

The situation is similar in the set-valued framework. Stochastic control problems in a multivariate setting, where the value function is now a set-valued function, have been studied in <cit.>; the corresponding backward recursion of the value function in the time consistent case is in analogy to the backward recursion of set-valued risk measures in the multi-portfolio time consistent case, as the value function (i.e., the risk measure itself) recurses backwards in time. Of course stochastic control problems can have a much more complicated structure as, e.g., dependencies on initial capital levels can complicate the numerical computations in the backward recursion, see <cit.>, compared to risk measures, where the input is the random vector at terminal time.









 §.§ Motivation


In this work we consider risk measurement in markets with transaction costs.  These market frictions result in the need to consider multivariate risk measures (i.e., risk measures for a vector-valued portfolio of “physical units” – the number of units of the d available assets or securities held – rather than the value of that portfolio in a single numéraire) because no single value of a portfolio or investment can be defined.  For instance, a mark-to-market value of the portfolio using the current mid-price between the bid and ask prices is strictly larger than the liquidation value of that same portfolio; both such values would result in different capital requirements and neither, necessarily, encodes the true risk of the portfolio.  Set-valued risk measures (see, e.g., <cit.>) were introduced in order to handle this problem by considering the set of all capital requirements, possibly held in multiple different assets, that make the portfolio acceptable. This eliminates the need to first convert the d-dimensional portfolio vector into a one-dimensional quantity and thus avoids the problems related to that.

We are focused on dynamic multivariate risk measures and questions surrounding time consistency.  For set-valued risk measures, the typical dynamic programming principle is called multiportfolio time consistency (see, e.g., <cit.>).  Such a property relates the risk between vector-valued portfolios over time by comparing the full set of acceptable capital allocations.  The usual risk measures, e.g., the superhedging risk measure, satisfy this time consistency property when the full space of assets are eligible to be held as capital.  However, typically, capital is required to be held in a numéraire, or cash, asset only.  
In such a setting, multiportfolio time consistency may no longer hold for the corresponding set-valued risk measure (this is the case, e.g., for superhedging). Furthermore, for a single eligible asset, the minimal capital requirement (often called the scalarization of the set-valued risk measure) becomes the logical object to consider, but it typically does not satisfy the scalar concept of time consistency either.

So the motivating questions of this paper are the following. If the eligible asset is just the cash asset, the set-valued risk measure does not satisfy multiportfolio time consistency, and the scalarization does not satisfy the usual time consistency, is there a more appropriate, weaker form of time consistency for this scalar risk measure? Can one use the fact that the set-valued risk measure, for the full set of eligible assets, is multiportfolio time consistent? The answer to both questions is yes and will be answered in this work, see in particular Lemma <ref> below.

We are specifically motivated to study the problem of scalar multivariate risk measures with a single eligible asset from the many works that consider the scalar superhedging price in a market with frictions.
We refer to, e.g., <cit.> for studies of the scalar superhedging price in a market with transaction costs and two assets. This has been extended to a general number of assets in <cit.>. The d dimensional version of the scalar superhedging price given by <cit.> under a sequence K of solvency cones (for a market with proportional transaction costs, see Section <ref> for further discussion) is as follows.
Under the appropriate no-arbitrage argument, the scalar superhedging price ρ^SHP(-X) in units of the first (cash) asset at time t=0 is given by

    ρ^SHP(-X)= sup_(,S)∈𝒬S_TX

for X ∈ L^∞(Ω,ℱ_T,;^d) which is the payoff in physical units. The set of dual variables is given by 𝒬 which denotes the set of all processes S = S along with their equivalent martingale measures  such that S_t^1≡ 1 and d/dt S_t∈ L^∞(Ω,t,;K_t)  for all t.
For the context of this paper, we note that Theorem 6.1 of <cit.> relates (<ref>) to the scalarization of the set-valued superhedging risk measure under a single eligible asset (the numéraire asset).
We will present a similar dual representation for general convex risk measures in a dynamic framework.  We then use this representation to define a new time consistency property, as detailed below, which is satisfied by, e.g., the superhedging risk measure.  This is in contrast to prior works on time consistency for scalar multivariate measures <cit.> whose time consistency property does not include the superhedging risk measure as an example. This agrees also with recent research on scalarizations of multivariate problems, as in <cit.>, indicating that the usual scalar concept of time consistency is way too strong to be satisfied for scalarizations of multivariate risk measures.
Though weaker forms of time consistency (as in, e.g., <cit.>) may be satisfied for the scalarization, we are interested in a weaker version that is strong enough to still allow a relation to the dynamic programming principle.


We call the weaker time consistency property defined in this work π-time consistency as it corresponds to the usual scalar time consistency for all the risk measure π^S_t appearing when fixing a pricing process S.  That is, the risk measure is time consistent in the usual way when the market model follows any (frictionless) no-arbitrage pricing process S. We will prove the equivalence of this weaker notion of time consistency and a certain type of backward recursion with respect to π^S_t. We will show that the superhedging risk measure in markets with transaction costs (expressed in the cash asset) as in <cit.> while not satisfying the usual scalar time consistency property, which also means it is not recursive with itself, will satisfy  π-time consistency and thus a backward recursion with respect to π^S_t.









§ SETUP


Consider a filtered probability space Ω,,ℱ, satisfying the usual conditions with 0 the trivial sigma algebra and = T. Let |·|_n be an arbitrary norm in ^n for n ∈; note that |·|_1 is equivalent to the absolute value operator.
Denote by 0tD = 0Ω,t,;D the set of equivalence classes of t-measurable functions X: Ω→ D ⊆^n for some n ∈.  Additionally, denote by ptD⊆0tD those random variables X ∈0tD such that X_p = ∫_Ω |X(ω)|_n^p d^1/p < +∞ for p ∈ (0,+∞) and X_∞ = _ω∈Ω |X(ω)|_n < + ∞ for p = +∞.  We further note that pD := pTD by definition.  
In particular, if p = 0, t is the linear space of the
equivalence classes of t-measurable functions X: Ω→^d.  For p > 0, t denotes the linear space of
t-measurable functions X: Ω→^d such that X_p < +∞ for p ∈
(0,+∞]. 
Note that an element X ∈t has components X_1,...,X_d in t for any choice of p ∈ [0,+∞].
For p ∈1,+∞ we will consider the dual pair t, t, where 1/p+1/q = 1 (with q = +∞ when p = 1 and q = 1 when p = +∞),  and endow it with the norm topology, respectively the σt,t-topology on t in the case p = +∞.
Throughout most of this paper we will focus on the case that p = +∞.

Throughout this work we will make use of the indicator functions which we will denote by 1_D: Ω→{0,1} for some D ∈.  These are defined so that 1_D(ω) = 1 if ω∈ D and 0 otherwise.
Additionally, throughout we will consider the summation of sets by Minkowski addition.
A set B ⊆ is called t-decomposable if 1_D B + 1_D^c B ⊆ B for any measurable D ∈t.

As in <cit.> and discussed in <cit.>, the portfolios in this paper are in “physical units” of an asset (in a market with d assets) rather than the value in a fixed numéraire, except where otherwise mentioned.  That is, for a portfolio X ∈t, the values of X_i (for 1 ≤ i ≤ d) are the number of units of asset i in the portfolio at time t.

Let M = ×{0}^d-1 denote the set of eligible portfolios, i.e., those portfolios which can be used to compensate for the risk of a portfolio. That is, for the purpose of this paper, we will assume that the first asset is the only asset available to compensate for risk (in typical examples this will be the cash asset).  For notational simplicity we define M_t = tM = t×{0}^d-1, a weak* closed linear subspace of t (see Section 5.4 and Proposition 5.5.1 in <cit.>).
We will denote M_+ := M ∩^d_+ = _+ ×{0}^d-1 to be the nonnegative elements of M.  We will additionally denote M_t,+ := M_t ∩t^d_+ = tM_+ = t_+×{0}^d-1 to be the nonnegative elements of M_t.

Denote the upper sets by 𝒫M_t;M_t,+ where 𝒫𝒵;C := D ⊆𝒵  |   D = D + C for some vector space 𝒵 and an ordering cone C ⊂𝒵.  Additionally, let 𝒢(𝒵;C) := D ⊆𝒵  |   D = D + C⊆𝒫(𝒵;C) be the upper closed convex subsets.







As noted previously, for the duality results below we will consider the weak* topology for p = +∞.  We will briefly describe the set of dual variables from the set-valued biconjugation theory as utilized in, e.g., <cit.>, and first defined in <cit.>.
First, define the space of probability measures absolutely continuous with respect to  as ℳ and those that are equivalent probability measures as ℳ^e.  The space of d-dimensional probability measures absolutely continuous with respect to  is thus denoted by ℳ^d.
Throughout we will use a -almost sure version of the -conditional expectation where ∈ℳ.  This is defined in, e.g., <cit.>.  Briefly, let X ∈, then define the conditional expectation

    Xt := ξ̅_t,T()Xt,

where 

    ξ̅_s,σ()[ω] := σ(ω)/s(ω)   if s(ω) > 0 
     1    else

for every ω∈Ω.  For a vector-valued probability measure ∈ℳ^d the result is defined component-wise, i.e., Xt = 1X_1t,…,dX_dt for any X ∈, and define ξ_s,σ() := ξ̅_s,σ(_1),…,ξ̅_s,σ(_d).
Note that, for any ∈ℳ and any times 0 ≤ t ≤ s ≤σ≤ T, = ξ̅_0,T() and ξ̅_t,σ() = ξ̅_t,s() ξ̅_s,σ() almost surely.

From this representation we can define the set of dual variables from the set-valued biconjugation theory <cit.> to be

    _t := (,w) ∈ℳ^d ×M_t,+\M_t  |   w_t^T(,w) ∈1^d_+.

In this representation we define 
    w_t^s(,w) = wξ_t,s()
 for any times 0 ≤ t ≤ s ≤ T and where x denotes the diagonal matrix with the components of x on the main diagonal.
Additionally, we employ the notation M_t = v ∈t  |  vu = 0  ∀ u ∈ M_t to denote the orthogonal space of M_t and C^+=v ∈t  |  vu≥ 0  ∀ u ∈ C to denote the positive dual cone of a cone C⊆t.






§ SCALARIZATIONS


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



 §.§ Definition



First, we wish to consider an axiomatic definition for the scalar conditional convex risk measures which will be the subject of this paper. These properties provide the classical interpretations that the risk measure is a capital requirement, that having more in each asset corresponds with lower risk, that diversification reduces risk, and that not being invested in the market carries with it a finite risk (possibly 0). Throughout this paper we will consider the unit vector e_1 := 1,0,...,0∈^d and the zero vector := 0,0,...,0∈^d.



A mapping ρ_t: →t is called a scalarized conditional convex risk measure if it satisfies the following properties for all X,Y ∈ and m ∈t:


  * Conditional cash invariance: ρ_t(X + m e_1) = ρ_t(X) - m;

  * Monotonicity: Y - X ∈^d_+⇒ρ_t(X) ≥ρ_t(Y);

  * Conditional convexity: ρ_t(λ X + (1-λ)Y) ≤λρ_t(X) + (1-λ)ρ_t(Y) for every λ∈∞t[0,1];


  * Finite at zero: ρ_t() ∈t.

A scalarized conditional convex risk measure is called coherent if it additionally satisfies:


  * Conditional positive homogeneity: ρ_t(λ X) = λρ_t(X) for every λ∈t_+.

A scalarized conditional convex risk measure is called _t-compatible if it additionally satisfies:


  * ρ_t(X) = _k ∈_tρ_t(X-k) for every X ∈, where _t = ∑_s = t^T ∞sK_s for s-measurable random convex cones K_s.


_t-compatibility is used to allow the possibility of trading at time points s ∈{t,...,T} according to, e.g., the bid-ask prices modeled by a sequence of solvency cones K.



The following two results relate the scalarized conditional risk measures ρ_t with set-valued risk measures R_t over a single eligible asset (provided by M_t).  Briefly, a set-valued risk measure is a mapping R_t from the space of contingent claims into sets of eligible portfolios, i.e., capital allocations, that make the initial claim acceptable to a risk manager or regulator.  These risk measures satisfy cash invariance (R_t(X + m) = R_t(X) - m for X ∈ and m ∈ M_t), monotonicity (R_t(X) ⊆ R_t(Y) if Y - X ∈^d_+), conditional convexity (R_t(λ X + (1-λ)Y) ⊇λ R_t(X) + (1-λ) R_t(Y) for X,Y ∈ and λ∈∞t[0,1]), and finiteness at zeros.  For more details on the set-valued conditional risk measure we refer to <cit.>.

Let R_t: →𝒢(M_t;M_t,+) := D ⊆ M_t   |   D = D + M_t,+ be a set-valued conditional convex (coherent) risk measure that is _t-compatible (i.e., R_t(X) = ⋃_k ∈_t R_t(X-k) for every X ∈), 
then the mapping ρ_t defined by

    ρ_t(X) := {m ∈t  |   m e_1 ∈ R_t(X)}  ∀ X ∈

is a scalarized conditional convex (resp. coherent) _t-compatible risk measure and, conversely,

    R_t(X) = (ρ_t(X) + t_+)e_1   ∀ X ∈.



First we will show that ρ_t(X) ∈t for any X ∈.  By assumption there exists some Z^-,Z^+ ∈ such that Z^-e_1 ∈ X - _t and Z^+e_1 ∈ X + _t.  This implies R_t(Z^- e_1) ⊆ R_t(X) ⊆ R_t(Z^+ e_1) by _t-compatibility. This implies

    ρ_t(X)    ≥ρ_t(Z^+ e_1) ≥ρ_t(Z^+_∞ e_1) ≥ -(ρ_t()_∞+Z^+_∞) > -∞
    ρ_t(X)    ≤ρ_t(Z^- e_1) ≤ρ_t(-Z^-_∞ e_1) ≤ρ_t()_∞+Z^-_∞ < +∞.

The remainder of the properties are proven in Theorem 3.15 and Corollary 3.17 of <cit.>.

Second, consider the converse statement R_t(X) = (ρ_t(X) + t_+)e_1 for every X ∈.
If ue_1 ∈ R_t(X) then u ≥ρ_t(X) by the definition of the scalarization.
The other direction follows if ρ_t(X)e_1 ∈ R_t(X).  To show this we use the result from <cit.> that

    R_t(X) = ⋂_w ∈M_t,+\M_tu ∈ M_t   |  ρ_t^M,w(X) ≤wu

where ρ_t^M,w denotes the scalarization ρ_t^M,w(X) := _m ∈ R_t(X)wm.
Then we note that 

    w(ρ_t(X)e_1)    = w_1 ρ_t(X) = w_1 u ∈t  |   ue_1 ∈ R_t(X)
       = w m   |   m ∈ M_t,  m ∈ R_t(X) = ρ_t^M,w(X),

which implies the result.







For the remainder of this paper we will assume that the conditions of Theorem <ref> are satisfied and that ρ_t is constructed as in (<ref>). 
We will additionally assume that _t is defined w.r.t. solvency cones K with K_T ⊇K̅ almost surely for some convex cone K̅⊆^d with K̅⊇^d_+ \{}.
  
The acceptance set A_t ⊆ is defined via the set-valued risk measure R_t, i.e., A_t = {X ∈  |  ∈ R_t(X)}.  From the acceptance set we can also define ρ_t as

    ρ_t(X) := {m ∈t  |   X + m e_1 ∈ A_t}  ∀ X ∈.
  
Under this setting ρ_t defines the minimal capital (in the first asset, e.g., Euros) necessary to hedge the risk of X when trades defined by a market model _t are allowed.


Let ρ_t: →t be a scalarized conditional convex risk measure.  Then it also satisfies the following two properties:


  * t-Lipschitz continuity: |ρ_t(X) - ρ_t(Y)| ≤X - Y__t,t almost surely for every X,Y ∈, and

  * Local property: 1_B ρ_t(X) = 1_B ρ_t(1_B X) for every X ∈ and B ∈t

where, as defined in <cit.>,

    X__t,s := {c ∈s  |   ce_1 ≥__t X ≥__t -ce_1}

for 0≤ s≤ t≤ T and with Y ≥__t X if Y - X ∈_t.


Let X,Y ∈ and B ∈t.


  * By definition X - Y__t,te_1 ≥__t X - Y.  By _t-compatibility this implies 
    ρ_t(X) ≥ρ_t(Y + X - Y__t,te_1) = ρ_t(Y) - X - Y__t,t.
  This provides the desired result by symmetry of X and Y.

  * By conditional convexity we recover 

    ρ_t(1_B X) = ρ_t(1_B X + 1_B^c) ≤ 1_B ρ_t(X) + 1_B^cρ_t(),
 
which implies 1_B ρ_t(1_B X) ≤ 1_B ρ_t(X).  To prove the converse

    ρ_t(X) = ρ_t(1_B [1_B X] + 1_B^c X) ≤ 1_B ρ_t(1_B X) + 1_B^cρ_t(X),

which implies 1_B ρ_t(X) ≤ 1_B ρ_t(1_B X).









 §.§ Dual representation



We will now consider the dual, or robust, representation for the scalar conditional risk measures. We present two equivalent formulations, both reliant on a lower semicontinuity property encoded in the following Fatou property.  The first of these two dual representations is intimately related to the dual representation of the underlying set-valued risk measures as discussed in Theorem <ref>.  See <cit.> for discussion on the dual representation for set-valued risk measures.  The second of these dual representations is a generalization of representation (<ref>) of the multivariate superhedging price derived in <cit.>.


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

    ρ_t(X) ≤lim inf_n →∞ρ_t(X_n)

for any ·_∞-bounded sequence X_n converging to X almost surely.



We will now provide the first dual representation for these scalarized conditional convex risk measures. This representation is similar to the results in <cit.>.  Recall the set of dual variables _t defined in (<ref>).  Additionally, we will make extensive use of the notation introduced in Section <ref> that w_t^s(,w) = wξ_t,s() for (,w) ∈_t with ξ_t,s() the vector of conditional Radon-Nikodym derivatives of  w.r.t. .



Let ρ_t be a scalarized conditional convex risk measure. Then ρ_t satisfies the Fatou property
if and only if for any X ∈ 

    ρ_t(X) = _(,m_⊥) ∈_t(e_1)-α_t(,m_⊥) - e_1 + m_⊥Xt,

where 

    _t(e_1) := (,m_⊥) ∈ℳ^d ×{0}×t^d-1_+  |   (,e_1 + m_⊥) ∈_t,   w_t^T(,e_1+m_⊥) ∈_t
 
and

    α_t(,m_⊥) := _Z ∈ A_te_1 + m_⊥-Zt.

If ρ_t is additionally a conditional coherent risk measure then (<ref>) can be reduced to

    ρ_t(X) = _(,m_⊥) ∈_t^ρ(e_1) -e_1 + m_⊥Xt,

for every X ∈ where _t^ρ(e_1) := (,m_⊥)   |  α_t(,m_⊥) = 0  -.





  * Assume the dual representation (<ref>) holds.  Let X ∈ and (X_n)_n ∈⊆ be a bounded sequence such that X = lim_n →∞ X_n almost surely. Therefore, YX_n →YX a.s. and in  for any Y ∈ (by the dominated convergence theorem).  Thus -w_t^T(,e_1+m_⊥)X_nt→-w_t^T(,e_1+m_⊥)Xt a.s. and in  for any (,m_⊥) ∈_t(e_1) as well.






    Thus we can see (as in the static setting, e.g., in <cit.>)
    
    ρ_t(X)    = _(,m_⊥) ∈_t(e_1)-α_t(,m_⊥) + e_1+m_⊥-Xt
       = _(,m_⊥) ∈_t(e_1)-α_t(,m_⊥) + lim_n →∞e_1+m_⊥-X_nt
       = _(,m_⊥) ∈_t(e_1)lim_n →∞-α_t(,m_⊥) + e_1+m_⊥-X_nt
       ≤lim inf_n →∞_(,m_⊥) ∈_t(e_1)-α_t(,m_⊥) + e_1+m_⊥-X_nt
       = lim inf_n →∞ρ_t(X_n).


  * Assume that the “Fatou property” is satisfied.  To prove the dual representation (<ref>) holds, we only need to prove that ρ_t(·) is proper and lower semicontinuous from the same logic as in Proposition A.1.7 of <cit.>.  This modification to the conditions in <cit.> can be accomplished because the requirements on the set-valued risk measures in that result are utilized, via <cit.> and its proof, to guarantee that ρ_t(·) is proper and lower semicontinuous. By ρ_t(X) ∈t for every X ∈, we can conclude ρ_t(·) is proper.
    Define C_z := Z ∈  |  ρ_t(Z)≤ z for any z ∈.  Define C_z^r := C_z ∩Z ∈  |  Z_∞≤ r for any r > 0 (in this case we will take Z_∞ := max_i = 1,...,dZ_i).  Take (X_n)_n ∈⊆ C_z^r converging to X ∈ in probability, and there exists a bounded subsequence (X_n_m)_m ∈ which converges to X almost surely.  Therefore ρ_t(X) ≤lim inf_m →∞ρ_t(X_n_m), and thus (by Fatou's lemma) X ∈ C_z^r.  That is C_z^r is closed in probability for any r > 0.  By <cit.>, C_z is weak* closed.  Now consider a net (X_i)_i ∈ I⊆ which converges to X ∈ in the weak* topology.  Let z_ϵ := ρ_t(X) - ϵ for any ϵ > 0. Then (C_z_ϵ)^c = Z ∈  |  ρ_t(Z) > z_ϵ is an open neighborhood of X for any ϵ > 0.  Since X_i → X, for any ϵ > 0 there exists a j_ϵ∈ I such that for any i ≥ j_ϵ we have X_i ∈ (C_z_ϵ)^c, i.e., ρ_t(X_i) > ρ_t(X) - ϵ.  Therefore lim inf_i ∈ Iρ_t(X_i)≥ρ_t(X), and the proof is complete.

Finally, the coherent case follows identically to Corollary 2.5 of <cit.>.




For the reader who is interested in the relation to set-valued risk measures, let us note that Proposition A.1.3 of <cit.> provides a sufficient condition on the underlying set-valued risk measure R_t (with any choice of eligible space M̃⊇ M) to guarantee that ρ_t satisfies the Fatou property. 



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



Let ρ_t be a scalarized conditional convex risk measure.  Then ρ_t satisfies the Fatou property 
if and only if for any X ∈ 

    ρ_t(X) = _(,S) ∈_t-β_t(,S) - S_TXt

where 
    β_t(,S) = _Z ∈ A_t -S_TZt
 and

    _t = (,(S_s)_s = t^T) ∈ℳ×∏_s = t^T s |  [         ∀ s = t,...,T, ∀ i = 1,...,d:; S_s,1≡ 1,  S_s,i_∞≤max{e_i__T,0 , 1},;                 S_s = S_Ts,   S_T ∈_T ]..



By Proposition <ref> the result trivially follows if


  * for every (,m_⊥) ∈_t(e_1) with α_t(,m_⊥) < +∞ there exists a (,S) ∈_t such that w_t^s(,e_1+m_⊥) = ξ̅_t,s() S_s for every time s ≥ t, and

  * for every (,S) ∈_t there exists a (,m_⊥) ∈_t(e_1) such that w_t^s(,e_1+m_⊥) = ξ̅_t,s() S_s for every time s. 


Let us show that the first property holds. Let (,m_⊥) ∈_t(e_1) with α_t(,m_⊥) < +∞.  Let Z_s := w_t^s(,e_1+m_⊥); note that (Z_s)_s = t^T is a -martingale.  Then we will define S_s by S_s,i = Z_s,i/Z_s,1   on {Z_s,1 > 0}
 1    else for all indices i = 1,...,d.  Note that S_s,1≡ 1 for all times s and S_t = e_1 + m_⊥∈t^d_+.  Further, define ∈ℳ by = Z_T,1/Z_t,1 = Z_T,1 (since Z_t = e_1+m_⊥).  Notice that {Z_s,1 = 0}⊆{Z_T,1 = 0} and by property of _t and Assumption <ref> we have that {Z_T,1 = 0} = {Z_T,i = 0} (as Z_T ∈_t^+) for any choice of index i, therefore we can see that

    S_T,is   = 1_{s > 0}/sS_T,i + 1_{s = 0} S_T,is
       = [ 1_{Z_s,1 > 0}Z_T,1/Z_s,11_{Z_T,1 > 0}Z_T,i/Z_T,1 + 1_{Z_T,1 = 0};          + 1_{Z_s,1 = 0}1_{Z_T,1 > 0}Z_T,i/Z_T,1 + 1_{Z_T,1 = 0} ]s
       = [ 1_{Z_s,1 > 0}1_{Z_T,1 > 0}Z_T,i/Z_s,1 + 1_{Z_s,1 > 0}1_{Z_T,1 = 0}Z_T,1/Z_s,1;          + 1_{Z_s,1 = 0}1_{Z_T,1 > 0}Z_T,i/Z_T,1 + 1_{Z_s,1 = 0}1_{Z_T,1 = 0} ]s
       = 1_{Z_s,1 > 0}1/Z_s,11_{Z_T,1 > 0} Z_T,is + 1_{Z_s,1 = 0}
       = 1_{Z_s,1 > 0}1/Z_s,11_{Z_T,i > 0} Z_T,is + 1_{Z_s,1 = 0}
       = 1_{Z_s,1 > 0}1/Z_s,1Z_T,is + 1_{Z_s,1 = 0}
       = 1_{Z_s,1 > 0}Z_s,i/Z_s,1 + 1_{Z_s,1 = 0} = S_s,i

for any s ∈{t,...,T} and any i = 1,...,d.  Finally 

    ξ̅_t,s() S_s    = Z_s,11_{Z_s,1 > 0}Z_s/Z_s,1 + 1_{Z_s,1 = 0} = 1_{Z_s,1 > 0} Z_s
       = 1_{ξ̅_t,s(_1) > 0} w_t^s(,e_1+m_⊥) ∈_t.

It remains to show that S_T ∈^d_+ (the case for s < T would then follow trivially): by definition e_i__T,0 e_1 ≥__T e_i ≥__T -e_i__T,0 e_1 for any index i and, by the definition of _t, we know e_i__T,0 < ∞.  This implies e_i__T,0≥ S_T,i≥ -e_i__T,0 (noting that _t⊆_T = Z ∈  |  ZK ≥ 0  - ∀ K ∈K_T), by dividing by  on the set { > 0} recovers S_T,i∈∞Ω,s,; with S_T,i_∞^≤e_i__T,0 (i.e., the ∞-norm of S_T,i under measure  is bounded).  Finally, by construction S_T,i = 1 on { = 0}, thus S_T,i_∞≤max{e_i__T,0 , 1}.  Since S is a -martingale, the bound S_s,i_∞≤max{e_i__T,0 , 1} holds as well.

To show the converse, i.e., the second property, let (,S) ∈_t.  Define m_⊥ = S_t - e_1 ∈{0}×1t^d-1.  And define = S_T,i/S_t,i   on S_t,i > 0
 1    else.  Then w_t^s(,e_1+m_⊥)_i = S_t,is = S_t,iS_T,i/S_t,i 1_{S_t,i > 0} + 1_{S_t,i = 0}s =  S_T,is = ξ̅_t,s() S_T,is = ξ̅_t,s() S_s,i for every index i = 1,...,d (because S_t,i = 0⊆ S_T,i = 0 by S a -martingale). 



For every choice of dual variables (,S) ∈_t we find that the dual norm S_T__T,t^* := {|S_TXt|   |  X__T,t≤ 1  -} = 1 almost surely.  








Let ρ_t be a scalarized conditional coherent risk measure.  Then ρ_t satisfies the Fatou property 
if and only if for any X ∈ 

    ρ_t(X) = _(,S) ∈_t^ρ -S_TXt

where _t^ρ = (,S) ∈_t   |  β_t(,S) = 0  -.


This follows directly from Theorem <ref> and Proposition <ref>. 



The d dimensional version of the dual representation of the scalar superhedging price (<ref>) given in Jouini, Kallal <cit.> is with respect to dual variables (,S) ∈ℳ×0^d_+ with S_s ≡ 1 for every s,  is a martingale measure for S, and S_T ∈_0.  This set of dual variables is a superset of _t from Theorem <ref>. From the proof of Theorem <ref>, it is clear that this larger set of dual variables could be used within this work as well.  However, for reasons that will be clear in subsequent results, e.g., Proposition <ref>, which is used extensively after, we restrict ourselves to the bounded prices as defined in _t. 





 §.§ Relevance




In the below proposition we will define a property so that the dual representation (<ref>) can be defined w.r.t. the dual variables 

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


Before continuing, note that (,S) ∈_t^e if and only if ∈ℳ^e is a martingale measure for S and S ∈∏_s = t^T sK_s such that S_s,1≡ 1 and S_s,i_∞≤max{e_i__T,0 , 1}.  This is in contrast to the definition of _t in which S_T ∈_t.  This modification can be accomplished as we are now only using equivalent measures  and since _t = ⋂_s = t^T Y ∈  |  Ysk_s ≥ 0  - ∀ k_s ∈sK_s. 





Let ρ_t be a scalarized conditional convex risk measure satisfying the Fatou property. Then

    ρ_t(X) = _(,S) ∈_t^e-β_t(,S) - S_TXt

for every X ∈ if and only if inf_Z ∈ A_tS_T Z > -∞ for some (,S) ∈_t^e.


If ρ_t(X) = _(,S) ∈_t^e-β_t(,S) - S_TXt for every X ∈ then, by t-decomposability of {S_TZ   |   (,S) ∈_t^e,   Z ∈ A_t} (recall from Section <ref> that a set B is called t-decomposable if 1_D B + 1_D^c B ⊆ B for any D ∈t) and the definition of the penalty function β_t, ρ_t() = sup_(,S) ∈_t^einf_Z ∈ A_tS_T Z > -∞ by ρ_t() ∈t.  In particular this implies that there exists some (,S) ∈_t^e such that inf_Z ∈ A_tS_T Z > -∞.

Conversely, let (,S̃) ∈_t^e such that inf_Z ∈ A_tS̃_T Z > -∞.  Since _t^e ⊆_t, the dual representation in Theorem <ref> implies that for every X ∈, ρ_t(X) ≥_(,S) ∈_t^e-β_t(,S) - S_TXt.
As in the proof of Proposition A.1.7 of <cit.> we will show the reverse inequality by considering the expectation.
Since {-β_t(,S) - S_TXt  |   (,S) ∈_t} is t-decomposable we are able to interchange the expectation and essential supremum.  
As done in <cit.>, for any ϵ∈ (0,1) and (,S) ∈_t, define
(_ϵ,S_ϵ) := (1-ϵ) (,S) + ϵ (,S̃) ∈_t^e.
Additionally, by definition of the penalty function, we can conclude
β_t(_ϵ,S_ϵ) ≤ (1-ϵ)β_t(,S) + ϵβ_t(,S̃).
Choose any X ∈

    ρ_t(X)   = sup_(,S) ∈_t-β_t(,S) - S_T X
       ≥sup_(,S) ∈_t^e-β_t(,S) - S_TX
       ≥sup_(,S) ∈_t-β_t(_ϵ,S_ϵ) - d_ϵ/dS_ϵ,TX
       ≥sup_(,S) ∈_t(1-ϵ) -β_t(,S) - S_TX
           + ϵ-β_t(,S̃) - d/dS̃_TX
       = (1-ϵ) ρ_t(X) + ϵ-β_t(,S̃) - d/dS̃_TX.

Since ρ_t(X)∈ then we take the limit as ϵ tends to 0 to recover that the final line above is equivalent to the first line, and the result is shown.





Let ρ_t be a scalarized conditional coherent risk measure satisfying the Fatou property. Then

    ρ_t(X) = _(,S) ∈_t^ρ,e -S_TXt

for every X ∈ if and only if _t^ρ,e := _t^e ∩_t^ρ≠∅, i.e., there exists some (,S) ∈_t^e such that β_t(,S) = 0 almost surely.


Recall from the logic of the proof of Corollary 2.5 of <cit.>, for any (,S) ∈_t we have that β_t(,S) < +∞ if and only if (,S) ∈_t^ρ.  The results follows from Proposition <ref> above.  If ρ_t(X) = _(,S) ∈_t^ρ,e -S_TXt then there exists some (,S) ∈_t^e such that inf_Z ∈ A_tS_T Z > -∞ (else ρ_t(·)≡ -∞).  Conversely, if inf_Z ∈ A_tS_T Z > -∞ for some (,S) ∈_t^e then we have

    ρ_t(X)    = _(,S) ∈_t^e 
    β_t(,S) < +∞-β_t(,S) - S_TXt
       = _(,S) ∈_t^ρ,e -S_TXt.





As in the literature for scalar risk measures, we will now introduce relevance (or sensitivity) and demonstrate that it implies the
existence of (,S) ∈_t^e such that inf_Z ∈ A_tS_T Z >
-∞.  See, e.g., <cit.> for previous literature on relevance of univariate conditional risk measures. 


A risk measure ρ_t is called relevant if 

    (ρ_t(-ϵ 1_D e_1) > ρ_t()) > 0

for every ϵ > 0 and every D ∈ with (D) > 0. 





Let ρ_t be a scalarized conditional convex risk measure satisfying the Fatou property. If ρ_t is relevant then there exists (,S) ∈_t^e such that inf_Z ∈ A_tS_T Z > -∞.


First, β_t(,S)   |   (,S) ∈_t is t-decomposable (by the definition of _t and β_t).  Therefore ∋-ρ_t() = _(,S) ∈_tβ_t(,S) = inf_(,S) ∈_tβ_t(,S). 
For any ϵ > 0 let the set _t^ϵ⊆_t be defined by

    _t^ϵ := (,S) ∈_t   |  β_t(,S) < -ρ_t() + ϵ -.


Now we will show that for any ϵ > 0 there exists a pair of dual variables (,S̃) ∈_t^e ∩_t^ϵ (and in particular this implies inf_Z ∈ A_tS̃_T Z > -∞).


  * We will show that _t^ϵ≠∅ for any ϵ > 0.  Let (_n,S_n)_n ∈⊆_t such that β_t(_n,S_n) ↘ -ρ_t() almost surely.  This sequence exists because β_t(,S)   |   (,S) ∈_t is t-decomposable, thus there exists a sequence (_n,S_n)_n ∈⊆_t such that 
    β_t(_n,S_n) ↘_(,S) ∈_tβ_t(,S) = -ρ_t()   -

    Define the t-measurable random variable τ_ϵ by
    
    τ_ϵ := minn   |  β_t(_n,S_n) < -ρ_t() + ϵ.

    It holds τ_ϵ < +∞ almost surely by construction of the sequence (_n,S_n)_n ∈, and thus (τ_ϵ = n)_n ∈⊆t defines a partition of Ω. We can define (_ϵ,S̅_ϵ) := ∑_n = 1^∞ 1_τ_ϵ = n (_n,S_n) ∈_t (trivially) with β_t(_ϵ,S̅_ϵ) ≤∑_n = 1^∞ 1_τ_ϵ = nβ_t(_n,S_n) < -ρ_t() + ϵ, i.e., (_ϵ,S̅_ϵ) ∈_t^ϵ.


  * We will show that if ρ_t is relevant then for every ϵ > 0 there exists an element (,S̃) ∈_t^ϵ such that d/d > 0 a.s.  Fix ϵ > 0 and let c_ϵ := sup( > 0)   |   (,S) ∈_t^ϵ.
Take a sequence (_n,S_n)_n ∈⊆_t^ϵ such that lim_n →∞(n > 0) = c_ϵ.  Define (,S̃) := ∑_n = 1^∞1/2^n (_n,S_n). It holds (,S̃) ∈_t^ϵ by convexity of β_t. In fact, d/d > 0 = ∪_n = 1^∞n > 0.  Therefore (d/d > 0) = c_ϵ.  It remains to show that c_ϵ = 1, and thus ∈ℳ^e.

    Assume c_ϵ < 1, and let D := d/d = 0.  Relevance implies

    (ρ_t(-ϵ 1_D e_1) > ρ_t()) > 0,

and by construction we have

    ρ_t(-ϵ 1_D e_1) = _(,S) ∈_t-β_t(,S) + ϵ1_Dt.

Therefore, there exists (,S̅) ∈_t such that the set

    B := β_t(,S̅) < -ρ_t() + ϵ1_Dt∈t

has positive probability, i.e., β_t(,S̅) < -ρ_t() + ϵ on B.

    Now we will define (,Ŝ) ∈_t^ϵ which is equal to (,S̅) on B.  Let (,S) ∈_t^ϵ (arbitrary) and let (,Ŝ) := 1_B (,S̅) + 1_B^c (,S) ∈_t^ϵ (because β_t(,Ŝ) = 1_B β_t(,S̅) + 1_B^cβ_t(,S)).

    By definition of B and since β_t(,S̅) ≥ -ρ_t() (by the dual representation of ρ_t()),     
    
    1_B 1_D d/d = 1_B 1_D d/dt > 0.

    This implies that d/d > 0∩ D≥d/d > 0∩ D ∩ B > 0.  Therefore 1/2 (,S̃) + 1/2 (,Ŝ) ∈_t^ϵ by convexity of _t^ϵ, and we have found a contradiction.



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



Let ρ_t be a scalarized conditional coherent risk measure satisfying the Fatou property.  ρ_t is relevant if and only if (<ref>) holds for every X ∈.


As shown above in Corollary <ref> and Lemma <ref>, if ρ_t is relevant then (<ref>) holds for every X ∈.  Conversely assume that ρ_t has dual representation with respect to the dual variables _t^ρ,e and assume ρ_t is not relevant, i.e., there exists some ϵ > 0 and D ∈ with positive probability such that ρ_t(-ϵ 1_D e_1) ≤ρ_t() = 0 almost surely.  
In particular this implies ϵ1_Dt≤ρ_t(-ϵ 1_D e_1) ≤ 0 for any (,S) ∈_t^ρ,e.
Therefore ϵ(D) ≤ 0 for every (,S) ∈_t^ρ,e.  However, this can only be true if (D) = 0, which implies (D) = 0 since ∈ℳ^e.  This is a contradiction.















§ TIME CONSISTENCY


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

    ρ_s(X) ≤ρ_s(Y) ⇒ρ_t(X) ≤ρ_t(Y)

for X,Y ∈ and t ≤ s.  
As we will demonstrate in Section <ref> the superhedging portfolios do not satisfy this ρ-time consistency.  Instead we will consider a decomposition of the scalarizations via their dual representations as defined in Theorem <ref>.  These details are provided in Section <ref>.  Then in Section <ref>, we consider a new time consistency property.  In particular, with this new property we determine the typical equivalence of time consistency and a recursive relation.



For the remainder of this paper we will assume that ρ_t is a relevant scalarized conditional convex risk measure satisfying the Fatou property, and corresponds with the acceptance set A_t as given in (<ref>).




 §.§ Overview

Before continuing to the main results of this work in regards to time consistency of the dynamic multivariate risk measures ρ, we will provide a brief overview over the topic, illustrate the importance of the derived results, set them in relation to existing results in the literature, and provide an economic interpretation.

As mentioned previously, the usual time consistency property, ρ-time consistency, is in general not satisfied for dynamic multivariate risk measures ρ given by (<ref>). In particular, it does not hold for the prototypical case of the superhedging risk measure as detailed in, e.g., Section <ref> below.  This is because of the frictions inherent to the risk measure in covering all risks with the cash asset alone, i.e., by considering a market with transaction costs.
In contrast, many prior works (e.g., <cit.>) on time consistency of multivariate risk measures have considered set-valued risk measures and a property called multiportfolio time consistency.  A set-valued risk measure is said to be multiportfolio time consistent if

    ⋃_X ∈𝒳 R_s(X) ⊇ R_s(Y)  ⇒ ⋃_X ∈𝒳 R_t(X) ⊇ R_t(Y)

for all 𝒳⊆, Y ∈, and t ≤ s.  This property relies on the choice of eligible assets M̃ taken in the image space for the set-valued risk measure R_t: →𝒢(M̃_t;M̃_t,+).  For instance, the set-valued superhedging risk measure is multiportfolio time consistent w.r.t. the full space of eligible assets (i.e., when one chooses M̃ = ^d) but not with the cash asset only (i.e., when M̃ = M=×{0}^d-1 as assumed within this paper), see Section <ref> below for details.  As with ρ-time consistency, this is due to the frictions inherent to the risk measure when restricting to covering risk with only the cash asset.

However, in Section <ref>, we will introduce a decomposition of the conditional risk measure ρ_t into a collection of multivariate risk measures in frictionless markets, each of which is defined by a price process satisfying a no-arbitrage condition. This decomposition allows us to recover the desired risk measure as the pointwise supremum over this collection of frictionless risk measures.  Given a price process S = (S_t)_t = 0^T ∈ (defined in (<ref>) below), we denote these frictionless multivariate risk measures as π^S.  The relationship between these frictionless multivariate risk measures and the typical (univariate) risk measures of, e.g., <cit.>, is presented in Section <ref>.  These new risk measures allow us to define a new notion of time consistency, called π-time consistency and provided in Definition <ref> below, which overcomes the obstacles related to market frictions in both ρ-time consistency and multiportfolio time consistency.  Briefly, if every no-arbitrage price S consistent with the market model  induces a risk measure π^S that is time consistent in the usual way, then we say the original risk measure ρ is π-time consistent.  Notably, the superhedging risk measure is π-time consistent.  Results on π-time consistency and its relation to multiportfolio time consistency of the underlying set-valued risk measure (appearing in (<ref>) and with possibly different choices of eligible assets) are provided in Sections <ref> and <ref>.  Though the proposed time consistency notion is weaker than ρ-time consistency, it is stronger than the weak notion of ρ-time consistency (i.e., where ρ_s(X) ≤ρ_s() implies ρ_t(X) ≤ρ_t() for every portfolio X ∈) as studied for univariate risk measures in, e.g., <cit.>. Section <ref> also contains a result on a backward recursion for π-time consistent scalar multivariate risk measures ρ_t  with respect to the underlying risk measure π^S.






 §.§ Preliminaries


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

    := S ∈∏_t = 0^T tK_t |  [     ∀ t = 0,...,T,  ∀ i ∈{2,3,...,d}:; S_t,1 = 1,  S_t,i_∞≤max{e_i__T,0 , 1} ]..


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

Let S ∈.  Define π_t^S: →t∪{-∞} by

    π_t^S(X) := _∈^e(S)-β_t(,S) - S_TXt

for any X ∈ where β_t is defined as in Theorem <ref> dependent on the acceptance set A_t and

    ^e(S) := ∈ℳ  |   (,S) ∈_0^e.


Note that π_t^S(X) ∈t∪{-∞} in general even though ρ_t(X) ∈t.  Though π_t^S(X) is trivially bounded from above by ρ_t(X), it is not integrable in general. Proposition <ref> provides a simple condition so that π_t^S(X) ∈t for any X ∈.



A sequence of prices S ∈ satisfies the no-arbitrage condition if

    NA∑_s = 0^TΓ_s(S_s)∩-Γ_T(S_T)⊆Γ_T(S_T)

where Γ_t(S_t) := {Z ∈t  |  S_tZ ≥ 0 }.


The motivation for calling this property “no-arbitrage” comes from the following proposition which relates the condition to the fundamental theorem of asset pricing.



Let S ∈, then S satisfies the no-arbitrage condition if and only if ^e(S) ≠∅.


This follows directly from the fundamental theorem of asset pricing by noting that our definition of no-arbitrage is equivalent to the usual one in frictionless markets.  This can be seen by:


  * A predictable process η is self-financing if and only if S_t(η_t+1 - η_t) ≤ 0 almost surely for every time t = 0,1,...,T-1 with η_0 = 0.  Equivalently this can be written as η_t+1 - η_t ∈ -Γ_t(S_t) for every time t = 0,1,...,T-1 with η_0 = 0.  Through summations, η_t is attainable through self-financing trading strategy if and only if η_t ∈ -∑_s = 0^t-1Γ_s(S_s).

  * The wealth of a portfolio η_t at time t is given by V_t := S_tη_t.  Therefore, in our notation, wealth satisfies the conditions V_t ≥ 0 and (V_t > 0) > 0 if and only if η_t ∈Γ_t(S_t) and η_t ∉-Γ_t(S_t) respectively.

Therefore it is immediately clear that (<ref>) is equivalent to stating that V_0 = 0, V_T ≥ 0 almost surely implies V_T = 0 almost surely.  Finally, we can replace the self-financing condition -∑_s = 0^T-1Γ_s(S_s) with the summation up to time T as these are equivalent conditions:


  * Assume ∑_s = 0^TΓ_s(S_s)∩-Γ_T(S_T)⊆Γ_T(S_T) then immediately by 0 ∈Γ_T(S_T) we can conclude ∑_s = 0^T-1Γ_s(S_s)∩-Γ_T(S_T)⊆Γ_T(S_T).

  * Conversely, assume ∑_s = 0^T-1Γ_s(S_s)∩-Γ_T(S_T)⊆Γ_T(S_T) and let X ∈∑_s = 0^TΓ_s(S_s)∩-Γ_T(S_T). We can decompose X = ∑_s = 0^T X_s where S_sX_s ≥ 0 almost surely and S_TX ≤ 0 almost surely.  Then we can easily see that

    0 ≥S_T X = S_T∑_s = 0^T-1 X_s + S_T X_T ≥S_T∑_s = 0^T-1 X_s.

Therefore ∑_s = 0^T-1 X_s ∈∑_s = 0^T-1Γ_s(S_s)∩-Γ_T(S_T)⊆Γ_T(S_T), i.e., S_T∑_s = 0^T-1 X_s = 0 almost surely.  This immediately implies that S_TX = 0 almost surely as well from (<ref>), i.e., X ∈Γ_T(S_T).







Let S ∈ then the following are equivalent:


  *  π_t^S() ∈t,

  *  there exists some ∈^e(S) such that β_t(,S) ∈t, and 

  *  S satisfies (<ref>) and there exists some X ∈t such that X ∈Ã_t^π = 0, where A_t^π := X ∈  |  π_t^S(X) ≤ 0 = A_t + ∑_s = t^T Γ_s(S_s) and Ã_t^π is a t-measurable random set such that A_t^π∩t = ∞tÃ_t^π.



First we will show that (<ref>.) is equivalent to (<ref>.), then we will show this is equivalent to (<ref>.).


  * First we will assume there exists some ∈^e(S) such that β_t(,S) ∈t. For such a  and for ρ_t defined by the same acceptance set A_t,

    t∋ -β_t(,S) ≤π_t^S() ≤ρ_t() ∈t.

Now we wish to prove the converse. We can trivially deduce a lower bound β_t(,S) ≥ -π_t^S() ∈t for every ∈^e(S).  To prove the upper bound we first note that {β_t(,S)   |  ∈^e(S)} is an t-decomposable set. This implies there exists a sequence (_n)_n ∈⊆^e(S) such that -β_t(_n,S) ↗π_t^S() almost surely.  Let us define τ^ϵ := min{n ∈  |   -β_t(_n,S) + ϵ > π_t^S()} < ∞ almost surely.  Then := ∑_n = 1^∞ 1_{τ^ϵ = n}_n ∈^e(S) (trivially utilizing monotone convergence) and 

    β_t(,S) ≤∑_n = 1^∞ 1_{τ^ϵ = n}β_t(_n,S) < -π_t^S() + ϵ∈t.


  * First, Ã_t^π exists by <cit.> since A_t^π is a nonempty, closed, and t-decomposable.

Given the existence of Ã_t^π, first we will assume that S satisfies (<ref>) and there exists some X ∈t such that X ∈Ã_t^π = 0.  Fix X ∈t such that X ∈Ã_t^π = 0, then there exists some ∈^e(S) (where ^e(S) ≠∅ by Proposition <ref>) such that β(,S) = _Z ∈ A_t -S_TZt < -S_tX ∈t by a conditional separating hyperplane argument.



Additionally, β_t(,S) ≥ -ρ_t() ∈t for any ∈^e(S) by construction.
Conversely, ^e(S) ≠∅ implies the prices S satisfy (<ref>) by Proposition <ref>.  And given β_t(,S) = _Z ∈ A_t -S_TZt∈t, we can immediately conclude that -β_t(,S) + ϵe_1 ∈t for any ϵ > 0 and -β_t(,S) + ϵe_1 ∈Ã_t^π = 0.





Fix S ∈ satisfying (<ref>), the following properties hold for π_t^S:


  * Conditional cash invariance: π_t^S(X + m) = π_t^S(X) - S_tm for any X ∈ and m ∈t;

  * Monotonicity: π_t^S(X) ≤π_t^S(Y) if S_TX ≥S_TY almost surely;

  * Conditional convexity: π_t^S(λ X + (1-λ)Y) ≤λπ_t^S(X) + (1-λ)π_t^S(Y) for every λ∈∞t[0,1];

  * Value consistent: π_t^S(X) = π_t^S([S_TX]e_1) for any X ∈.

If A_t is additionally a conditional cone then π_t^S is conditional positive homogeneous.


Trivially from the definition of π_t^S and that S_t = S_Tt for any ∈^e(S) and S_T,1 = 1 almost surely.


The following proposition provides a necessary and sufficient condition for π_t^S(X) ∈t for every X ∈.  Proposition <ref> relates this condition to the no-arbitrage condition (<ref>) for the frictionless prices S.


Let S ∈ and X ∈ then, for any time t, π_t^S(X) ∈t if and only if  π_t^S() ∈t for any time t.


Let X = X_1_∞,…,X_d_∞ and assume π_t^S() ∈t. 

    π_t^S(X)    ≤π_t^S(-X) = π_t^S() + S_tX∈t
    π_t^S(X)    ≥π_t^S(X) = π_t^S() - S_tX∈t.

The converse direction is trivial from the same inequalities.



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

    _t := {S ∈  |  π_t^S() ∈t},
 
i.e., S ∈_t if and only if S satisfies the no-arbitrage condition (<ref>) and there exists some X ∈ such that X ∈Ã_t^π = 0.

The following proposition shows the decomposition of the dual representation (<ref>) of ρ_t into two components based on the two dual variables (,S).  π_t^S(X), as defined in (<ref>), runs over the dual variables  only.  But, it is shown in Proposition <ref> below, it is enough to take the essential supremum with respect to S over the set _t only. This means it is enough to consider in the dual representation of ρ_t just those π_t^S(X) that map into t, i.e., to consider those frictionless prices that satisfy the no-arbitrage condition (<ref>).  This will be of particular importance when considering the new time consistency concept presented in Sections <ref> and <ref>.


 
For any time t and any X ∈ 

    ρ_t(X) = _S ∈_tπ_t^S(X).



Let S ∈ then ∈^e(S) implies (^t,(S_s)_s = t^T) ∈_t^e where d^t/d := ξ̅_t,T().  By the definition of ^t and the -almost sure definition of the conditional expectation, we recover that -β_t(,S) - S_TXt = -β_t(^t,S) - ^tS_TXt for any X ∈.  
Conversely, let (,(S_s)_s = t^T) ∈_t^e and define Ŝ∈ by Ŝ_s := S_Ts =  S_s    if  s ≥ t 
S_ts   if  s < t.  Immediately it is clear that ∈^e(Ŝ).  By definition of Ŝ, we recover that -β_t(,S) - S_TXt = -β_t(,Ŝ) - Ŝ_TXt for any X ∈.

Using this result, we determine the equality of the first and second lines below:

    ρ_t(X)    = _(,S) ∈_t^e-β_t(,S) - S_TXt
    
       = _S ∈_∈^e(S)-β_t(,S) - S_TXt
       = _S ∈π_t^S(X).


From this and Proposition <ref> we immediately find 

    ρ_t(X) ≥_S ∈
    π_t^S() ∈tπ_t^S(X) ≥_(,S) ∈_t^e
    β_t(,S) ∈t-β_t(,S) - S_TXt.

We now wish to show the reverse inequality.  Since {-β_t(,S) - S_TXt  |   (,S) ∈_t^e} is t-decomposable we can find a sequence (_n,S_n)_n ∈⊆_t^e such that -β_t(_n,S_n) - nS_n,TXt↗ρ_t(X) ∈t almost surely.  Define τ^ϵ := min{n ∈  |  β_t(_n,S_n) + nS_n,TXt < -ρ_t(X) + ϵ} < ∞ almost surely for any ϵ > 0, and let (,S̃) := ∑_n = 1^∞ 1_{τ^ϵ = n} (_n,S_n) ∈_t^e (trivially using monotone convergence).  Then

    t∋ -ρ_t() ≤β_t(,S̃) ≤ -ρ_t(X) - S̃_TXt + ϵ∈t.

Since such a (,S̃) ∈_t^e can be found for any ϵ > 0, a sequence of those can be used to get convergence as ϵ→ 0 and the proof is complete.









 §.§ Relation to univariate scalar risk measures

In this section we wish to relate the terms π^S for a fixed S ∈ introduced in the prior section to the univariate dynamic risk measures as studied in, e.g., <cit.>.  This relationship will then be used later to directly prove some properties for π^S.



Let S ∈_t, then ϕ_t^S: →t defined by 

    ϕ_t^S(Z) := π_t^S(Ze_1)

for any Z ∈ is a convex (coherent) univariate scalar risk measure.  Additionally, π_t^S(X) = ϕ_t^S(S_TX) for any X ∈.


First we will show that ϕ_t^S is a traditional scalar risk measure:


  * ϕ_t^S(0) = π_t^S() ∈t.

  * Let Z ∈ and m ∈t, then ϕ_t^S(Z + m) = π_t^S([Z + m]e_1) = π_t^S(Ze_1) - (S_te_1)m = ϕ_t^S(Z) - m.

  * Let Z_1,Z_2 ∈ where Z_1 ≥ Z_2 almost surely.  Then Z_1e_1 ≥ Z_2e_1 almost surely, and by definition ϕ_t^S(Z_1) = π_t^S(Z_1e_1) ≤π_t^S(Z_2e_1) = ϕ_t^S(Z_2).

  * Convexity and positive homogeneity follow immediately.

The final statement follows by π_t^S(X) = π_t^S([S_TX]e_1) = ϕ_t^S(S_TX) for any X ∈.



We will now consider the primal and dual representations for ϕ_t^S, defined in (<ref>).  Afterwards we will consider the representations for the stepped risk measures ϕ_t,s^S for t < s.  These stepped risk measures are useful for equivalent properties of time consistency, as discussed in the next section.


Let S ∈_t, then the primal and dual representations for ϕ_t^S, defined in (<ref>), are given by:


  * ϕ_t^S(Z) = m ∈t  |   Z + m ∈S_TA_t^π for any Z ∈ where A_t^π = {X ∈  |  π_t^S(X) ≤ 0};

  * ϕ_t^S(Z) = _∈^e(S)-β_t(,S) - Zt for any Z ∈.



First, since ϕ_t is a traditional scalar risk measure, we immediately know a primal and dual representation exist: for any Z ∈

    ϕ_t^S(Z)    = m ∈t  |   Z + m ∈ A_t^ϕ
       = _∈ℳ-β_t^S() - Zt.

where A_t^ϕ = Z ∈  |  ϕ_t^S(Z) ≤ 0 and β_t^S() = _Z ∈ A_t^ϕ -Zt for every ∈ℳ.
Therefore showing A_t^ϕ = S_TA_t^π is sufficient to prove this result.  We note that A_t^π = A_t + ∑_s = t^T Γ_s(S_s) in order to find

    A_t^ϕ   = Z ∈  |  ϕ_t^S(Z) ≤ 0
       = Z ∈  |  π_t^S(Ze_1) ≤ 0
       = Z ∈  |   Ze_1 ∈A_t + ∑_s = t^T Γ_s(S_s)
       = S_TX   |   X ∈A_t + ∑_s = t^T Γ_s(S_s)∩ M_T
       = S_TX   |   X ∈A_t + ∑_s = t^T Γ_s(S_s)
       = S_TA_t^π.

Equation (<ref>) follows from S_T,1 = 1 almost surely for any S ∈_t.  Equation (<ref>) follows since X ∈ A_t^π if and only if [S_TX]e_1 ∈ A_t^π.

We conclude this proof with a consideration of the structure of the penalty function β_t^S.  Let ∈ℳ, then

    β_t^S()    = _Z ∈ A_t^ϕ -Zt
       = _X ∈ A_t^π -S_TXt
       = _X ∈ A_t -S_TXt + ∑_s = t^T _k_s ∈Γ_s(S_s) -S_Tk_st
       = β_t(,S)    on {∈^e(S)}
    ∞   else

Since the dual representation is taken on the set with β_t^S() < ∞ this implies

    ϕ_t^S(Z) = _∈^e(S)-β_t(,S) - Zt
 
for any Z ∈.




Let S ∈_t and t < s. Define ϕ_t,s^S: s→t as the restriction of the domain of ϕ_t^S to s.  Then primal and dual representations are defined via the acceptance set and penalty function for primal and dual representations given in:


  * A_t,s^ϕ = S_sA_t,s^π where A_t,s^π := A_t^π∩s;

  * β_t,s^S() = _X ∈ A_t,s^π -S_sXt for any ∈^e(S).



Since ϕ_t,s^S is a univariate stepped risk measure, if we prove the result for the acceptance set then the penalty function follows immediately.  First we will show that A_t,s^ϕ := A_t^ϕ∩s⊆S_sA_t,s^π where A_t,s^π := A_t^π∩s.  Take Z ∈ A_t,s^ϕ, then π_t^S(Ze_1) = ϕ_t^S(Z) ≤ 0 thus Ze_1 ∈ A_t,s^π.  By S_s,1 = 1 almost surely, Z = S_s(Ze_1) ∈S_sA_t,s^π.  Now we will show that A_t,s^ϕ⊇S_sA_t,s^π.  Let X ∈ A_t,s^π, then

    ϕ_t,s^S(S_sX) = ϕ_t^S(S_sX) = π_t^S([S_sX]e_1) = π_t^S(X) ≤ 0

where the last equality follows directly from the definition of π_t^S and X ∈s.



We would like to note that, unlike in the full case ϕ_t^S, the stepped risk measure need not satisfy the dual representation with respect to the penalty function from the scalarized conditional convex risk measure ρ_t.  That is, it is possible that ϕ_t,s^S(Z) ≠_∈^e(S)-β_t,s(,S) - Zt for some Z ∈s.







 §.§ π-time consistency

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


Fix S ∈.  We say π^S is time consistent if for all times t < s and X,Y ∈ it holds

    π_s^S(X) ≤π_s^S(Y) ⇒π_t^S(X) ≤π_t^S(Y).

The dynamic risk measure ρ is called π-time consistent if π^S is time consistent for every price sequence S ∈^* := ⋂_t ≥ 0_t.





Fix S ∈^*, then π^S is time consistent if and only if ϕ^S is time consistent (defined in the usual way for univariate scalar risk measures).


Let π^S be time consistent and let Z_1,Z_2 ∈ such that ϕ_s^S(Z_1) ≥ϕ_s^S(Z_2) for times t < s:

    ϕ_s^S(Z_1) ≥ϕ_s^S(Z_2)    ⇒π_s^S(Z_1e_1) ≥π_s^S(Z_2e_1)
       ⇒π_t^S(Z_1e_1) ≥π_t^S(Z_2e_1) ⇒ϕ_t^S(Z_1) ≥ϕ_t^S(Z_2).


Conversely, let ϕ^S be time consistent and let X_1,X_2 ∈ such that π_s^S(X_1) ≥π_s^S(X_2):

    π_s^S(X_1) ≥π_s^S(X_2)    ⇒ϕ_s^S(S_TX_1) ≥ϕ_s^S(S_TX_2)
       ⇒ϕ_t^S(S_TX_1) ≥ϕ_t^S(S_TX_2) ⇒π_t^S(X_1) ≥π_t^S(X_2).



We will now use the equivalence between the time consistency of π^S and ϕ^S to determine equivalent properties for time consistency of π^S. 

Fix S ∈^*, then the following are equivalent for a normalized and convex π^S (i.e., π_t^S() = 0 for all times t ≥ 0).


  * π^S is time consistent;

  * π_t^S(X) = π_t^S(-π_s^S(X)e_1) for all t < s and X ∈; 

  * A_t^π = A_t,s^π + A_s^π where A_t,s^π = A_t^π∩s for all t < s;

  * β_t(,S) = β_t,s^S() + β_s(,S)t for all t < s and all probability measures ∈^*(S), where we define

    ^*(S) := ∈^e(S)   |  β_0(,S) < ∞;


  * For all ∈^*(S) and all X ∈, the process

    V_t^(X) := π_t^S(X) + β_t(,S),    t ≥ 0

is a -supermartingale.

In each case the dynamic risk measure admits a robust representation in terms of the set ^*(S) defined in (<ref>), i.e.,

    π_t^S(X) = _∈^*(S)-β_t(,S) - S_TXt

for all X ∈ and all times t ≥ 0.


These results all follow immediately from the equivalent properties of ϕ^S-time consistency and its equivalence to π^S-time consistency (see Proposition <ref>).  To show the equivalence of the recursive formulation and the summation of acceptance sets we consider Lemma <ref> below as the equivalence between acceptance set versions of time consistency for ϕ^S and π^S is less direct.



Consider the setting of Theorem <ref> but for a non-normalized π^S.  Then π^S is time consistent if and only if π_t^S(X) = π_t^S([π_s^S()-π_s^S(X)]e_1) for all t < s and X ∈.


Now consider coherent π^S only.  Note that, immediately, the coherence of π^S implies normalization.  Additionally, in such a setting, the penalty function is provided by an indicator function; as such we can describe the probability measures of interest, defined in (<ref>), by

    ^*(S) = ∈^e(S)   |  β_0(,S) = 0.


Fix S ∈^*, then the following are equivalent for a coherent π^S.


  * π^S is time consistent;

  * π_t^S(X) = π_t^S(-π_s^S(X)e_1) for all t < s and X ∈;

  * A_t^π = A_t,s^π + A_s^π where A_t,s^π = A_t^π∩s for all t < s;

  * The set ^*(S) is stable;

  * For all ∈∈^e(S)   |  β_0(,S) = 0 and all X ∈, the process π_t^S(X)_t = 0^T is a -supermartingale.

In each case the dynamic risk measure admits a robust representation in terms of the set ^*(S), i.e.,

    π_t^S(X) = _∈^*(S) -S_TXt

for all X ∈ and all times t ≥ 0.




Fix S ∈^* and let 0 ≤ t ≤ s ≤ T.


  * Let D ⊆s such that D + Γ_s(S_s) ⊆ D.  Then X ∈ D + A_s^π if and only if -π_s^S(X) ∈S_sD.

  * A_t,s^ϕ + A_s^ϕ = S_TA_t,s^π + A_s^π





  * First let X ∈ D + A_s^π.  Immediately we can decompose X = X_D + X_s such that X_D ∈ D and X_s ∈ A_s^π.  Then π_s^S(X) = π_s^S(X_D + X_s) = π_s^S(X_s) - S_sX_D ≤ -S_sX_D.  That is, -π_s^S(X) ≥S_sX_D and by property of D + Γ_s(S_s) ⊆ D this implies -π_s^S(X)e_1 ∈ D or -π_s^S(X) ∈S_sD by S_s,1 = 1.  Conversely let -π_s^S(X) ∈S_sD for some X ∈.  We can then define X = -π_s^S(X)e_1 + (X + π_s^S(X)e_1).  We conclude that -π_s^S(X)e_1 ∈ D and by definition of the acceptance set we see that X + π_s^S(X)e_1 ∈ A_s^π, thus the proof is complete.

  * First consider Z ∈ A_t,s^ϕ + A_s^ϕ.  By <cit.>, this implies -π_s^S(Ze_1) = -ϕ_s^S(Z) ∈ A_t,s^ϕ = S_sA_t,s^π. Therefore, Ze_1 ∈ A_t,s^π + A_s^π and thus Z ∈S_TA_t,s^π + A_s^π by S_T,1 = 1. Conversely, X ∈ A_t,s^π + A_s^π if and only if -ϕ_s^S(S_TX) = -π_s^S(X) ∈S_sA_t,s^π = A_t,s^ϕ.  By <cit.> this is equivalent to S_TX ∈ A_t,s^ϕ + A_s^ϕ, thus the proof is complete.



We conclude this section by considering the backwards composition to construct a π-time consistent risk measure.


Let S ∈^* and consider a discrete time setting.  Define the backwards composition of the portfolio returns Z ∈ by

    ϕ̃_T^S(Z)    = -Z,     ϕ̃_t^S(Z) = ϕ_t^S(-ϕ̃_t+1^S(Z))  ∀ t < T.

Equivalently we can define the backwards composition of the portfolio X ∈ by

    π̃_T^S(X)    = -S_TX,     π̃_t^S(X) = π_t^S(-π̃_t+1^S(X)e_1)  ∀ t < T.

Then ϕ̃^S, equivalently π̃^S, is time consistent.  
Additionally, the scalarization 

    ρ̃_t(X) := _S ∈^*π̃_t^S(X)  ∀ X ∈

is π-time consistent.


First, the time consistency of ϕ̃^S and π̃^S follow trivially by an induction argument.
Second, ρ̃ is π-time consistent immediately by its construction via {π̃^S  |   S ∈^*}.












 §.§ Relation to multiportfolio time consistency

In this section we will relate the concept of multiportfolio time consistency defined in <cit.> for set-valued risk measures to time consistency properties and representations for scalar risk measures ρ with a single eligible asset.  



Let A be a dynamic acceptance set.  Let the set-valued risk measure R (with any choice of eligible space M̃⊆^d), defined by

    R_t(X) := m ∈tM̃  |   X + m ∈ A_t  ∀ X ∈,

be multiportfolio time consistent, then ρ defined as in (<ref>) is π-time consistent.


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


We will prove the result with full eligible space M̃ = ^d, the general case follows by the same logic.  Let S ∈^*, t < s and X,Y ∈ 
then

    π_s^S(X) ≤π_s^S(Y)
       ⇒S_su   |   u ∈s,   X + u ∈ A_s^π⊇S_su   |   u ∈s,   Y + u ∈ A_s^π
       ⇒u ∈s  |   X + u ∈A_s + ∑_r = s^T Γ_r(S_r)
               ⊇u ∈s  |   Y + u ∈A_s + ∑_r = s^T Γ_r(S_r)
       ⇒u ∈s  |   X + u ∈ A_s + ∑_r = s^T Γ_r(S_r)⊇u ∈s  |   Y + u ∈ A_s + ∑_r = s^T Γ_r(S_r)
       ⇒u ∈s  |   X + u ∈ A_s + ∑_r = t^T Γ_r(S_r)⊇u ∈s  |   Y + u ∈ A_s + ∑_r = t^T Γ_r(S_r)
       ⇒⋃_X̂∈ X - ∑_r = t^T Γ_r(S_r) R_s(X̂) ⊇⋃_Ŷ∈ Y - ∑_r = t^T Γ_r(S_r) R_s(Ŷ)
       ⇒⋃_X̂∈ X - ∑_r = t^T Γ_r(S_r) R_t(X̂) ⊇⋃_Ŷ∈ Y - ∑_r = t^T Γ_r(S_r) R_t(Ŷ)
       ⇒u ∈t  |   X + u ∈ A_t + ∑_r = t^T Γ_r(S_r)⊇u ∈t  |   Y + u ∈ A_t + ∑_r = t^T Γ_r(S_r)
       ⇒π_t^S(X) ≤π_t^S(Y).




In fact, we will now use the above result relating multiportfolio time consistency to π-time consistency in order to present a recursive definition of ρ with respect to the decomposed form π^S.

Consider the setting of Lemma <ref> for normalized acceptance sets A (i.e., A_t = X ∈  |  ∈ R_t(X) for set-valued risk measure such that R_t(X) = R_t(X) + R_t() for every X ∈ for all times t).  Then
ρ satisfies the recursive relation

    ρ_t(X) = _S ∈_tπ_t^S([π_s^S()-π_s^S(X)]e_1)

for all portfolios X ∈ and times t < s.


This is an immediate result of Proposition <ref> and the recursive form from Theorem <ref> for non-normalized π_s^S (see Remark <ref>).  Proposition <ref> is used to guarantee that π_s^S()-π_s^S(X) ∈s for every S ∈_t.


Briefly, we will provide some interpretations and insights for the recursive relation (<ref>).  The recursive relation for ρ_t indicates that the risk measure can be related to the dynamic programming principle for (π^S_t)_t = 0^T over all frictionless prices S satisfying the no-arbitrage condition (<ref>).
To clarify what this does not mean, assume for simplicity ρ_t(X) = π_t^S^t_X(X) for some S^t_X ∈_t for every X ∈ and every time t, i.e., the essential supremum is attained at S^t_X.  Though it is tempting to assume this is sufficient to recover the usual recursion ρ_t(X) = ρ_t(ρ_s()-ρ_s(X)) equivalent to (<ref>), the fact that S^t_X depends on both the time t and portfolio X provides the recursion

    ρ_t(X) = π_t^S^t_X([π_s^S^t_X()-π_s^S^t_X(X)]e_1) ≠π_t^S^t_X([π_s^S^s_()-π_s^S^s_X(X)]e_1) = ρ_t(ρ_s()-ρ_s(X)).

More generally, the essential suprema over S ∈_t need not limit in the same direction for different times t and portfolios X. Since one is unable to merge the essential suprema at time t and s, the usual recursion ρ_t(X) = ρ_t(ρ_s()-ρ_s(X)) does not hold.
In fact, the recursive relation (<ref>) can be viewed as (ρ_t)_t = 0^T satisfying the dynamic programming principle for (π^S_t)_t = 0^T for a fixed scalarization S. Recall now, that (ρ_t)_t = 0^T itself is a fixed scalarization of (R_t)_t = 0^T (in the direction of the cash asset). In <cit.> we show that scalarized risk measures are in general only recursive with itself if the scalarizations change over time in a certain way (see also <cit.>). Since here the cash asset and thus the scalarization is fixed, (ρ_t)_t = 0^T is not recursive, but by Corollary <ref> satisfies the weaker recursive property (<ref>) with respect to a fixed scalarization of (π^S_t)_t = 0^T. 

The above recursive form for ρ simplifies when π_s^S() = 0 for every S ∈_t, e.g., if ρ is a coherent risk measure. The following proposition provides further results under the same setting as Corollary <ref> above.  In particular, we find a relation between the sets _t over time, and specifically we can deduce that ^* = _0.  



Consider the setting of Lemma <ref> for closed and normalized acceptance sets A.  Then
_t ⊆_s for any times t < s and ^* = _0. 


Assume S ∈_t.  Let X ∈ such that (X ∈Ã_t^π) = 0 (the existence of which is guaranteed by Proposition <ref>).  Then by multiportfolio time consistency we find that:

    0    = (X ∈Ã_t^π) = X ∈A_t + ∑_r = t^T Γ_r(S_r)
       = X ∈A_t ∩ M_s + A_s + ∑_r = t^T Γ_r(S_r)
       ≥X ∈A_s + ∑_r = s^T Γ_r(S_r) = (X ∈Ã_s^π).

Therefore the choice of X ∈ also satisfies the condition such that S ∈_s. In the above, we let  denote the operator which generates the random set associated with the closure of the set of random variables as with Ã_t^π for A_t^π defined in Proposition <ref>.  The existence of these random sets is guaranteed by <cit.>. Finally, by construction in Definition <ref>, ^* = ⋂_t ≥ 0_t = _0 by the monotonicity of these sets of frictionless price processes.





We will conclude our discussion of time consistency by relating the usual definition of time consistency of the scalar mappings ρ to multiportfolio time consistency of the set-valued risk measures R under the same single eligible asset space.

Consider the setting of Theorem <ref>.  R is multiportfolio time consistent if and only if ρ is ρ-time consistent (i.e., time consistent as defined in (<ref>)).


First, by Theorem <ref>, we know that R_t(X) = ρ_t(X) + t_+e_1 for all times t and portfolios X ∈.

Let R be multiportfolio time consistent. For any t < s and any X,Y ∈,

    ρ_s(X) ≤ρ_s(Y)    ⇒ R_s(X) ⊇ R_s(Y) ⇒ R_t(X) ⊇ R_t(Y) ⇒ρ_t(X)
    ≤ρ_t(Y).


Let ρ be time consistent. Additionally let t < s, X ∈, and ⊆,

    R_s(X) ⊆⋃_Y ∈ R_s(Y)    ⇒u ≥ρ_s(X) ⇒∃ Y ∈:  u ≥ρ_s(Y)
       ⇒∃ Y ∈: ρ_s(X) ≥ρ_s(Y) ⇒∃ Y ∈: ρ_t(X) ≥ρ_t(Y)
       ⇒∃ Y ∈:  R_t(X) ⊆ R_t(Y) ⇒ R_t(X) ⊆⋃_Y ∈ R_t(Y)



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

If ρ is ρ-time consistent, then it is also π-time consistent.


This follows directly from Lemmas <ref> and <ref>.














§ EXAMPLES



Due to Lemma <ref>, we immediately know that every multiportfolio time consistent set-valued risk measure generates a multivariate scalar risk measure ρ which is π-time consistent.  Thus we refer to <cit.> for some examples.  In this section we will focus on results on the superhedging risk measure with a single eligible asset, followed by an example for composed risk measures like the composed average value at risk.  For these examples we will demonstrate the concept of π-time consistency and show that the prior, strong, time consistency property (ρ_s(X) ≤ρ_s(Y) ⇒ρ_t(X) ≤ρ_t(Y)) is too strong of a concept.



 §.§ Superhedging under proportional transaction costs



Let us first consider a market with proportional transaction costs only. As introduced in Section <ref> and discussed in more details in <cit.>, the market model will be defined by a sequence of solvency cones K.  We will assume that the market satisfies the robust no-arbitrage property and Assumption <ref>.  Such a superhedging risk measure was studied at time 0 in the discrete time setting in, e.g., <cit.>.  By the definition of the superhedging risk measure, with trading at discrete times t ∈{0,1,...,T}, we can define the dual representation via

    ρ^SHP_t(X) = _(,S) ∈_t^SHP-S_TXt

where

    _t^SHP := (,S) ∈_t^e   |  ξ̅_t,s() S_s ∈1sK_s^+ ∀ s ≥ t.

In particular, for the construction of the decomposition π_t^S we care about

    _t^SHP(S) :=  EMM(S) := ∈ℳ^e   |   S  is a -mtg   if  S ∈
    ∅   else.

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

    A_t^ϕ   := Z ∈  |  Zt≥ 0  ∀∈_t^SHP(S)
       = Z ∈  |  Zt≥ 0  ∀∈ EMM(S).

Immediately this implies that A_t^ϕ is the acceptance set of the scalar superhedging risk measure, and therefore we recover that ϕ^S is time consistent (and thus so is π^S by Proposition <ref>) as is expected since the set-valued superhedging risk measure is multiportfolio time consistent in the case with full eligible space M̃ = ^d (see, e.g., <cit.>).
Thus, ρ^SHP is π-time consistent by its definition (Definition <ref>), respectively by Lemma <ref>.


Corollary <ref> provides a recursive relation for ρ^SHP, which we would like to note is similar to, but distinct from, that utilized in <cit.>. 


We conclude by demonstrating that the superhedging risk measure ρ^SHP, even though it is π-time consistent, is itself not ρ-time consistent.  This follows by Lemma <ref>. The underlying set-valued risk measure R is a closed and conditionally coherent risk measure with acceptance sets A_t = ∑_s = t^T ∞sK_s for all times t and M = ×{0}^d-1.  It is clear from the definition of the acceptance sets and the space of eligible portfolios that, in general, A_t ≠ A_t ∩ M_s + A_s.  Thus, by <cit.>, we can immediately conclude that R is not multiportfolio time consistent with this choice of eligible assets, and by Lemma <ref> ρ^SHP is not ρ-time consistent as defined in (<ref>).




 §.§ Superhedging under convex transaction costs




Continuing with the superhedging example, let us now consider a market with convex transaction costs, e.g., one that also includes price impact.  As discussed in <cit.>, we will model such a market with convex solvency regions C.  We will assume that this market model satisfies the no-scalable robust no-arbitrage condition and has recession cones satisfying Assumption <ref>.  

As in the case with proportional transaction costs above, we can introduce the dual representation, in the form of those provided in <cit.>, for the superhedging price ρ^SHP_t via

    ρ^SHP_t(X) = _(,S) ∈_t^e∑_s = t^T σ_t^s(sS_s) - S_TXt,

where the penalty function is defined as the support function of the solvency regions

    σ_t^s(Y) := _Z ∈sC_sYZt.

In particular, the penalty function is finite only if sS_s ∈1sC_s for all times s ∈{t,...,T}.



If we consider a conical market model K, then the penalty function is determined by 
    σ_t^s(Y) = 0    on {Y ∈K_s}
     -∞   on {Y ∉K_s}.
 Thus we immediately recover the superhedging price presented in Section <ref> as a special case.  Therefore we recover the formulation from Jouini, Kallal <cit.> as seen in (<ref>) in the static case as well.



As with the case under proportional transaction costs only, it immediately follows from Lemma <ref> and the multiportfolio time consistency of the set-valued superhedging risk measure with full eligible space M̃ = ^d (see, e.g., <cit.>) that ρ^SHP is π-time consistent.  

As in the case with proportional transaction costs only, though we find that the superhedging price is π-time consistent, it is not ρ-time consistent (with respect to definition (<ref>)).  This follows by Lemma <ref> as the superhedging risk measure is not multiportfolio time consistent when the space of eligible assets M is only the cash asset. 





 §.§ Composed risk measures



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

    R̃_t(X)    := R_t,t+1(R̃_t+1(X)) = ⋃_Z ∈R̃_t+1(X) R_t,t+1(-Z)   ∀ t ∈{0,1,...,T-1}
    R̃_T(X)    = R_T(X).

This was studied for, e.g., the set-valued average value at risk in <cit.>.  As with the superhedging example above, often we consider the full space of eligible assets M̃ = ^d as it is natural when considering trading in a market model <cit.> (as is studied in this paper) or with systemic risk measures <cit.>.  In that setting, by Lemmas <ref> and <ref>, the scalarization of the composed risk measure ρ̃_t(X) :={m ∈t  |   m e_1 ∈R̃_t(X)} for X ∈ is π-time consistent but not ρ-time consistent.


For illustrative purposes, let us briefly consider the composed average value at risk with full space of eligible assets M̃ = ^d.  We refer to <cit.> for details on the set-valued composed average value at risk.
In particular, consider the stepped average value at risk with levels λ^t ∈t[ϵ,1]^d for the stepped risk measure from time t to t+1 and some lower threshold ϵ > 0.  The scalarization ρ̃_t: →t of the composed average value at risk at time t, defined by (<ref>), can be given by the dual representation:

    ρ̃_t(X)    = _S ∈_t_∈^λ(S) -S_TXt
    ^λ(S)    = ∈ℳ^e   |   S_s,i≥λ_i^s ξ̅_s,s+1() S_s+1,i ∀ s = 0,1,...,T-1, i = 1,2,...,d.

Note that any probability measure ∈^λ(S) for some S ∈_t (and any fixed time t) will also be a dual variable for the univariate composed average value at risk with sequence of levels (λ_1^t)_t = 0^T-1 as described in <cit.>.  As discussed above, this multivariate risk measure is π-time consistent but not ρ-time consistent.




plain



