Acceptability Indices of Performance for Bounded Càdlàg Processes
aa r X i v : . [ q -f i n . M F ] N ov Acceptability Indices of Performance for Bounded C`adl`agProcesses
Christos E. Kountzakis ∗ , Damiano Rossello † November 7, 2019
Abstract
Indices of acceptability are well suited to frame the axiomatic features of many performance mea-sures, associated to terminal random cash flows. We extend this notion to classes of c`adl`ag processesmodelling cash flows over a fixed investment horizon. We provide a representation result for boundedpaths. We suggest an acceptability index based both on the static Average Value-at-Risk functionaland the running minimum of the paths, which eventually represents a RAROC-type model. Somenumerical comparisons clarify the magnitude of performance evaluation for processes.K
EY WORDS : Acceptability Indices; C`adl`ag Processes; Banach Lattice; Intra-Horizon Risk; Perfor-mance Measures for Processes; RAROCJEL C
LASSIFICATION
C02 · C63 · G17 · G21 · G23
A financial performance aims to evaluate the return characteristics of a given investment portfolio, and tofulfill the risk and asset allocation constraints provided by investors. The resulting measurement can beused to judge the quality of managerial skillfulness, since fund managers viewed as competitors shouldbe able to process those piece of information not reflected by market prices, then providing an actualvalue-added service. The balance between reward and risk is condensed into a performance measure asthe popular Sharpe ratio (SR), which can be used to rank investment portfolios according to these twocharacteristics. Other performance measures are alternative to SR, accounting for stylized facts aboutfinancial returns such as asymmetrical and fat-tailed distributions. See [4] for a textbook treatment ofseveral performance measures. ∗ Department of Mathematics, University of the Aegean, 83200 Samos Greece, email: [email protected] † Corresponding Author : Department of Economics and Business, University of Catania, 95129 Catania, Italy, email: [email protected] indices of ac-ceptability of nonnegative financial performance for a given terminal random cash flow. The basic ideais intimately linked to the well known analysis in Atzner et al. (1999) of coherent risk measure andtheir acceptance sets, and the subsequent analysis in Carr et al. (2001) of arbitrage pricing and accep-tance hedging: There is a continuum of degrees of acceptability of a position, varying with differentlevels of stressed scenarios supporting positive expectations of cumulative terminal cash flow. Thanks tothe duality between coherent risk measures and performance measures, a class of equivalent probabil-ity measures, or the corresponding set of their Radon-Nykodim derivatives, give the acceptability of atrade’s cash flow. In other words, an index of acceptability derived from a coherent risk measure mustbe proportional to the amount of stress tolerated and must yield nonnegative values exceeding floors,so that nonnegative expected cash flows are attained. This fits well the regulatory-capital requirementsand the pertaining economic-capital modelling used in practice, for example by banks undertaking exante improvements in business-performance tracking through the use of risk adjusted return on capital(RAROC), as the ratio of expected final return to the economic capital measured by a coherent risk mea-sure or by the Value-at-Risk (VaR). Not surprisingly, an acceptability index (AI) may be expressed justin ratio form, as the forerunner static SR.On the other hand, the industry of fund management claims the use of performance measures such asthe Calmar ratio (CR) designed to account for the risk associated to cash flow resulting from the wholeinvestment’s path over a fixed horizon. If this is the case, SR is redefined by set the reward measure in thenumerator equal to the expected terminal cash flow, and using the expected maximum drawdown over thehorizon as a risk measure in the denominator. The risk-adjusted return then takes into account the futureevolution of the market value of the position, not just the terminal one. Albeit this kind of performancemeasure is widespread among practitioners, it cannot be directly placed into the realm of static AIsstudied in [11]. Static risk measures does not embed the cash flow’s path experienced over time. On theother hand, dynamic risk measures have been developed to account for this. First, static risk measuresare turned into conditional ones to account for the information available at the risk assessment. Then, onsome filtered probability space risk assessment is updated as time elapses and new information arrives, sothat a sequence of conditional risk measures depicts a dynamical framework based on different notionsof time consistency , see [1] and the references therein for a detailed review. It is possible to turn thingsaround, and define a coherent monetary risk measure as in Cheridito et al. (2004), yielding a numericalevaluation rather than a sequence of conditional risk measures (random variables). For the special caseof a finite sequence of adapted cash flows see [18, Section 3.2].In the present paper we provide a framework for AIs of performance put in duality with coherent mone-tary risk measures for bounded c`adl`ag processes. To recover the information lost in the static setting, onerecords all possible stressed scenarios during the holding period of a financed position, which is repre-sented by a path rather than a random variable. However, we do not develop a dynamic setup to processinformation, rather we give a representation based on a static index acting on processes which preservesthe main features of an AI: Quasi-concavity, positive homogeneity, monotonicity and Fatou continuity.2ventually, this includes the one-period AI as a particular case. Furthermore, we propose an example ofAI for processes related to the one-time step RAROC. Our contribution is similar to that of Bielecki et al.(2014), or the more recent Bielecki et al. (2017), which define coherent risk and AIs for paths in a newlyproposed dynamical setting. These authors studied the duality between dynamic coherent risk measures ρ : { , . . . , T } × D × Ω → R and dynamic coherent acceptability indices ρ : { , . . . , T } × D × Ω → [ , ∞ ] , where { , . . . , T } is a set of dates and D is a set of (adapted) real-valued stochastic processes modellingcash flows. These authors impose two additional properties to α and ρ called independence of the past and dynamic consistency . Nevertheless, this framework cannot deal with performance measures suchas the aforementioned CR. Our contribution enable us to overcome this limitation, and to deal withpath-transformations like taking the maximum drawdown or the running minimum of a cash flow’s path.Eventually information about time resolution of an investment process (from the spatial viewpoint) is notlost. On the generalization of the AIs to the continuous-time setting see Biagini and Bion-Nadal (2015).The paper proceeds as follows. Section 2 introduces the essential toolkit for treating some classes ofc`adl`ag processes as models of total cumulative cash flows evolving within a finite horizon, and definesthe AIs of performance in this extended framework. The duality concerning such classes are briefly re-viewed, together with additional results on their lattice structure used in proving the main representationof this paper. Section 3 is devoted to the generalization of the static AI of [11] from the domain L ∞ to the collection of bounded c`adl`ag processes. The new AI is obtained in a straightforward manner, byproperly combining the contributions [11, 10]. The multi-period analogue of the static system of sup-porting kernels is depicted in terms of bi-variate processes reproducing the Radon-Nikodym derivativescorresponding to a static system of kernels. Section 4 deals with the analysis of second order stochasticdominance compatible with the extended AI. Section 5 studies its arbitrage and expectation consistency.Section 6 contains the main example of an AI for processes based on the one-period RAROC, as theratio of the expected terminal cumulative cash flow to the Average Value-at-Risk of the cash flow’s run-ning minimum. Section 6 provides numerical comparisons of simulated values in order to appreciate themagnitude of AIs and other performance measures. Section 8 contains some concluding remarks. In this paper we model the whole evolution of financial outcomes over a finite time-interval rather thanthe terminal cumulative cash flow typically handled in performance analysis. Here X = ( X t ) t ∈ [ , T ] is thestochastic process modeling the random cash flow resulting from dynamic trading over the investmenthorizon [ , T ] , where T > . We are given a filtered probability space ( Ω , F , ( F t ) t ∈ [ , T ] , P ) satisfying theusual conditions, i.e., the basis space ( Ω , F , P ) is complete, the filtration ( F t ) t ∈ [ , T ] is right-continuous,and the initial information F contains all the P -null events of F . Almost surely equal random variablesare identified as well as indistinguishable processes on the filtered space, X t ( ω ) = Y t ( ω ) for almost all ω ∈ Ω and all t > . Comparisons among processes are understood in the latter sense. For example, X > Y means that X t is greater than or equal to Y t , for all dates t and for almost all ω . As usual we set L p : = L p ( Ω , F , P ) and following [10, 13] we denote R the vector space of (the P -a.s. and for every t ∈ , T ] equivalence classes of) c`adl`ag processes that are adapted to the filtration. For the characterizationwe develop in Section 3, the model X of a (discounted) cash flow evolving within the horizon is thatof a bounded c`adl`ag process belonging to the stricter class R ∞ , i.e. X ∗ : = sup t ∈ [ , T ] | X t | ∈ L ∞ . This is aBanach space equipped with the norm k X k R ∞ : = k X ∗ k ∞ , where k·k ∞ is the usual norm on L ∞ . Definition 1.
A performance measure α : R ∞ → R is an AI for processes if it satisfies the followingproperties:(1) Acceptable cash flows at a level x > A x : = (cid:8) X ∈ R ∞ (cid:12)(cid:12) α ( X ) > x (cid:9) . In the current context, this is a family describing acceptability for any level x . The convexity ofany A x is equivalent to the quasi-concavity of α , namely α ( λ X + ( − λ ) Y ) > x for any λ ∈ [ , ] provided X , Y are such that α ( X ) > x and α ( Y ) > x . Taking x = min { α ( X ) , α ( Y ) } quasi-concavityimplies that a diversified position performs better than its components.(2) Acceptable cash flows are valued monotonically, X Y ⇒ α ( X ) α ( Y ) , thus α is an increasing map and Y is at least as acceptable as X . (3) The acceptance set A x is required to be a convex cone, because α is not meant to be an investmentcriterion but rather it measures to what extent moving away from marginal trades supporting therandom cash flow X results in a new investment direction based on alternative pricing kernels.Hence α ( λ X ) = α ( X ) , for λ > , i.e. scale invariance is required and the performance of an investment should not depend upon theinitial endowment. In other words, λ X is based on a trade in the same direction of X , and then ithas the same level of acceptance.(4) The acceptability functional is required to be upper Fatou-continuous for X ∈ R ∞ ,lim sup n → ∞ α ( X n ) α ( X ) , and for every bounded sequence ( X n ) n ∈ N ⊂ R ∞ of paths such that ( X n ) ∗ converges in probabilityto X ∗ , i.e. ( X n − X ) ∗ P → . This implies that α ( X ) > x provided that α ( X n ) > x for every n ∈ N and x > . The bounded sequence can be taken as k X n k R ∞ • M -norm, whenever X , Y > k sup { X , Y }k = max {k X k , k Y k} ; • L -norm, whenever X , Y > k X + Y k = max {k X k , k Y k} . Thus, a norm complete Riesz space equipped with an M -norm is an AM-space, while a norm completeRiesz space equipped with an L -norm is an AL-space. A useful result states that a Banach lattice is anAM-space (resp. an AL-space) if and only if its dual is an AL-space (resp. an AM-space), see [2, Chs8, 9] for more details. The class R p , for p ∈ [ , ∞ ] , generalizes R ∞ to those c`adl`ag processes such that X ∗ ∈ L p + . This is also a Banach space with norm k X k R p : = k X ∗ k p , see Appendix A for a brief review. Proposition 1. R p is a Banach lattice, for p ∈ [ , ∞ ] . Moreover, R ∞ is an AM-space with order unit.Proof. On R p let us consider the partial ordering X > Y ⇔ X t ( ω ) > Y t ( ω ) for any t ∈ [ , T ] , and for P -almost all ω ∈ Ω . Using this partial ordering, if X , Y ∈ R p and | X | > | Y | , which implies that | X t ( ω ) | > | Y t ( ω ) | for any t ∈ [ , T ] , and for P -almost all ω ∈ Ω , this also implies that the random variables X ∗ , Y ∗ ∈ L p + satisfy the inequality k X k R p = k X ∗ k p > k Y ∗ k p = k Y k R p . Hence, R p is a Banach lattice for p ∈ [ , ∞ ] . To show that R ∞ is an AM space with unit, it suffices toprove that such an order unit is the stochastic process = ( t ) t ∈ [ , T ] , where t ( ω ) = , P -a.s. In order toprove it, we have to show that R ∞ = ∪ ∞ n = [ − n , n ] , where [ − n , n ] denotes the following order intervalof R ∞ : [ − n , n ] = { X ∈ R ∞ : n > X > − n } , with respect to the partial ordering of R ∞ defined above.The inclusion | X t | > sup t ∈ [ , T ] | Y t | in L ∞ , and consequently ∪ ∞ n = [ − n , n ] ⊆ R ∞ is obvious. For theopposite inclusion, for any X ∈ R ∞ , we may define the stochastic process Y ∈ R ∞ , such that Y > X andmoreover Y t = ([ | X t | + ]) t .Another class of Banach lattices related to the geometry of c`adl`ag processes is A q , for q ∈ [ , ∞ ] , con-taining the bi-variate processes A : [ , T ] × Ω → R such that A = ( A pr t , A op t ) t ∈ [ , T ] has right-continuouscoordinates with finite variation, A pr being predictable with A pr0 = , while A op being optional and purelydiscontinuous, see Appendix A. We have Var ( A pr ) + Var ( A op ) ∈ L q , where Var ( · ) is he usual variation ofa process. The related positive cone A q + contains those bi-variate processes A ∈ A q such that A pr , A op > B q + : = (cid:8) A ∈ A q + (cid:12)(cid:12) h , A i > (cid:9) , where is the element X ∈ R p such that X ∗ ( ω ) = ( ω ) = , P -a.s. The partial ordering implied by A q + on A q is defined as C > D ⇐⇒ C − D ∈ A q + , and makes A q + a Banach lattices. More generally we have:5 roposition 2. A q is a Banach lattice. Moreover, A it is an AL-space.Proof. The partial ordering defined on A q , is the following: A > B ⇔ ˆ A > ˆ B , whereˆ A : = Var ( A pr ) + Var ( A op ) ∈ L q , ˆ B : = Var ( B pr ) + Var ( B op ) ∈ L q . Thus, if | A | > | B | this is equivalent to ˆ | A | > ˆ | B | . Thus, A q is a Banach lattice. This also implies k ˆ | A |k = k ˆ A k > k ˆ | B |k = k ˆ B k , which means that A is an AL-space, since L is an AL-space, as well.For conjugate exponents p = ∞ and q = , the duality between the spaces R ∞ and A plays an importantrole in our representation of AIs for processes, namely Theorems 1 and 2 in Section 3, as well as otherresults in Sections 3, 5 and 6. The dual pair h R ∞ , A i is based on the dual pairing h X , A i defined on R ∞ × A , see Appendix A. In our main result (Theorem 1) we replace the infimum over classical non-negativeexpectations with respect to equivalent probability measures (viz. their Radon-Nikodym derivatives),with positive increasing dual processes with unit expected variation. Thus, for every x ∈ R + , the x -increasing family ( D x ) x ∈ R + of [11, Theorem 1] is now replaced by an x -increasing family ( Q x σ ) x ∈ R + , where each Q x σ is a subset of the class D σ defined in [10], containing the bi-variate processes A ∈ A that are in addition nonnegative, increasing and such that E [ Var ( A pr ) + Var ( A op )] = , see also AppendixA. We give the analogue of [11, Theorem 1], to characterize an AI having a numerical value x ∈ R + suchthat the bounded c`adl`ag path X attains a positive bilinear form h X , A i (which is the analogue of theexpectation in the one-period case), under each bi-variate process A from the subset Q σ ⊂ D σ (whichis the analogue of the Radon-Nykodim derivative of the absolute continuous probability measure givingthe acceptability in the one-period case) corresponding to the level x . There is a one-parameter class ofsuch sets.
Theorem 1.
A map α : R ∞ → [ , ∞ ] is an AI for processes if and only if there exists a family of x-increasing family ( Q x σ ) x ∈ R + such that the representation (1) α ( X ) = sup (cid:26) x ∈ R + (cid:12)(cid:12)(cid:12)(cid:12) inf A ∈ Q x σ h X , A i > (cid:27) holds, with inf ∅ = ∞ and sup ∅ = . Proof. ( ⇐ ) We claim that α defined by (1) satisfies the four properties of an AI. First, we check property(1) of Definition 1 that α values only acceptable cash flows X ∈ R ∞ evolving during the horizon [ , T ] which belong to a convex level set A x for any x ∈ R + . Indeed, assuming that X and in addition Y ∈ R ∞ α which is > x , then choosing y < x for any bi-variate process A ∈ Q y σ we have avalue of the linear functional h· , A i , corresponding to each cash flow, which must be > . Taking aconvex combination for λ ∈ [ , ] we then have h λ X + ( − λ ) Y , A i > , and for every A in the biggestclass Q x σ we thus have inf A h λ X + ( − λ ) Y , A i > x ∈ R + suchthat α ( λ X + ( − λ ) Y ) > x which proves the convexity of the level set A x (or equivalently the quasi-concavity of the index α ). For the monotonicity, given two elements of R ∞ such that X Y , the stochasticintegral together with the expectation operator defining the bilinear form used in (1) are monotone, then h Y , A i > h X , A i > t ∈ [ , T ] | X nt | P → sup t ∈ [ , T ] | X t | for a bounded sequence ( X n ) n ∈ N of elements X n ∈ R ∞ and some X ∈ R ∞ . This implies (cid:18) Z ( , T ] X nt − d A pr t + Z [ , T ] X nt d A op t (cid:19) P → (cid:18) Z ( , T ] X t − d A pr t + Z [ , T ] X t d A op t (cid:19) , thus by the Lebesgue’s Dominated Convergence theorem we havelim n → ∞ h X n , A i = h X , A i > lim sup n → ∞ inf A ∈ Q y σ h X , A i > A ∈ Q y σ . Now, for any such y < x , any A ∈ Q y σ and any n ∈ N we have α ( X n ) > x by constructionso that the previous implies α ( X ) > x too, which is the equivalent formulation of the Fatou property forthe AI.Before completing the proof, we observe that representation (1) of Theorem 1 is equivalently given by(2) α ( X ) = sup (cid:8) x ∈ R + (cid:12)(cid:12) ρ x ( X ) (cid:9) , as pointed out in [11]. Indeed, each functional on R ∞ defined by ρ x ( X ) : = − inf A ∈ Q x σ h X , A i , x ∈ R + is by [10, Corollary 3.5] a coherent risk measure for processes. For x y , passing from Q x σ to the biggerset Q y σ the value of ρ x ( X ) will increase to ρ y ( X ) , for any X ∈ R ∞ . Then, the supremum in (1) of Theorem1 will increase too and obviously the equivalent representation given in (2) holds true. Conversely, for afamily ( ρ x ( X )) x ∈ R + of coherent risk measures for processes X ∈ R ∞ , which is increasing in x as a map x ρ x ( X ) for a fixed X , any set Q x σ corresponding to a risk measure ρ x ( X ) in the family must be biggeranytime x increases, due to the representation (2). Remark 1.
The acceptability set A x introduced in Definition 1 of Section 2 is equivalently given by A x : = (cid:8) X ∈ R ∞ (cid:12)(cid:12) ρ x ( X ) (cid:9) . Thus, for every x ∈ R + we have a whole family which is clearly decreasing in x . As a consequence thenumerical value of an AI for processes can be recast as α ( X ) = sup (cid:8) x ∈ R + (cid:12)(cid:12) X ∈ A x (cid:9) . There are several levels x at which the performance of a trade can be measured by valuing its riskinessin an acceptable way.
7n order to prove the ‘if part’ we need the following characterizations. Based on the AM-AL dualitybetween R ∞ and A , we in addition see that for any x ∈ R + the coherent risk measure for processes ρ x ( X ) = inf { m ∈ R | m · + X ∈ A x } , for every X ∈ R ∞ , has the following dual representation : − ρ x ( X ) = inf π ∈ A x π ( X ) , where A x = { f ∈ A : f ( X ) > , each X ∈ A x } , is the polar set of A x in A . Finally, recall that a subsetof a vector space is called a wedge if it is convex and it has the property that for any x lying in the setwe also have that λ · x belong to the same set, for every λ ∈ R + . Putting all things together, we have thatproving ( ⇒ ) of Theorem 1 amounts to prove the following: Theorem 2.
For any AI α : R ∞ → [ , ∞ ] , with the property that every level set A x of α is a wedge, thereexists an increasing family ( Q x σ ) x ∈ R + of functional sets lying in A such that x y implies Q x σ ⊂ Q y σ and α ( X ) = sup (cid:26) x ∈ R + (cid:12)(cid:12)(cid:12)(cid:12) inf π ∈ Q x σ π ( X ) > (cid:27) holds.Proof. The level sets of α are A x = { X ∈ R ∞ | α ( X ) > x } . For these sets, A y ⊂ A x holds, if y x . Forthe equivalent polar sets Q x σ = A x lying in A , if x z this implies Q x σ ⊂ Q z σ . Thus, any X ∈ R ∞ liesin some A x . This implies that ρ x ( X )
0, hence − ρ x ( X ) >
0. From the equivalent dual representationof the coherent risk measure ρ x , then α ( X ) = sup { x ∈ R + | − ρ x ( X ) } and we are done.The family ( Q x σ ) x ∈ R + (viz. system of supporting kernels in [11]) can be characterized as(3) Q x σ = { A ∈ Q σ |h X , A i > , ∀ X ∈ R ∞ , α ( X ) > x > } . Then, we have the following maximality property:
Lemma 1.
For any AI α , there exists a family ( Q x σ ) x ∈ R + supporting the representation (1) and definedby (3), such that if ( E x σ ) x ∈ R + is a different x-increasing family satisfying (1), then it holds E x σ ⊂ Q x σ forany x ∈ R + . Proof.
As in the proof of [11, Proposition 1] we use a squeezing argument and show that equation (1)can be split in two opposite inequalities. Let us suppose that α ( X ) , defined by (1) and supported by some x -increasing family ( E x σ ) x ∈ R + , is strictly greater than sup { x ∈ R + | inf A ∈ Q x σ h X , A i > } , for any process X ∈ R ∞ . Then α ( X ) > y > sup (cid:26) x ∈ R + (cid:12)(cid:12)(cid:12)(cid:12) inf A ∈ Q x σ h X , A i > (cid:27) , y ∈ R + . But this implies the existence of A ∈ Q y σ which makes negative the bilinear form insidethe supremum, contradicting the definition of Q y σ in (3). To show the reverse inequality, let us supposethat E x σ ⊃ Q x σ . Then, we can find A ∈ E x σ which again makes negative the bilinear form and at the sametime makes α ( X ) > x ∈ R + , contradicting the definition (1) of α . Then, α ( X ) > sup (cid:26) x ∈ R + (cid:12)(cid:12)(cid:12)(cid:12) inf A ∈ Q x σ h X , A i > (cid:27) and ( Q x σ ) x ∈ R + is a maximal family.We state a Lemma which will be useful in the identification of typical AIs for processes, provided that afamily ( Q x σ ) x ∈ R + is meant to supports α as given by Lemma 1. Recall that the space Q σ ⊂ A inheritsthe norm k · k A and then it is a Banach space. Lemma 2.
Define an AI α by (1). Let ( Q x σ ) x ∈ R + be a family of convex k · k A -closed subsets of Q σ thatare minimal with respect to intersection, i.e. Q x σ : = ∩ y > x Q y σ for any x ∈ R + . Then ( Q x σ ) x ∈ R + supports α in the representation (1).Proof. Let ( E x σ ) x ∈ R + be the x -increasing family supporting α . For some x ∈ R + , take a nonempty k · k A -closed and convex set Q x σ ⊂ E x σ . This enable us to find some y > x and B ∈ E x σ such that B / ∈ Q y σ . Thus,by the Hahn-Banach Separation Theorem we further find X ∈ R ∞ such that h X , B i < < inf A ∈ Q x σ h X , A i , but this implies α ( X ) > y > x > B ∈ Q y σ and by the maximality stated in Lemma 1 we are done.It is worth noting that by choosing T = , R ∞ = L ∞ ( Ω , F , P ) and one gets the static AI as in [11]. The consistence of a performance measure for processes α with the Second order Stochastic Dominance(SSD), hardly depends upon the definition of SSD itself on the spaces R ∞ . To get the equivalent notionin this space of stochastic processes we require (as in the static setting) that if a trade with random cashflow X ∈ R ∞ has given a greater ‘utility’ than another Y ∈ R ∞ , then it should have a higher performancetoo, α ( X ) > α ( Y ) . Whence, we need to adapt the notion of expected utility in order to characterize thispreference relation via SSD.Given a couple of stochastic cash flows X , Y ∈ R ∞ , we recall that X ∗ = sup t ∈ [ , T ] | X t | = X ∗ + + X ∗− isthe corresponding random variable in L ∞ + and similarly for Y . Therefore, the binary relation defined on R ∞ × R ∞ by X < SSD Y ⇐⇒ X ∗ < SSD Y ∗ ,
9s the analogue of the SSD in the one-time step setting, where what matter are the terminal cash flows.Here instead, we consider the path-dependency using the running maximum of the reflected (at the origin)processes X ∗ and Y ∗ . As a consequence, we rewrite the above SSD relation as X < SSD Y ⇐⇒ Z z F X ∗ ( s ) d s Z z F Y ∗ ( s ) d s , for every z ∈ ( , ∞ ) . For the quote on terminal wealth, see [20, p. 671]. In the same paper, we find the characterization of SSDin terms of expected utility. Hence, we define the expected utility on R ∞ in the following sense: Definition 2.
For a random cash flow evolving in time X ∈ R ∞ , the version of expected utility forprocesses over the horizon [ , T ] is given by E ( U ( X )) : = E ( U ( X ∗ )) , where X ∗ ∈ L ∞ + , and U : R → R ∪ { ∞ } is some concave, non-decreasing function. In Section 3 we introduced the family of coherent risk measures for processes ρ x ( X ) = inf { m ∈ R | m · + X ∈ A x } , for any X ∈ R ∞ , with the dual representation ρ x ( X ) = sup π ∈ A x π ( − X ) , where A x = { f ∈ A | f ( X ) > , for any X ∈ A x } is the polar set of A x in A . If we suppose A , B areacceptability subsets of R ∞ , such that A ⊂ B , then B ⊂ A holds for the equivalent polar sets in A . The above dual representation equals − ρ x ( X ) = inf π ∈ A x π ( X ) . Moreover, X ∈ A x implies − ρ x ( X ) > ( ρ x ) x ∈ R + of coherent risk measures for processes which is monotone withrespect to R ∞ + , where R ∞ + contains those bounded c`adl`ag paths X > , this entails R ∞ + = A and conse-quently if x > A x ⊂ A . For such AIs arbitrage consistency holds, because α ( X ) = sup (cid:26) x ∈ R + (cid:12)(cid:12)(cid:12)(cid:12) inf π ∈ A x π ( X ) > (cid:27) = sup { x ∈ R + | − ρ x ( X ) > } = ∞ . Hence, we proved the following:
Theorem 3.
The AI α : R ∞ → [ , ∞ ] defined through a family of monotone coherent risk measures forprocesses in R ∞ + and the order unit of R ∞ is Arbitrage Consistent with respect to R ∞ + . Now we come to the extension of the expectation consistency stated in [11] for the static case. We againtransfer the properties of AIs to the dual system (cid:10) R ∞ , A (cid:11) from the dual system (cid:10) L ∞ , L (cid:11) . efinition 3. An AI α : R ∞ → [ , ∞ ] is called expectation consistent, if and only if w ( X ) > , then α ( X ) >
0. The functional w is the one which corresponds to ∈ L . Proposition 3.
An AI for processes α , defined on R ∞ is expectation consistent if the level set of zero is R ∞ + . Proof.
If the above condition holds, since A x ⊂ A , for every x ∈ R + , where as usual A x = { X ∈ R ∞ | α ( X ) > x } , we notice that if w ( X ) >
0, this implies α ( X ) > . Assume that X ∈ R ∞ describes the continuous-time cumulative random return over a finite horizon, andwithout loss of generality that the interest rates are zero (avoiding to treat excess returns). We propose tocharacterize the following AI:(4) α ( X ) : = E ( X T ) ρ ( X ) , where the denominator represents a coherent risk measure for adapted bounded c`adl`ag processes, withthe convention α ( X ) = ∞ whenever ρ ( X ) . The above measure is reminiscent of the SR given by E ( X T ) sd ( X T ) , where the denominator is the usual standard deviation of the terminal total cumulative return; thenumerator measures the expected reward of the underlying investment just at the horizon. Thus, equation(4) is a RAROC-type of performance measure provided that the expectation in the numerator is positive.To see why α ( X ) is an AI for processes, we need to find the bi-variate process picked from Q x σ which isconsistent with the representation (1). The right choice is the convex combination:˜ A : = + x B + x + x A , x ∈ R + , A , B ∈ D σ where B = ( B pr t , B op t ) t ∈ [ , T ] : = ( , I { u t } ) t ∈ [ , T ] . In fact, we have the chain of equivalences for x > α ( X ) > x ⇐⇒ E ( X T ) ρ ( X ) > x ⇐⇒ E ( X T ) > − x · inf A ∈ D σ h X , A i⇐⇒ + x E ( X T ) + x + x inf A ∈ D σ h X , A i > ⇐⇒ inf A ∈ D σ (cid:20) + x E ( X T ) + x + x h X , A i (cid:21) > ⇐⇒ inf A ∈ D σ (cid:28) X , + x · B + x + x · A (cid:29) > ⇐⇒ inf ˜ A ∈ Q x σ h X , ˜ A i > .
11e use Lemma 2 for the closeness feature of the sets Q x σ supporting this RAROC-type measure. In theconvex combination defining ˜ A , the first term B projects the whole random return X onto the terminaldate T through the expectation; the second term A pertains to the representation of the coherent measurefor processes ρ ( X ) . Note that E ( X T ) < ∞ by the assumption X ∈ R ∞ . Remark 2.
The functional ρ ( X ) : = − inf A ∈ D σ h X − E ( X T ) , A i is obviously a monetary coherent riskmeasure for X ∈ R ∞ . By Remark 1 in Section 3 it induces a reward-risk separation for acceptability atlevel x ∈ R + , because X ∈ A x implies the non-negativity of the corresponding ρ x ( X ) and thus α ( X ) = E ( X T ) ρ ( X ) > x ⇐⇒ E ( X T ) + inf A ∈ D σ h X − E ( X T ) , A i > . If one chooses the coherent risk measure for bounded paths(5) ρ ( X ) = AV@R γ ( inf t ∈ [ , T ] X t ) , then the above AI can be made operational. Here AV@R γ is the Average Value-at-Risk at the level γ ∈ ( , ] . In fact, a risk measure ρ ( X ) for bounded processes can be viewed as R ∞ θ −−→ L ∞ ˜ ρ −−→ R + , the composition of a path-transformation with a one-period risk measure ˜ ρ applied to the resulting ran-dom variable θ ( X ) . Obviously, ρ ( X ) would be a coherent monetary risk measure for bounded processesif and only if ˜ ρ is a coherent monetary risk measure for single-period cumulative returns, and θ trans-forms the paths of X in such a way the properties studied in [10] are preserved. Equivalently, the com-bined effect of a path-transformation and a static risk measurement is(6) ρ ( X ) : = ˜ ρ ( θ ( X )) = − inf Z ∈ L + , E ( Z )= E [ θ ( X ) · Z ] . In the current setting, θ is the running minimum of X . It is important to note that other types of path-transformations can be taken into account, see [10, Examples 5.2, 5.5] where θ ( X ) = T R T X t d t and itmight be viewed as the continuous-time arithmetic average price of the underlying of an Asian option.Anyway, the acceptability of the proposed ρ ( X ) stems from the coherence of the static ˜ ρ = AV@Rtogether with the monotonicity of the running minimum (properties (1) and (2) are not destroyed). More-over, the law invariance of AV@R implies that of α . Clearly, it is also expectation consistent but neverarbitrage consistent. Recall thatAV@R γ ( inf t ∈ [ , T ] X t ) = γ Z γ VaR s ( inf t ∈ [ , T ] X t ) d s , where as usual the VaR is defined as the negative of the s -quantile of the running minimum’s distribution, − inf (cid:8) x ∈ R | P ( inf t ∈ [ , T ] X t x ) > s (cid:9) , and γ ∈ ( , ] . The widespread CR(7) CR ( X ) : = E ( X T ) E ( sup t ∈ [ , T ] D t ) , for X ∈ R p , and p ∈ [ , ∞ ] ,
12s a classical performance measure depending on the whole investment’s path, but fails to be acceptableas we see below. Here D = ( D t ) t ∈ [ , T ] is the drawdown process over [ , T ] of the random return X , definedas D t : = sup u ∈ [ , t ] X u − X t , i.e., it is the drop of X from its running maximum, while the denominator inequation (7) is the maximum drawdown, i.e. the greatest drop of X from its running maximum over thewhole horizon (the supremum of X reflected at its running supremum). From now on we assume positiveperformance indices for processes, whenever E ( X T ) > θ ( X ) = sup t ∈ [ , T ] D t which is not monotone. One can try to replace the expectation E = ˜ ρ (as the one-period risk functional)with the tail-conditional expectation (i.e. the AV@R defined on the right tail of the distribution of themaximum drawdown) which is a one-period coherent risk measure, but the lack of monotonicity of theaforementioned θ ( X ) destroys acceptability (only convexity is preserved). Remark 3.
For two bounded c`adl`ag cash flows X , Y ∈ R ∞ such that X law = Y , the concept of law invariantAI developed in [11] in the static case can be translated int the current setting by limiting ourselves to thecase of RAROC-type AIs (4). After the corresponding path-transformation is made, the sameness in lawof any couple of bounded c`adl`ag cash flows then translates to θ ( X ) law = θ ( Y ) , thus their transformed pathsentail random variables sharing the same probability distribution. We can appeal to [11, Theorem 5].Firstly, the coherent risk measure for processes ρ ( X ) in the representation of α ( X ) can be based on the weighted VaR , i.e. the spectral representation R ( , ] AV@R γ ( θ ( X )) µ ( d γ ) with a Borel probability mea-sure µ on the unit interval. In fact, this in turn is equivalent to the representation of the path-dependentrisk measure (6) through a concave distortion, ˜ ρ ( θ ( X )) = − R R y d ( Ψ x ( F θ ( X ) ( y ))) . Then, for every x ∈ R + one defines AI as in the static case by specifying the concave distortion Ψ x ( u ) : = min { γ − u , } with γ = + x and µ = δ + x ; the choice of Z additionally needs E (( Z − u ) + ) Φ x ( u ) for every u ∈ R + suchthat Φ is the convex conjugate of the concave distortion. As a by product, α ( X ) is also consistent withSSD as developed in Section 4. To give more insight on the behavior of the performance indices discussed in the previous Section, wepresent here a simulation exercise based on the following ingredients: • Generation of two L´evy processes X , Y to describe possible patterns of continuously compoundedreturns over the horizon [ , T ] , with T = • Determination of six empirical distribution functions F X T n , F Xn , F DXn , F Y T n , F Yn , F DYn for the final re-turns X , Y at T and the corresponding running minimums and maximum drawdowns within [ , T ] ; • Estimation of the sample counterparts CR n , α n , γ of the CR and the ratio (5) presented in Section 6,under the two alternative distributional assumptions;13 Comparison of the numerical values deduced from the estimated statistics.We admittedly carry on the numerical simulation in the unbounded case, albeit as usual the discretizationscheme gives bounded sample paths. Thus, modulo any asymptotic consideration we use the results fromthe simulated paths as an approximation for the bounded case. Due to our main interest in the final stepof the above simulation recipe, we choose two parsimonious models of L´evy processes:1. A Brownian motion X with constant drift µ > σ > Y with the same diffusion part, as suggested by Kou [16].Other models are available for simulating different L´evy processes such as stable, variance gamma, hy-perbolic, stable etc. We rest on the simpler which do not require either the use of special functions norany series representation. Moreover, we keep to the minimum the difficulty of have no explicit L´evymeasure associated to the processes. Recall that the Kou’s model for Y contains four additional param-eters: the annual intensity λ of a homogeneous Poisson process counting the number of jumps for thenon-diffusion part; η , η > < p < X and Y over a equally-spaced discretization of the horizon by a grid of 1000 time points, through the usual Euler scheme for thecorresponding stochastic differential equations with annual µ = . , annual σ = . , annual λ =
10 and η = . , η = . . This setting entails a mean number of 10 jumps per year with average size 2 .
2% andjump volatility 4 .
47% as suggested by [16]. The simulation is straightforward because of the indepen-dence between the diffusion and the jump part. We compute the final values, the running minimums andthe maximum drawdowns of the simulated paths, then we repeat this last step 1000 times as well to get theestimated empirical distribution functions ˆ F X T n , ˆ F Xn , ˆ F DXn , ˆ F Y T n , ˆ F Yn , ˆ F DYn , with n = . Analyzing TableTable 1: Summary Statistics of Simulated Valuesˆ F mXn ˆ F DXn ˆ F mYn ˆ F DYn ˆ F X T n ˆ F Y T n Skewness − . − . − . − − . − . − . − . − . quasi symmetric distributionis that of the simulated final return under the Brownian’s law. Indeed, even when this is the assumptionone has non-symmetric distributions concerning the running minimum, the maximum drawdown and the14igure 1: Empirical density of the Brownian-return’s running minimum. −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0
Figure 2:
Empirical density of the jump-diffusion-return’s running minimum. −0.25 −0.2 −0.15 −0.1 −0.05 0
Figure 3:
Empirical density of the Brownian-return’s maximum drawdown.
Empirical density of the jump-diffusion-return’s maximum drawdown.
Figure 5:
Empirical density of the final Brownian return. −0.2 −0.1 0 0.1 0.2 0.3 0.4
Figure 6:
Empirical density of the final jump-diffusion return. −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 X is assumed to be a benchmark of normal market values,while the jump-diffusion model Y allows for leptokurtic and asymmetric returns. In presence of shocks,price changes due to good or bad news result in return movements as either overreaction or underreactionof market, according to fat tails and high peak of the jumps distribution.Considering the sample ( X iT ) i = ,..., n from the simulation of terminal paths, the sample ( DX i ) i = ,..., n fromthe simulation of the maximum drawdown and the sample ( X i ) i = ,..., n from the simulation of the runningminimum, we compute the following estimators in the Brownian case: • CR n = / n ∑ ni = X iT / n ∑ ni = DX i ; • α n , γ = / n ∑ ni = X iT / k ∑ ni = k X ( k ) , with k : = [ n γ ] the greatest integer less than or equal to n γ and ( X ( i ) ) i = ,..., n beingthe corresponding ordered sample; we set γ = .
01 and γ = . • SR Bn = / n ∑ ni = X iT (cid:16) n ∑ ni = X i − ( n ∑ ni = X i ) (cid:17) / ; • SR Jn = / n ∑ ni = X iT (cid:16) n ∑ ni = DX i − ( n ∑ ni = DX i ) (cid:17) / . The last two estimators refer to SR where the standard deviation is based on the running minimum andthe maximum drawdown, respectively. The estimators in the case of the jump-diffusion Y are obtainedin the same way. All the proposed performance indices provide a reward measurement in terms of fi-nal expected return. CR and its relatives (viz. the last tow rows of Table 2) entail overestimation ofTable 2: Simulated Values of Performance IndicesBrownian motion Jump-diffusionCalmar ratio ˆCR n γ = .
05 ˆ α n , γ γ = .
01 ˆ α n , γ B E ( final return ) sd ( running minimum ) J E ( final return ) sd ( maximum drawdown ) α γ have smaller values. This maybe suggests a more prudentperformance evaluation by taking into account adverse downside scenarios under the proper distribu-tional assumption towards a forecast of phenomenon like margin calls, rebalancing of trading positions,counter-party risks, fund redemption. In this perspective, the index based on (4) with coherent monetary17isk measure for processes being the one proposed in Section 6, gives a sensible trade-off between ac-ceptability as a theoretical feature and the ability to capture extremes changes in the return profile duringthe whole holding period. Furthermore, the resulting measurement is compatible with the heightenedneed for performance-tracking tools that embed economic cost of risk, and the consequential use of theRAROC by financial institutions. Remark 4.
When random cash flows are asymmetric and fat-tailed, performance evaluation using SR-type indices is questionable. Investors’ preference for certain portfolio compositions heavily relies onmore than the first two moments of the cash flow’s distribution, and clear enough they could be wealth-seeking and risk-adverse but prefer portfolios with higher volatility. In these situations, the
AV@R be-haves better as a static risk measure and assuming a tangential portfolio (in the sense of minimum
AV@R for a fixed tail probability γ ) it constitute a sensible substitute for the volatility given by the stan-dard deviation or the alike, when portfolio optimization comes into play and asset allocation or rankingof investment funds concerns can be tackled more efficiently. Observe in addition how all the estimatorsabove are biased for finite samples but have asymptotic efficiency as n grows more and more. Intra-horizon risk measurement has recently gained increasing consideration among academics and prac-titioners. For example, VaR may be improved in the time dimension by focusing on the return’s distri-bution within the whole investment horizon. On the fund management side, performance analysis withdrawdown is well developed and the evaluation and even the ranking of investment portfolios by mea-sures such as CR is popular among fund managers. These indices are based on the risk-adjustment ofreturns by recording all the information about the evolution of processes modelling profit and losseswithin a fixed horizon. Motivated by these well documented facts and in addition by the growing useof the RAROC as a tool of banks’ backward-looking manner at business performance-tracking and riskmanagement, we propose a performance measurement for processes. Therefore, we look on one handat the approach of Cherny & Madan (2009) to static performance evaluation driven by the concept ofAIs, and on the other hand to the work of Cheridito et al. (2004) which generalizes static coherent riskmeasures to monetary risk measures for processes. Our main contribution is to extend the representationof an AI as applied not to random variables giving terminal cash flow, but instead to stochastic processesmodelling the evolution of cash flows over a finite horizon. The domain of an AI is now the class ofbounded c`adl`ag processes, with new representation results. Our contribution does not overlap that ofBielecki et al. (2014), since they treat dynamic coherent AIs with a focus on time-consistency, whilewe develop static AIs for processes. Notwithstanding, our framework can embody information about thewhole investment process even if sequential conditioning is ruled out. We propose an AI expressed asthe ratio of expected terminal cash flow to a coherent monetary risk measure for processes. Eventually,we compare different numerical values of this acceptability ratio and other performance measures, toappreciate the embedding of information about stressed scenarios concerning the whole horizon. Ourratio could be a sensible compromise between acceptability (as a desirable aggregate property) and de-18endency on the whole cash flow’s path (as the quest of practitioners). This is also compatible with thewidespread use of RAROC and fixes the lack of acceptability of CR. The research agenda on this topicwill include the generalization of our AI to the unbounded case (e.g. the space R ), as well as a morecomprehensive definition of law invariance of AIs for processes. Also a study of the asymptotic behaviorof AIs when the horizon becomes infinite is desirable. A Appendix: Duality Relations for Processes
The main results of Section 3 are based on the generalization of acceptability from spaces of randomvariables to spaces of stochastic processes, partly introduced in Section 2. To keep the present paper asmuch self-contained as possible, we list in this Appendix some facts about these spaces of processes andthe corresponding duality relations following closely [10]. For p ∈ [ , ∞ ] the collection R p : = X : [ , T ] × Ω → R X c`adl`ag ( F t ) -adapted k X k R p < ∞ is a Banach space. Recall that increasing processes A : [ , T ] × Ω → R (i.e. adapted, with positive right-continuous and increasing paths) induces a measure d A t ( ω ) . In case A has right-continuous paths withfinite variation, its unique decomposition A = A + − A − into two right-continuous increasing processesinduces P -a.s. positive measures on [ , T ] with disjoint support. The total variation of such process isthe random variable Var ( A ) : = A + T + A − T . Moreover, if A is optional (i.e. a measurable on [ , T ] × Ω equipped with the σ -algebra generated by the adapted c`adl`ag processes) then A + , A − are optional. When A is predictable (i.e. measurable on [ , T ] × Ω equipped with the σ -algebra generated by the adaptedcontinuous processes) then A + , A − are predictable. Thus, for q ∈ [ , ∞ ] we have the collection A q : = A : [ , T ] × Ω → R A = ( A pr , A op ) right-continuous, finite variation A pr predictable, A pr0 = A op optional, purely discontinuousVar ( A pr ) + Var ( A op ) ∈ L q This collection equipped with the norm k A k A q : = k Var ( A pr ) + Var ( A op ) k q is a Banach space. The subset A q + containing those A ∈ A q with the predictable and optional parts being non-negative and increasing.The bilinear form h X , A i : = E (cid:20) Z ( , T ] X t − d A pr t + Z [ , T ] X t d A op t (cid:21) defined on R p × A q for p , q ∈ [ , ∞ ] such that p − + q − = , is also continuous and put in duality thespaces R p and A q . Indeed, it is well known that |h X , A i| k X k R p k A k A q . Appendix: Coherent Monetary Risk Measures for Processes
The apparatus introduced in Appendix A, is employed for extending the structure theorem for coherentrisk measures ρ : L ∞ → R : Given the σ ( L ∞ , L ) -closed acceptance set C = { X ∈ L ∞ | ρ ( X ) } for adiscounted terminal cash flow X , then ρ ( X ) = − inf Q ∈ D E Q ( X ) = − inf Z ∈ L + , E ( Z )= E ( X · Z ) , for a certain set D of probability measures absolutely continuous with respect to P , with correspondingRadon-Nikodym derivatives Z = d Q d P . The typical proof of this result uses C to support the representationitself, see [12] for a thorough treatment of static (also dynamic) risk measures based of the correspondingcoherent monetary utility functional φ = − ρ . Now, static coherent risk measures are in duality with staticAIs since α ( X ) = sup { x ∈ R + | ρ x ( X ) } , where ρ x is an indexed family of coherent risk measures whose representation is supported by an x -increasing family ( D x ) x ∈ R + of absolutely continuous probability measures, together with the correspond-ing Radon-Nikodym derivatives and acceptance sets. In the present paper we do a similar constructionby indexing the set D σ : = (cid:8) A ∈ A + (cid:12)(cid:12) k A k A = (cid:9) , and then working with the bilinear form h X , A i rather than the classical expectation E ( X · Z ) . Obviously,in this extended framework we have that ρ ( X ) = − inf A ∈ D σ h X , A i , X ∈ R ∞ , with corresponding σ ( R ∞ , A ) -closed acceptance set C = { X ∈ R ∞ | ρ ( X ) } , see [10, Corollay 3.5].In our paper we clearly attach a numerical acceptability level x ∈ R + to the supporting set D σ to entailthe duality with AIs for bounded c`adl`ag processes. Finally, a few facts to note. First, the space R ∞ isinvariant with respect to the probability measure P or its equivalents. Second, for p , q ∈ [ , ∞ ] such that p − + q − = A q can be identified with the topological dual ( R p ) ∗ of the space R p , while A ⊂ ( R ∞ ) ∗ . Thus [10, Theorem 3.3, Corollary 3.5] provide those coherent monetary utility functionalson R ∞ that can be represented with vectors of A . Acknowledgements
The authors are grateful to two anonymous Referees and an Associate Editor for their helpful suggestionsand critical comments, that help us to improve the presentation of the paper.
References [1] Acciaio, B. and Penner, I. (2011). Dynamic Risk Measures. In J. Di Nunno and B. Øksendal (Eds.)
Advanced Mathematical Methods for Finance,
Chapter 1, 11–44. Springer.202] Aliprantis, C.D., Border, K.C. (2006). Infinite Dimensional Analysis (A Hitchhiker’s Guide).Springer, 3rd Ed.[3] Aliprantis, C.D., Tourky, R. (2007). Cones and Duality. American Mathematical Society[4] Amenc, N., Le Sourd, V. (2003). Portfolio Theory and Performance Analysis. Wiley,[5] Bielecki, T., Cialenco, I., Zhang, Z. (2014). Dynamic Coherent Acceptability Indices and theirApplications to Finance.
Math. Fin. [6] Bielecki, T., Cialenco, I., Pitera, M. (2017). A Survey of Time Consistency of Dynamic Risk Mea-sures and Dynamic Performance Measures in Discrete Time: LM-Measure Perspective.
Probab.Uncertain Quant. Risk
Math. Oper. Res. [8] Biagini, S., Bion-Nadal, J. (2015). Dynamic Quasi Concave Performance Measures.
Journ. ofMathematical Econom. , pp. 143-153[9] Carr, P., Geman, H., Madan, D.B. (2001). Pricing and Hedging in Incomplete Markets. Journ. ofFinancial Econom. , pp. 131-167[10] Cheridito, P., Delbaen, F., Kupper, M. (2004). Coherent and Convex Monetary Risk Measures forBounded C`adl`ag Processes. Stoch. Proc. and their Applications , pp. 1-22[11] Cherny, A., Madan, D. (2009). New Measures for Performance Evaluation.
The Review of FinancialStudies (7), pp. 2571-2606[12] Delbaen, F. (2012). Monetary Utility Functions. Osaka University CSFI Lecture Notes Series 3.Osaka University Press[13] Dellacherie, C., Meyer, P.A. (1982). Probabilities and Potential B. North-Holland (Ch. V-VIII)[14] Inoui, A. (2003). On the Worst Conditional Expectation. Journ. of Math. Analysis and Appl. , ,pp. 237-247[15] Jaschke, S., K ¨uchler, U. (2001). Coherent Risk Measures and Good Deal Bounds. Finance andStochastics , , pp. 181-200[16] Kou, S.G. (2002). A Jump-Diffusion Model for Option Pricing. Management Science (8), pp.1086-1101[17] Konstantinides, D.G., Kountzakis, C.E. (2011). Risk Measures in Ordered Normed Linear Spaceswith Non-Empty Cone-Interior. Insurance: Math. and Economics , pp. 111-122[18] Pflug, G.C. and R ¨omisch, W. (2007). Modeling, Measuring and Managing Risk. World Scientific,Hackensack. 2119] Polyrakis, I.A. (2008). Demand Functions and Reflexivity. Journ. of Math. Analysis and Appl. ,pp. 695-704[20] Shalit, H., Yitzhaki, S. (1994). Marginal Conditional Stochastic Dominance.
Management Science40