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arXiv:1303.6675v1 [math.FA] 26 Mar 2013 The Natural Banach Space for Version Independent Risk Measures Alois Pichler * March 28, 2013 Abstract Risk measures, or coherent measures of risk are often considered on the space L , and important theorems on risk measures build on that space. Other risk measures, among them the most important risk measure—the Average Value-at-Risk—are well defined on the larger space L 1 and this seems to be the natural domain space for this risk measure. Spectral risk measures constitute a further class of risk measures of central importance, and they are often considered on some L p space. But in many situations this is possibly unnatural, because any L p with p>p0, say, is suitable to define the spectral risk measure as well. In addition to that risk measures have also been considered on Orlicz and Zygmund spaces. So it remains for discussion and clarification, what the natural domain to consider a risk measure is? This paper introduces a norm, which is built from the risk measure, and a Banach space, which carries the risk measure in a natural way. It is often strictly larger than its original domain, and obeys the key property that the risk measure is finite valued and continuous on that space in an elementary and natural way. Keywords: Risk Measures, Rearrangement Inequalities, Stochastic Dominance, Dual Rep- resentation Classification: 90C15, 60B05, 62P05 1 Introduction This paper addresses coherent measures of risk (risk measures, for short) and the natural domain (the natural space), where they can be considered. Coherent measures of risk have been introduced in the seminal paper [4] in an axiomatic way and have been investigated in a series of subsequent papers in mathematical finance since then. In the actuarial literature, however, risk measures and axiomatic treatments have been considered already earlier, for example in Denneberg ([10]) and in this journal by Wang et al. ([27]). We state the axioms (cf. [5]) for a convex risk measure ρ, mapping R-valued random variables into the real numbers R or to +. Here, the initial axioms have been adapted to follow the interpretation of loss instead of profit—the common modification in insurance—in the usual and appropriate way. * Norwegian University of Science and Technology. Contact: [email protected] 1
Transcript
  • arX

    iv:1

    303.

    6675

    v1 [

    mat

    h.FA

    ] 2

    6 M

    ar 2

    013

    The Natural Banach Space for Version Independent Risk

    Measures

    Alois Pichler

    March 28, 2013

    Abstract

    Risk measures, or coherent measures of risk are often considered on the space L, and

    important theorems on risk measures build on that space. Other risk measures, among them

    the most important risk measurethe Average Value-at-Riskare well defined on the larger

    space L1 and this seems to be the natural domain space for this risk measure. Spectral risk

    measures constitute a further class of risk measures of central importance, and they are often

    considered on some Lp space. But in many situations this is possibly unnatural, because any

    Lp with p > p0, say, is suitable to define the spectral risk measure as well. In addition to

    that risk measures have also been considered on Orlicz and Zygmund spaces. So it remains for

    discussion and clarification, what the natural domain to consider a risk measure is?

    This paper introduces a norm, which is built from the risk measure, and a Banach space,

    which carries the risk measure in a natural way. It is often strictly larger than its original

    domain, and obeys the key property that the risk measure is finite valued and continuous on

    that space in an elementary and natural way.

    Keywords: Risk Measures, Rearrangement Inequalities, Stochastic Dominance, Dual Rep-

    resentation

    Classification: 90C15, 60B05, 62P05

    1 Introduction

    This paper addresses coherent measures of risk (risk measures, for short) and the natural domain(the natural space), where they can be considered. Coherent measures of risk have been introducedin the seminal paper [4] in an axiomatic way and have been investigated in a series of subsequentpapers in mathematical finance since then. In the actuarial literature, however, risk measures andaxiomatic treatments have been considered already earlier, for example in Denneberg ([10]) and inthis journal by Wang et al. ([27]).

    We state the axioms (cf. [5]) for a convex risk measure , mapping R-valued random variablesinto the real numbers R or to +. Here, the initial axioms have been adapted to follow theinterpretation of loss instead of profitthe common modification in insurancein the usual andappropriate way.

    Norwegian University of Science and Technology.

    Contact: [email protected]

    1

    http://arxiv.org/abs/[email protected]

  • (M) Monotonicity: (Y1) (Y2) whenever Y1 Y2 almost surely;

    (H) Positive homogeneity: (Y ) = (Y ) whenever > 0;

    (C) Convexity: ((1 )Y0 + Y1) (1 ) (Y0) + (Y1) for 0 1;

    (T) Translation equivariance1: (Y + c) = (Y ) + c if c R.

    The main observation in this paper starts with the fact that the risk measure can be associated ina natural way with a seminorm, which is a norm in important cases. It is an elementary propertythat the risk measure is continuous with respect to the norm introduced.

    We investigate this new norm for specific risk measures, starting with spectral risk measures. Itturns out that the domain, where the spectral risk measure can be defined in a meaningful way, isalways strictly larger than L. The respective space is a Banach space, and we study its topology,which can be compared with Lp spaces. However, the topology always differs from the topology ofan Lp space (cf. [13]).

    A risk measure being a convex functionhas a convex conjugate function, and the FenchelMoreau theorem allows recovering the initial function, the initial risk measure in our situation.The convex conjugate function involves the dual of the initial space, for this reason it is essential tounderstand the dual of the Banach space associated with the risk measure. The norm on the dualspace measures the growth of the random variable by involving second order stochastic dominancerelations.

    It is elaborated moreover in this paper that a risk measure cannot be defined in a meaningfulway on a space larger than L1.

    The domain and the co-domain of spectral risk measures

    The axioms characterizing risk measures have been stated above without giving the domain and theco-domain precisely. Indeed, important results are well known when considering as a function onL, : L R: the results include Kusuokas representation (cf. [18] and (3) below) and resultson continuity. We state the following example.

    Proposition 1. Every R-valued risk measure on L is Lipschitz-continuous with constant 1, itsatisfies | (Y2) (Y1)| Y2 Y1.

    Proof. See, e.g., [14, Lemma 4.3] for a proof.

    In many situations, for example when considering the trivial risk measure () := E () or theAverage Value-at-Risk, the domain L is not satisfactory large enough, the domain L1 is perhapsmore natural and convenient to consider in this situation.

    Depending on the domain chosen for a risk measure, the co-domain is often specified to be R,or the extended reals R {}, in some publications even R {, }. In this context it shouldbe emphasized that there is an intimate relationship between the properties continuity of a riskmeasure and its range, the following important result clarifies the connections:

    Proposition 2. Consider a R {}-valued, lsc. risk measure defined on Lp, 1 p < ,satisfying (M), (C) and (T). Suppose further that { < } has a nonempty interior. Then isfinite valued and continuous on the entire Lp.

    1In an economic or monetary environment this is often called Cash invariance instead.

    2

  • The proof is contained in [23] and in [24], Proposition 6.7. The preceding discussion of the latterreference also contains the following reformulation of the statement, which is more striking: A riskmeasure satisfying (M), (C) and (T) is either finite valued and continuous on the entire Lp, or ittakes the value + on a dense subset.

    Both results suggest to consider R (i.e. R\ {}) valued risk measures solely, because theseare precisely the finite valued and continuous risk measures.

    Outline of the paper: The following Section 2 introduces the associated norm and elaborates itselementary property. The subsequent section, Section 3, addresses an elementary risk measure, thespectral risk measure. This risk measure is elementary, as every version independent risk measurecan be built from spectral risk measures.

    A space is introduced, which we call the space of natural domain, which is as large as possibleto carry a spectral risk measure. It is verified that the associated space is a Banach space. Thenew norm can be used in a natural way to extend the domain of elementary risk measures, and itis elaborated which Lp spaces the space of natural domain comprises.

    This section contains moreover the remarkable result, that there is no finite valued risk measureon a space larger than L1.

    We study further the topological dual of the Banach space introduced (Section 5). It turns outthe dual norm can be characterized by use of the Average Value-at-Risk, the simplest risk measure,and by second order stochastic dominance. The investigations are pushed further to more generalrisk measures, and an even more general Banach space to carry a general risk measure is highlightedin Section 6.

    2 The norm associated with a risk measure

    The results presented in this paper start along with the observation that a risk measure inducesa (semi-)norm in the following elementary way.

    Definition 3. Let L be a vector space of R-valued random variables on (,F , P ) and : L R {,} be a risk measure. Then

    := (||)

    is called associated norm, associated with the risk measure .

    Remark 4. If no confusion may occur we shall simply write to refer to .

    The following proposition verifies that is indeed a seminorm on the appropriate vectorspace.

    Proposition 5 (Finiteness, and the seminorm property). Let be a risk measure on a vector spaceof R-valued random variables. Then = (||) is a seminorm on L := {Y : (|Y |) < } and is finite valued on L.

    Proof. We show first that that is R-valued on L = {Y : (|Y |) < }. For this observe thatY |Y |, and by monotonicity thus (Y ) (|Y |) = Y . Moreover it holds that (0) = 0 2 andthus

    0 = 2

    (

    1

    2Y +

    1

    2(Y )

    )

    2

    (

    1

    2 (Y ) +

    1

    2 (Y )

    )

    = (Y ) + (Y ) ,

    2Otherwise, (0) = (2 0) = 2 (0) would imply 1 = 2, a contradiction.

    3

  • such that (Y ) (Y ). Now Y |Y | and, again by monotonicity, (Y ) (Y ) (|Y |) = Y . Summarizing thus | (Y )| Y , such that is finite valued on L.

    Note that Y = (| Y |) = (|| |Y |) = || (|Y |) = || Y ,

    and thus is positively homogeneous.Next it follows from monotonicity, positive homogeneity and convexity that

    Y1 + Y2 = (|Y1 + Y2|) (|Y1|+ |Y2|) = 2

    (

    1

    2|Y1|+

    1

    2|Y2|

    )

    2

    (

    1

    2 (|Y1|) +

    1

    2 (|Y2|)

    )

    = (|Y1|) + (|Y2|)

    = Y1+ Y2 ,

    and this is the triangle inequality.

    The next proposition elaborates, that the risk measure is continuous with respect to its associ-ated norm. This consistency result on continuity generalizes Proposition 1.

    Proposition 6 (Continuity). Let be a risk measure, defined on a vector space of R-valued randomvariables. Then is Lipschitz continuous with constant 1 with respect to the seminorm = (||).

    Proof. As for continuity note that

    (Y2) = 2

    (

    1

    2Y1 +

    1

    2(Y2 Y1)

    )

    2

    (

    1

    2 (Y1) +

    1

    2 (Y2 Y1)

    )

    (Y1) + (|Y2 Y1|)

    by convexity and monotonicity. It follows that (Y2) (Y1) Y2 Y1 . Interchanging the rolesof Y1 and Y2 reveals that

    | (Y2) (Y1)| Y2 Y1 ,

    the assertion. To accept that the Lipschitz constant 1 cannot be improved consider the particularchoices Y1 := 0 and Y2 := 1 in view of translation equivariance (T).

    3 Spectral risk measures

    Among the initial attempts to introduce premium principles to price insurance contracts are dis-torted probabilities, a concept which can be summarized nowadays by distorted acceptability func-tionals (cf. [20]) or spectral risk measures. Spectral risk measuresor the weighted Value-at-Risk(cf. [8]), which is a more suggestive termhave been considered for example in [2, 1]. This riskmeasure involves the Value-at-Risk at level p,

    [email protected] (Y ) := F1Y (p) := inf {y : P (Y y) p} ,

    which is the left-continuous, lower semi-continuous (lsc.) quantile; the spectral risk measure (orweighted [email protected]) then is the functional

    (Y ) :=

    1

    0

    (u)[email protected] (Y ) du, (1)

    4

  • mapping a random variable Y to a real number, if the integral exists.The function : [0, 1] R+0 , called the spectrum or spectral function, is a weight function. To

    build a reasonable premium principle the function should obey some properties to be consistentwith the axioms imposed on risk measures: first, associating Y with loss, should evaluate tononnegative reals, R+0 . Higher losses should be weighted higher, thus should be nondecreasing.And finally, as represents a weight function, it is natural to request

    1

    0 (u) du = 1.An important, elementary spectral risk measure satisfying all axioms above is the Average

    Value-at-Risk, which is specified by the spectral function

    (u) :=

    {

    0 if u < 1

    1 else,

    that is

    [email protected] (Y ) :=1

    1

    1

    [email protected] (Y ) du ( < 1) , (2)

    and for = 1 the Average Value-at-Risk per definition is

    [email protected] (Y ) := lim1

    [email protected] (Y ) = ess supY ( = 1) .

    The domain of spectral risk measures

    It is obvious that the Average Value-at-Risk ( < 1) may be well defined on L1, with the resultthat

    |[email protected] (Y )| 1

    1 E |Y | =

    1

    1 Y 1 <

    (

    Y L1)

    ,

    that means that [email protected] is finite valued whenever Y L1. This is not the case, however, for = 1:

    a restriction to the smaller space L L1 is necessary in order to ensure that [email protected] is finitevalued,

    |[email protected] (Y )| Y < (Y L) .

    Even more peculiarities appear when considering the spectral function (u) := 121u . Clearly,

    Lq whenever q < 2, but / L2. Hlders inequality can be employed to insure that is finitevalued on Lp (p > 2, 1

    q+ 1

    p= 1), because

    | (Y )| q

    ( 1

    0

    F1Y (u)p

    )

    1p

    =1

    2

    (

    2

    2 q

    )1q

    Y p ,

    and the constant 12

    (

    22q

    )1q

    again exceeds every finite bound whenever q approaches 2 from below.

    So what is a good space to consider ? Any Lp (p > 2) guarantees that is finite valued

    and continuous, but L2 is obviously too large. The nave choice

    p>2 Lp does not have a satisfying

    norm, or topology neither. (See, for different configurations, [6, 7].)

    5

  • Further properties and importance of spectral risk measures

    A well known and essential representation of risk measures was elaborated by Kusuoka in [18] (see[17] for the statement presented below). Kusuokas result considers risk measures on L which areversion independent (also: law invariant), i.e. which satisfy (Y ) = (Y ) whenever Y and Y

    share the same law, that is if P (Y y) = P (Y y) for every y R.

    Theorem 7 (Kusuokas representation). A version independent risk measure on L of an atom-less probability space (, F , P ) has the representation

    (Y ) = supM

    1

    0

    [email protected] (Y ) (d) , (3)

    where M is a collection of probability measures on [0, 1].

    Kusuoka representation of a spectral risk measure. The Kusuoka representation of a

    spectral risk measure is provided by the probability measure ((a, b]) := b

    ad () on [0, 1],

    where is the nondecreasing function

    (p) := (1 p) (p) +

    p

    0

    (u) du (0 p 1), (p) := 0 (p < 0) , (4)

    which satisfies (1) = 1 and d (p) = (1 p) d (p). It holds that

    (Y ) =

    1

    0

    [email protected] (Y ) (d) , (5)

    which exposes the Kusuoka representation of a spectral risk measure (cf. [25]).

    Kusuoka representation by spectral risk measures. Conversely, any measure (providedthat ({1}) = 0) of the representation (3) can be related to the function

    () =

    0

    1

    1 u(du), (6)

    and it holds that 1

    0

    [email protected] (Y ) (d) =

    1

    0

    ()[email protected] (Y ) d = (Y ) ,

    which is a spectral risk measure.But even the requirement ({1}) = 0 can be dropped: indeed, there is a set S of continuous

    (and thus bounded) spectral functions on [0, 1], such that the relation

    (Y ) = supM

    1

    0

    [email protected] (Y ) (d) = supS

    1

    0

    [email protected] (Y ) () d = supS

    (Y ) (7)

    holds (cf. [19]). This again exposes the importance of spectral risk measures, as every versionindependent risk measure can be built from spectral risk measures by (7).

    Recall that Kusuokas representation builds on the space L. But again it is not clear, if, andto which larger space this risk measure can be extended, because every might allow a differentdomain.

    6

  • 4 The space of natural domain, L

    Let be a nonnegative, nondecreasing, integrable function with 1

    0 (u)du = 1. For Y a randomvariable we consider the function

    (Y ) =

    1

    0

    (u) F1Y (u) du

    already defined in (1). For L1 (which is a minimal requirement to insure that 1

    0(u)du = 1),

    is certainly well defined for Y L, but for other random variables the integral possibly diverges.And it might diverge to +, to , or be even of the indefinite form . The followingdefinition respects the finiteness of the spectral risk measure in view of Proposition 5.

    Definition 8. The natural domain corresponding to a spectral risk measure induced by aspectral function is

    L :={

    Y L0 : Y < }

    ,

    whereY := (|Y |) .

    Note that |Y | 0 is positive, such that F1|Y | () 0 is positive as well and the condition

    (|Y |) < makes perfect sense for any measurable random variable Y L0.

    Remark 9. The seminorm has the representation

    Y =

    0

    (

    F|Y |(y))

    dy

    in terms of the cdf. F|Y | directly, without involving the inverse F1|Y | ( () :=

    1

    (u)du).

    Proposition 10. = (||) is a norm on L.

    Proof. It was already shown in Proposition 5 that is a seminorm. What remains to beshown is that separates points. For this recall that is positive, nondecreasing, and sat-

    isfies 1

    0 (p) dp = 1, and F|Y | () is a nondecreasing and positive function as well. Hence if 1

    0 (p)F1|Y | (p) dp = 0, then F

    1|Y | () 0, that is Y = 0 almost everywhere. The function

    thus separates points in L and hence is a norm.

    The next theorem already elaborates that the set L is large enough and at least contains Lp,

    whenever Lq (and the exponents are conjugate, 1p+ 1

    q= 1).

    Theorem 11 (Comparison with Lp). Let be fixed.

    (i) If Lq for some q [1,] with conjugate exponent p, then

    L Lp L L1

    andY 1 Y q Y p (8)

    whenever Y Lp.

    7

  • (ii) For bounded (i.e. L) it holds moreover that L = L1, the norms are equivalent andsatisfy

    Y 1 Y Y 1 .

    It follows in particular from (ii) that P (A) 1A 1 for measurable sets A, and Y =Y 1 for the function being constantly 1 ( = 1).

    Proof. Note that 1

    0 (u) du = 1 and () is nondecreasing, hence there is a u (0, 1) such that

    (u) 1 for u < u and (u) 1 for u > u. Note as well that u

    01 (u) du =

    1

    u (u) 1 du.

    Then it follows that

    u

    0

    (1 (u))F1|Y | (u) du

    u

    0

    (1 (u))F1|Y | (u) du

    =

    1

    u

    ( (u) 1)F1|Y | (u) du

    1

    u

    ( (u) 1)F1|Y | (u) du,

    because F1|Y | () is increasing. After rearranging thus

    Y 1 = E |Y | = 1

    0

    F1|Y | (u) du

    1

    0

    F1|Y | (u) (u) du = (|Y |) = Y ,

    which is the first assertion. The inclusion L L1 is immediate as well, as Y < implies that

    Y 1 < .The remaining inequality

    Y =

    1

    0

    F1|Y | (u) (u) du

    ( 1

    0

    (u)q)

    1q

    ( 1

    0

    F1|Y | (u)p

    )

    1p

    = q (E |Y |p)

    1p

    is Hlders inequality.

    Remark 12. The inequality Y 1 Y is also a direct consequence of Chebyshevs sum inequality

    in its continuous form, which states that 1

    0 f (u) du 1

    0 g (u) du 1

    0 f (u) g (u) du whenever f and

    g are both nondecreasing (choose f = and g = F1|Y | ; cf. [15]).

    Theorem 13 (Comparability of L-spaces). Suppose that

    c := sup0

  • Proof. To accept (10) define the functions Si () := 1

    i (u) du (i = 1, 2), then by RiemannStieltjes

    integration by parts and as u 7 F1|Y | (u) is nondecreasing,

    Y 2 =

    1

    0

    F1|Y | (u)2 (u) du =

    1

    0

    F1|Y | (u) dS2 (u)

    = F1|Y | (u)S2 (u)

    1

    0+

    1

    0

    S2 (u) dF1|Y | (u) = F

    1|Y | (0) +

    1

    0

    S2 (u) dF1|Y | (u)

    F1|Y | (0) + c

    1

    0

    S1 (u) dF1|Y | (u)

    = F1|Y | (0) + c F1|Y | (u)S1 (u)

    1

    0 c

    1

    0

    F1|Y | (u) dS1 (u)

    = F1|Y | (0) (c 1) + c

    1

    0

    F1|Y | (u)1 (u) du c Y 1 ,

    because F1|Y | (0) 0 and c 1 (choose = 0 in (9)).

    To accept that c is the smallest constant satisfying (10) just consider the random variable

    Y = 1Ac , for which Y = (1Ac) = 1

    P (A) (u) du. The assertion follows, as the measurable set

    A may be chosen arbitrarily.

    It is a particular consequence of (10) that

    [email protected] (|Y |) [email protected] (|Y |) 1 11 2

    [email protected] (|Y |) ,

    which holds whenever 1 2 < 1. It should be noted, however, that [email protected] (Y ) [email protected] (Y ) [email protected] (Y ) in general.

    The following representation result for spectral risk measures is well known for in an appro-priate space. We extend it to L, the result will be used in the sequel.

    Proposition 14 (Representation of the spectral risk measure). has the equivalent representa-tion 3

    (Y ) = sup {EY (U) : U is uniformly distributed} (11)

    on L.

    Remark 15. For the Average Value-at-Risk it holds in particular that

    [email protected] (Y ) = sup

    {

    EY Z : EZ = 1, 0 Z 1

    1

    }

    (12)

    in view of the spectral function (2).

    Proof. Consider the random variable Z = (U) for a uniformly distributed random variable U ,then P (Z ()) = P ( (U) ()) P (U ) = , that is [email protected] (Z) (). But as

    1 = 1

    0 () d 1

    0 [email protected] ( (U)) d = E (U) = 1

    0 (p) dp = 1 it follows that

    [email protected] (Z) = () . (13)

    3A random variable U is uniformly distributed if P (U u) = u whenever u [0, 1].

    9

  • Now F1Y () is an increasing function, and so is (). By the HardyLittlewood rearrangementinequality (cf. [16] and [20, Proposition 1.8] for the respective rearrangement inequality, sometimesalso referred to as HardyLittlewoodPlya inequality, cf. [9]) it follows thus that

    EY (U) 1

    0

    F1Y () () d.

    However, if Y and U are coupled in a co-monotone way, then equality is attained, that is EY (U) =

    1

    0 F1Y () () d. This proves the statement in view of the definition of the spectral risk

    measure, (1).

    The next theorem demonstrates that the spaces L really add something to Lp spaces, the space

    L is strictly larger than Lp.

    Theorem 16 (L is larger than Lp). The following hold true:

    (i) Suppose that Lq for some 1 q < . Then the space of natural domain L is strictlylarger than Lp, Lp $ L ( 1p +

    1q= 1).

    (ii) In particular the space of natural domain L is (always) strictly larger than L, L $ L

    (q = 1).

    Remark 17. It should be noted that the statement of the latter theorem does not hold for L:In this situation is well defined on L

    1, and L = L1 by the preceding Theorem 11, (i).

    Proof. To prove the first assertion assume that Lq for 1 < q < . Consider the uniquely

    determined numbers t0 := 0 < t1 < t2 < < 1 for which tn

    0(u)qdu =

    qq(p+1)

    nj=1

    1jp+1

    and

    observe that tn

    tn1(u)qdu =

    qq(p+1)

    1np+1

    .4 Define the function

    (u) :={

    n if tn1 u < tn,

    let U be uniformly distributed and consider the random variable

    Y := (U)q1 (U) . (14)

    Note, by (11), that

    (Y ) = E (U)Y = E (U) (U)q1

    (U) = E (U)q (U)

    =

    1

    0

    (u)q (u) du =

    n=1

    tn

    tn1

    (u)q n du

    =qq

    (p+ 1)

    n=1

    n

    np+1=

    qq (p+ 1)

    n=1

    1

    np=

    qq

    (p)

    (p+ 1)< ,

    4 (p) :=

    n=1

    1

    npis Riemanns Zeta function, the series converges whenever p > 1.

    10

  • because p > 1. Next,

    Y pp = E |Y |p =

    1

    0

    (u)(q1)p (u)p du

    =

    1

    0

    (u)q (u)p du =

    n=1

    tn

    tn1

    (u)q np du

    =qq

    (p+ 1)

    n=1

    np

    np+1=

    qq (p+ 1)

    n=1

    1

    n= .

    Hence, Y L, but Y / Lp.

    The second statement of the theorem is actually the first statement with q = 1, but the aboveproof needs a modification: To accept it define, as above, an increasing sequence of values byt0 := 0 < t1 < t2 < < 1 satisfying

    tn

    0(t)dt 1 2n. Note, that

    tn

    tn1

    (u)du

    1

    tn1

    (u) du = 1

    tn1

    0

    (u)du 21n.

    Define moreover the increasing function

    () :=

    n=0

    1[tn, 1] ()

    (i.e. (t) = n if tn1 t < tn) and observe that whenever t 1.Now let U be a uniformly distributed random variable and set Y := (U). Then

    (Y ) =

    1

    0

    (u)(u)du =

    n=1

    tn

    tn1

    (u)(u)du

    =

    n=1

    n

    tn

    tn1

    (u)du

    n=1

    n 21n = 4 < ,

    so Y L. But Y / L, because P (Y n) 1 tn1 > 0 by definition of .

    Remark 18. Notably the preceding proof applies for the random variable Y = (U)q1

    (U)

    in (14) equally well whenever 1 < p, such that L is larger than Lp by an entire infinite

    dimensional manifold.

    It was demonstrated above that the space L is contained in L1. The above inequality (8),

    1 , allows to prove an even much stronger result: a finite valued risk measure cannotbe considered on a space larger than L1. This is the content of the following theorem, which wascommunicated to the author by Prof. Alexander Shapiro (Georgia Tech). In brief: it does not makesense to consider risk measures on a space larger than L1.

    Theorem 19. Let L L0 be a vector space collecting R-valued random variables on ([0, 1] , B, )(the standard probability space equipped with its Borel sets) such that L % L1 and |Y | L, if Y L.Then there does not exist a version independent, finite valued risk measure on L.

    11

  • Proof. Suppose that : L R is a version independent, and finite valued risk measure on L. Re-stricted to L, Kusuokas theorem (Theorem 7) applies and takes the form () = supS ().Choose Y L\L1, that is E |Y | = , or

    p

    0F1|Y | (u) du whenever p 1.

    Next, pick any S . Define Yn := min {n, |Y |} and observe that (Yn) (|Y |) by mono-tonicity. Note that Yn L

    and hence, by Kusuokas representation, (8) and the particular choiceof Y ,

    (|Y |) (Yn) (Yn) = Yn Yn1

    P (|Y |n)

    0

    F1|Y | (u) du ,

    as n . Hence, is not finite valued on L.

    Theorem 20. (L, ) is a Banach space over R.

    Proof. It remains to be shown that (L, ) is complete. For this let (Yk)k be a Cauchy sequencefor . By (8) the sequence (Yk)k is a Cauchy sequence for 1 as well, and from completenessof L1 it follows that there exists a limit Y L1. We shall show that Y L.

    It follows from convergence in L1 that (Yk)k converges in distribution, that is FYk (y) FY (y)for every point y where FY is continuous and moreover F

    1|Yk| () F

    1|Y | () (cf. [26, Chapter 21]).

    Now

    Y = (|Y |) =

    1

    0

    (t)F1|Y | (t) dt =

    1

    0

    (t) limk

    F1|Yk| (t) dt

    =

    1

    0

    lim infk

    (t)F1|Yk| (t) dt lim infk

    1

    0

    (t)F1|Yk| (t) dt = lim infkYk

    by Fatous Lemma, which is applicable because F1|Yk| () 0.

    As (Yk)k is a Cauchy sequence one may pick k N such that Yk Yk < 1 for all k > k

    ,and hence Yk Yk + Yk Yk < Yk + 1 < by the triangle inequality. Thesequence (Yk)k thus is uniformly bounded in its norm. Hence,

    Y lim infk

    Yk Yk + 1 < ,

    that is Y L and L thus is complete.

    Example 21. Consider the spectrum () = 121 . It should be noted that L

    p>2 Lp, and

    provides a reasonable norm on that set.Restricted to Lp, for some p > 2, the open mapping theorem (cf. [22] or [3]) insures that the

    norms are equivalent, that is there are constants c1 and c2 such that

    c1 Y p Y c2 Y p (Y Lp L).

    The latter inequalities hold just for Y Lp, but not for Y L.

    Proposition 22. Measurable, simple (step) functions are dense in L, and in particular L is

    dense in L.

    Proof. Given Y L and > 0, find t0 (0, 1) such that t0

    0 F1Y (u)(u)du k, which is

    possible because the sequence is Cauchy. It follows that

    Y S

    lim infk

    YkS YkS + 1 < ,

    and hence Y LS , that is LS is complete.

    Theorem 34. The risk measure S is finite valued on LS , it is moreover continuous with respectto the norm

    Swith Lipschitz constant 1.

    Proof. The assertion follows from the more general Proposition 6.

    Comparison of different LS spaces. The norm of the identity

    id :(

    LS1, S1)

    (

    LS2, S2)

    is

    id = sup2S2

    inf1S1

    sup0 0 : 2 S2 1 S1 :

    1

    2 (u) du c

    1

    1 (u) du for all (0, 1)

    }

    .

    20

  • Examples

    We give finally two examples for which the norm S

    induced by a set of spectral functions Scoincides with the norm p on L

    p. Note, that this is contrast to the space L, as Theorem 16insures that L is strictly larger than L

    p.

    Example 35 (Higher order semideviation). The psemideviation risk measure for 0 < 1 is

    (Y ) := EY +

    (Y EY )+

    p.

    Then LS = Lp, where S is an appropriate spectrum to generate = S , and the norms S and

    p are equivalent.

    Proof. The generating set S is provided in [24] and in [25], the higher order semideviation riskmeasure takes the alternative form

    (Y ) = S (Y ) = supLq

    (

    1

    q

    )

    EY +

    q (Y ) .

    It is evident that S (|Y |) (

    1 q

    )

    Y 1+ Y p (1 + ) Y p, such that S is finite valued

    for Y Lp. We claim that the natural domain is LS = Lp. For this suppose that Y LS \L

    p, i.e.Y 1 < , but Y p = . So it holds that

    S (Y ) supLq

    (Y )

    q= sup

    ZLqEY

    Z

    Zq= Y p =

    by Lp Lq duality, hence Y / LS and thus LS = Lp.

    It follows by the open mapping theorem that the norms are equivalent.

    Example 36. Theorem 16 states that L % L, that is to say L is strictly larger than L. Thisis not the case any more for the space LS : for this consider just the risk measure

    (Y ) := sup

  • 8 Acknowledgment

    The author is indebted to Prof. Alexander Shapiro (Georgia Tech) for numerous discussions on thisand other subjects, not only during the work on this paper. In particular Theorem 19 is attributedto Prof. Shapiro.

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    23

    IntroductionThe norm associated with a risk measureSpectral risk measuresThe space of natural domain, LThe Dual of the natural domain LThe general natural domain space LSSummaryAcknowledgment


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